tag:blogger.com,1999:blog-57880512559384857462012-01-29T12:13:20.612-06:00A window through the walls of our classroom. This is an interactive learning ecology for students and parents in my AP Calculus class. This ongoing dialogue is as rich as YOU make it. Visit often and post your comments freely.dkuropatwahttp://www.blogger.com/profile/08462283847470560887noreply@blogger.comBlogger2701100tag:blogger.com,1999:blog-5788051255938485746.post-90713043407118636492008-06-26T15:30:00.000-05:002008-06-26T15:30:00.606-05:00<img src="http://static.flickr.com/73/161775713_4ed2bb3663_m_d.jpg" align="right" hspace="8">We had our graduation exercises today. A gentle push into the world for all of you. I hope you're leaving with the keys to your future in your hand.<br /><br /><b><i>I'm so glad we've had this time together,<br /><br />Just to have a laugh or learn some math,<br /><br />Seems we've just got started and before you know it,<br /><br />Comes the time we have to say, "So Long!"</b></i><br /><br />So long everybody! Watch this space in the fall for pointers to new blogs for each of my classes. <br /><br />Farewell, Auf Wiedersehen, Adieu, and all those good bye things. ;-)<div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-9071304340711863649?l=apcalc07.blogspot.com' alt='' /></div>dkuropatwahttp://www.blogger.com/profile/08462283847470560887noreply@blogger.com3tag:blogger.com,1999:blog-5788051255938485746.post-10535395541068781612008-06-17T09:07:00.008-05:002008-06-17T10:15:20.337-05:00The exam is over and we did a little survey in class. The results are below; 7 students participated. If you'd like to add another comment on what you see here <a href="mailto:dkuropatwa@wsd1.org?subject=AP Calculus Survey June 2008&body=Hi there, I'd like to comment on the survey. Here are my thoughts:">email me</a> or leave a comment below this post.<br /><br />Without any further ado, here are the results of our class's survey. Please share your thoughts by commenting (anonymously if you wish) below .....<br /><br /><b><i>Classroom Environment</i></b><br />The questions in this section were ranked using this 5 point scale:<br /><br /><table border="3" bordercolor="blue"><tr align="center"><td>Strongly Disagree</td><td>Disagree</td><td>Neutral</td><td>Agree</td><td>Strongly Agree</td></tr><tr align="center"><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td></tr></table><br /><br />The <b>bold</b> numbers after each item are the average ratings given by the entire class.<br /><br />1. The teacher was enthusiastic about teaching the course. <b>4.86</b><br /><br />2. The teacher made students feel welcome in seeking help in/outside of class. <b>4.71</b><br /><br />3. My interest in math has increased because of this course. <b>4.14</b><br /><br />4. Students were encouraged to ask questions and were given meaningful answers. <b>4.57</b><br /><br />5. The teacher enhanced the class through the use of humour. <b>4.43</b><br /><br />6. Course materials were well understood and explained clearly by the teacher. <b>4.29</b><br /><br />7. Graded materials fairly represented student understanding and effort. <b>4.29</b><br /><br />8. The teacher showed a genuine interest in individual students. <b>3.71</b><br /><br />9. I have learned something that I consider valuable. <b>4.86</b><br /><br />10. The teacher normally came to class well prepared. <b>4.43</b><br /><br /><b><i>Overall Impression of the Course</i></b><br />The questions in this section were ranked using this 5 point scale:<br /><br /><table border="3" bordercolor="blue"><tr align="center"><td>Very Poor</td><td>Poor</td><td>Average</td><td>Good</td><td>Very Good</td></tr><tr align="center"><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td></tr></table><br /><br />1. Compared with other high school courses I have taken, I would say this course was: <b>4.57</b><br /><br />2. Compared with other high school teachers I have had, I would say this teacher is: <b>5.00</b><br /><br />3. As an overall rating, I would say this teacher is: <b>4.86</b><br /><br /><b><i>Course Characteristics</i></b><br /><br />1. Course difficulty, compared to other high school courses:<br /><br /><table border="3" bordercolor="blue"><tr align="center"><td>Very Easy<br>0%</td><td>Easy<br>0%</td><td>Average<br>28.6%</td><td>Difficult<br>28.6%</td><td>Very Difficult<br>42.9%</td></tr></table><br /><br />2. Course workload, compared to other high school courses:<br /><br /><table border="3" bordercolor="blue"><tr align="center"><td>Very Easy<br>0%</td><td>Easy<br>14.3%</td><td>Average<br>28.6%</td><td>Difficult<br>28.6%</td><td>Very Difficult<br>28.6%</td></tr></table><br /><br />3. Hours per week required outside of class:<br /><br /><table border="3" bordercolor="blue"><tr align="center"><td>0 to 2<br>28.6%</td><td>2 to 3<br>0%</td><td>3 to 5<br>14.3%</td><td>5 to 7<br>42.9%</td><td>over 7<br>14.3%</td></tr></table><br /><br />4. Expected grade in the course:<br /><br /><table border="3" bordercolor="blue"><tr align="center"><td>F<br>0%</td><td>D<br>14.3%</td><td>C<br>0%</td><td>B<br>57.1%</td><td>A<br>28.6%</td></tr></table><br /><br /><b><i>AP Exam Preparation</i></b><br />The <b>bold</b> numbers after each item are the average ratings given by the entire class.<br /><br />How prepared were you to write this exam? <b>75.6%</b><br /><br />How much effort did you put into preparing for this exam? <b>73.1%</b><br /><br />How good a job did your teacher do preparing you for this exam? <b>90.1%</b><br /><br />Did you have enough preparation using your calculator?<br /><br /><center><table border="3" bordercolor="blue"><tr align="center"><td>Yes<br>100%</td><td>No<br>0%</td></tr></table></center><br /><br />Did you have enough preparation without using your calculator?<br /><br /><center><table border="3" bordercolor="blue"><tr align="center"><td>Yes<br>100%</td><td>No<br>0%</td></tr></table></center><br /><br />Was your teacher too hard or too easy on you?<br /><br /><center><table border="3" bordercolor="blue"><tr align="center"><td>Too Hard<br>0%</td><td>Too Easy<br>42.9%</td><td>Just Right<br>57.1%</td></tr></table></center><br /><br /><b><i>Specific Feedback</i></b><br />[<b>Ed. Note:</b> Numbers in parentheses indicate the number of students, over 1, that gave the same answer.]<br /><br />What was your best learning experience in this course? <br /><br /><table border="3" bordercolor="blue"><tr align="center"><td>Enthusiasm, effort, and humour by the teacher (2)<br />Workshop classes<br />Pre-Tests (3)<br /><a href="http://expertvoices08.blogspot.com">Developing Expert Voices</a> project<br />This blog<br />Everything except the Developing Expert Voices project<br />Teaching others and explaining concepts<br />Lectures<br />Exam Prep<br />Scribe posts were awesome<br />Technology<br />SMARTboard<br />Group learning</td></tr></table><br /><br />What was your worst learning experience in this course?<br /><br /><table border="3" bordercolor="blue"><tr align="center"><td><a href="http://expertvoices08.blogspot.com">Developing Expert Voices</a> project (2)<br />Falling behind due to the fast progress of other classmates<br />Not being able to do homework in class<br />The <a href="http://apcalc07.pbwiki.com">wiki solutions manual</a><br />Lectures were not visual enough<br />When substitute teachers wouldn't let us talk to each other</td></tr></table><br /><br />What changes would you suggest to improve the way this course is taught?<br /><br /><table border="3" bordercolor="blue"><tr align="center"><td>Return tests more quickly<br />Go over solutions to every test question always<br />Give regular "marks updates" so I know when I need to invest a lot or a little effort<br />Cancel the Developing Expert Voices project<br />More practice work for home posted to the blog<br />Cancel the <a href="http://apcalc07.pbwiki.com">wiki assignment</a><br />Slower explanations that hit all key points<br />Tell students how fast their Developing Expert Voices dues dates creep up on them<br />More group learning, I learned a lot in groups<br />Have a review class every couple of units</td></tr></table><br /><br />It's interesting to compare the items that were considered both the worst and best learning experiences. Also, take a look at the list of worst learning experiences compared to suggestions for next year. Help me do a better job next year by commenting on what you see here ....<div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-1053539554106878161?l=apcalc07.blogspot.com' alt='' /></div>dkuropatwahttp://www.blogger.com/profile/08462283847470560887noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-79858404407431346812008-06-15T23:15:00.000-05:002008-11-18T18:01:08.634-06:00<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://farm2.static.flickr.com/1162/1390706332_8572d1c1f8_d.jpg"><img style="margin: 0pt 0pt 10px 10px; float: right; cursor: pointer; width: 320px;" src="http://farm2.static.flickr.com/1162/1390706332_8572d1c1f8_d.jpg" alt="" border="0" /></a>First an update on this podcast: While we have received few comments on this or any of our class blogs the number of times the audio files have been downloaded is remarkable ...<br /><br /><blockquote><a href="http://adifference.blogspot.com/2008/04/student-voices-episode-1-jessie.html">Episode 1: Jessie</a> 2440 downloads<br /><br /><a href="http://adifference.blogspot.com/2008/04/student-voices-episode-2-timmathy.html">Episode 2: Tim_MATH_y</a> 1766 downloads<br /><br /><a href="http://adifference.blogspot.com/2008/05/student-voices-episode-3-chris-craig.html">Episode 3: Chris, Craig, Graeme</a> 1367 downloads</blockquote><br /><br />Thanks to all our listeners. We might get one more published during this school year but this may be the last until September. In any case feel free to let us know your thoughts about what you heard; every comment is appreciated.<br /><br />In this episode of <a href="http://adifference.blogspot.com/search/label/Student%20Voices">Student Voices</a> Justice, Lawrence, and Richard talk about how they put together <a href="http://expertvoices08.blogspot.com/2008/06/team-lrj-studios-present.html">their Developing Expert Voices project</a> and what they learned in the process: how they they best learn math, how it can best be taught, and many other incidental things like team work and organizational skills.<br /><br />They have titled their project with one of my favourite reminders to all my students: Mathematics is the Science of Patterns. If you watch any of the video content they created you'll hear several "in jokes", listen for them. Without any further ado, here is the podcast. A copy of the poster they made for their work is below.<br /><br /><center><embed src="http://www.archive.org/download/DarrenKuropatwaStudentVoices_Episode4Justice_Lawrence_andRichard/SV04JusticeLaurenceRichardPC40SW08.mp3" type="audio/mpeg" autostart="false" loop="false" bgcolor="#ecf5fd" height="74" width="144"></embed><br />(<a href="http://www.archive.org/download/DarrenKuropatwaStudentVoices_Episode4Justice_Lawrence_andRichard/SV04JusticeLaurenceRichardPC40SW08.mp3">Download File</a> 12.2Mb, 25 min. 30 sec.)<br /></center><br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://4.bp.blogspot.com/_tlF3qs3v9cw/SFXm56L5XhI/AAAAAAAAAXY/6BZ6-EQewm8/s1600-h/Poster.jpg"><img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;" src="http://4.bp.blogspot.com/_tlF3qs3v9cw/SFXm56L5XhI/AAAAAAAAAXY/6BZ6-EQewm8/s400/Poster.jpg" border="0" alt=""id="BLOGGER_PHOTO_ID_5212326026297302546" /></a><br /><br /><div align="right">Photo Credit: <a href="http://www.flickr.com/photos/zhzheka/1390706332/">Shadow singer</a> by flickr user EugeniusD80</div><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-7985840440743134681?l=apcalc07.blogspot.com' alt='' /></div>dkuropatwahttp://www.blogger.com/profile/08462283847470560887noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-58923437004258338452008-05-12T23:22:00.002-05:002008-05-12T23:26:22.532-05:00<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://farm2.static.flickr.com/1162/1390706332_8572d1c1f8_d.jpg"><img style="float:right; margin:0 0 10px 10px;cursor:pointer; cursor:hand;width: 320px;" src="http://farm2.static.flickr.com/1162/1390706332_8572d1c1f8_d.jpg" border="0" alt="" /></a><br />In this episode of Student Voices three Advanced Placement Calculus students, Chris, Craig, and Graeme, talk about a wiki assignment they did to prepare for the exam. Then the conversation transitions to a discussion of the many things they learned while doing their <a href="http://expertvoices08.blogspot.com">Developing Expert Voices</a> project. It ends with a challenge, the result of which will be featured in a future podcast.<br /><p><br />Let Chris, Craig, and Graeme know what you thought about the podcast by leaving a comment here on this post.<br /><p><br /><center><br /> <embed src="http://www.archive.org/download/DarrenKuropatwaStudentVoices_Episode3Chris_Craig_Graeme/SV03ChrisCraigGraemeapcalc07.mp3" width="144" height="74" type="audio/mpeg" autostart="false" loop="false" bgcolor="#ECF5FD"> </embed><br /><br><br />(<a href="http://www.archive.org/download/DarrenKuropatwaStudentVoices_Episode3Chris_Craig_Graeme/SV03ChrisCraigGraemeapcalc07.mp3">Download File</a> 31.8Mb, 26 min. 30 sec.)<br /></center><br /><p><br />The video mentioned near the end of the podcast is called <a href="http://www.youtube.com/watch?v=K2cYWfq--Nw">Daft Hands</a>. Here it is:<br /><p><br /><center><br /><object width="425" height="355"><param name="movie" value="http://www.youtube.com/v/K2cYWfq--Nw&hl=en"></param><param name="wmode" value="transparent"></param><embed src="http://www.youtube.com/v/K2cYWfq--Nw&hl=en" type="application/x-shockwave-flash" wmode="transparent" width="425" height="355"></embed></object><br /></center><br /><p><br /><div align="right">Photo Credit: <a href="http://www.flickr.com/photos/zhzheka/1390706332/">Shadow singer</a> by flickr user EugeniusD80</div><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-5892343700425833845?l=apcalc07.blogspot.com' alt='' /></div>dkuropatwahttp://www.blogger.com/profile/08462283847470560887noreply@blogger.com1tag:blogger.com,1999:blog-5788051255938485746.post-6239925851802174662008-05-05T13:42:00.001-05:002008-05-05T13:47:51.575-05:00Here they are ...<br /><br /><center><br /><div style="width:425px;text-align:left" id="__ss_389144"><object style="margin:0px" width="425" height="355"><param name="movie" value="http://static.slideshare.net/swf/ssplayer2.swf?doc=apcalc07080505-1210012961976266-8"/><param name="allowFullScreen" value="true"/><param name="allowScriptAccess" value="always"/><embed src="http://static.slideshare.net/swf/ssplayer2.swf?doc=apcalc07080505-1210012961976266-8" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="355"></embed></object><div style="font-size:11px;font-family:tahoma,arial;height:26px;padding-top:2px;"><a href="http://www.slideshare.net/?src=embed"><img src="http://static.slideshare.net/swf/logo_embd.png" style="border:0px none;margin-bottom:-5px" alt="SlideShare"/></a> | <a href="http://www.slideshare.net/dkuropatwa/ap-calculus-slides-may-5-2008?src=embed" title="View 'AP Calculus Slides May 5, 2008' on SlideShare">View</a> | <a href="http://www.slideshare.net/upload?src=embed">Upload your own</a></div></div><br /></center><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-623992585180217466?l=apcalc07.blogspot.com' alt='' /></div>dkuropatwahttp://www.blogger.com/profile/08462283847470560887noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-72405478079757634112008-04-27T19:15:00.005-05:002008-04-27T19:50:13.784-05:00<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://farm2.static.flickr.com/1162/1390706332_8572d1c1f8_d.jpg"><img style="float:right; margin:0 0 10px 10px;cursor:pointer; cursor:hand;width: 320px;" src="http://farm2.static.flickr.com/1162/1390706332_8572d1c1f8_d.jpg" border="0" alt="" /></a><br />In this episode Timothy came back to school on Friday afternoon to talk about his week attending the miniUniversity program at the University of Winnipeg. He talks about the differences he finds between teaching and learning at high school and university and describes learning in the university classroom using a thought provoking metaphor, listen for it. Also, we have a cameo appearance by two very special people at the very end.<br /><p><br />Please feel free to leave Tim_MATH_y your comments here on this post.<br /><p><br /><center><br /> <embed src="http://ia360905.us.archive.org/1/items/DarrenKuropatwaStudentVoices_Episode2Tim_math_y/SV02TimMathYapcalc07_64kb.mp3" width="144" height="74" type="audio/mpeg" autostart="false" loop="false" bgcolor="#ECF5FD"> </embed><br /><br><br />(<a href="http://ia360905.us.archive.org/1/items/DarrenKuropatwaStudentVoices_Episode2Tim_math_y/SV02TimMathYapcalc07_64kb.mp3">Download File</a> 7.2Mb, 15 min. 3 sec.)<br /></center><br /><p><br /><div align="right">Photo Credit: <a href="http://www.flickr.com/photos/zhzheka/1390706332/">Shadow singer</a> by flickr user EugeniusD80</div><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-7240547807975763411?l=apcalc07.blogspot.com' alt='' /></div>dkuropatwahttp://www.blogger.com/profile/08462283847470560887noreply@blogger.com3tag:blogger.com,1999:blog-5788051255938485746.post-9702690320374674422008-04-25T15:46:00.002-05:002008-04-25T16:03:41.003-05:00Here they are ...<br /><br /><center><br /><div style="width:425px;text-align:left" id="__ss_372723"><object style="margin:0px" width="425" height="355"><param name="movie" value="http://static.slideshare.net/swf/ssplayer2.swf?doc=apcalc07080425-1209156548024628-8"/><param name="allowFullScreen" value="true"/><param name="allowScriptAccess" value="always"/><embed src="http://static.slideshare.net/swf/ssplayer2.swf?doc=apcalc07080425-1209156548024628-8" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="355"></embed></object><div style="font-size:11px;font-family:tahoma,arial;height:26px;padding-top:2px;"><a href="http://www.slideshare.net/?src=embed"><img src="http://static.slideshare.net/swf/logo_embd.png" style="border:0px none;margin-bottom:-5px" alt="SlideShare"/></a> | <a href="http://www.slideshare.net/dkuropatwa/ap-calculus-slides-april-25-2008?src=embed" title="View 'AP Calculus Slides April 25, 2008' on SlideShare">View</a> | <a href="http://www.slideshare.net/upload?src=embed">Upload your own</a></div></div><br /></center><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-970269032037467442?l=apcalc07.blogspot.com' alt='' /></div>dkuropatwahttp://www.blogger.com/profile/08462283847470560887noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-18344003024664476292008-04-23T12:47:00.000-05:002008-04-23T12:48:24.331-05:00Here they are ...<br /><br /><center><br /><div style="width:425px;text-align:left" id="__ss_369611"><object style="margin:0px" width="425" height="355"><param name="movie" value="http://static.slideshare.net/swf/ssplayer2.swf?doc=apcalc07080423-1208972633719644-8"/><param name="allowFullScreen" value="true"/><param name="allowScriptAccess" value="always"/><embed src="http://static.slideshare.net/swf/ssplayer2.swf?doc=apcalc07080423-1208972633719644-8" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="355"></embed></object><div style="font-size:11px;font-family:tahoma,arial;height:26px;padding-top:2px;"><a href="http://www.slideshare.net/?src=embed"><img src="http://static.slideshare.net/swf/logo_embd.png" style="border:0px none;margin-bottom:-5px" alt="SlideShare"/></a> | <a href="http://www.slideshare.net/dkuropatwa/ap-calculus-slides-april-23-2008?src=embed" title="View 'AP Calculus Slides April 23, 2008' on SlideShare">View</a> | <a href="http://www.slideshare.net/upload?src=embed">Upload your own</a></div></div><br /></center><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-1834400302466447629?l=apcalc07.blogspot.com' alt='' /></div>dkuropatwahttp://www.blogger.com/profile/08462283847470560887noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-82939164025239252422008-04-23T08:39:00.002-05:002008-04-23T08:42:47.628-05:00Hello there. Van scribing for Monday's class. Not much happened. Substitute teacher gave us a worksheet to hand in for Wednesday's class, and it's worth 100 marks each, 5 questions total.<br /><br />That's about it.<br /><br />Next scribe... Dino...<div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-8293916402523925242?l=apcalc07.blogspot.com' alt='' /></div>Vanhttp://www.blogger.com/profile/07679810325423377354noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-20270549545808329632008-04-19T13:29:00.001-05:002008-11-18T18:01:08.906-06:00<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://4.bp.blogspot.com/_tlF3qs3v9cw/SAo3CljRiII/AAAAAAAAAWw/ATUn9MTojWw/s1600-h/SVKidsOfConversation.jpg"><img style="float:right; margin:0 0 10px 10px;cursor:pointer; cursor:hand;" src="http://4.bp.blogspot.com/_tlF3qs3v9cw/SAo3CljRiII/AAAAAAAAAWw/ATUn9MTojWw/s320/SVKidsOfConversation.jpg" border="0" alt=""id="BLOGGER_PHOTO_ID_5191022038077245570" /></a><br />I was talking to Jessie, one of my Applied Math students, earlier this week while helping her review over the lunch hour. I found her comments so compelling I asked her (and later her parents) if I could record and publish her comments so other students could hear what she had to say. I've long thought students need to hear from other students how they best learn to help them all learn.<br /><p><br />This is the first in what I hope will be a series of podcasts called <b>Student Voices</b>. I'm hoping to have one of these short conversations with a student published each week. If you'd like to volunteer to be featured in one of these just let me know.<br /><p><br />In this episode Jessie shares how she uses <a href="http://am40sw08.blogspot.com">her class blog</a> to learn and describes her personal "tipping point" from being confused to understanding Statistics very well. She also discusses the value of learning conversations and how sometimes being a "teacher" and sometimes a student helps her learn.<br /><p><br />Please feel free to leave Jessie your comments here or <a href="http://am40sw08.blogspot.com/2008/04/student-voices-podcast-episode-1-jessie.html">on this post on her class blog</a>.<br /><p><br /><center><br /> <embed src="http://www.archive.org/download/DarrenKuropatwaStudentVoices_Episode1Jessie/SV01JessieAM40SW00_64kb.mp3" width="144" height="74" type="audio/mpeg" autostart="false" loop="false" bgcolor="white"> </embed><br /><br><br />(<a href="http://www.archive.org/download/DarrenKuropatwaStudentVoices_Episode1Jessie/SV01JessieAM40SW00_64kb.mp3">Download File</a> 5.6Mb, 11 min. 40 sec.)<br /></center><br /><p><br /><div align="right">Photo Credit: <a href="http://flickr.com/photos/11739182@N03/1263985679/">Kids of conversation</a> by flickr user Kris Hoet</div><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-2027054954580832963?l=apcalc07.blogspot.com' alt='' /></div>dkuropatwahttp://www.blogger.com/profile/08462283847470560887noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-91935591612970587582008-04-19T13:00:00.002-05:002008-04-19T13:39:14.754-05:00<span style="font-family:trebuchet ms;">So now we are well into exam review and for good reason. (There are only 6 classes left until the exam).<br /><br /><span style="color: rgb(102, 0, 0);">On Thursday our class consisted of five accumulation problems from the AP exams of five different years.</span><br /><br /><span style="color: rgb(102, 0, 0);">We completed them in groups and then Mr. K. went over them in class. The questions were not extremely hard, but there were a few tricky parts. I believe after Mr. K. discussed them in full, we understood where we had made our mistakes.<br /><br /><span style="color: rgb(153, 51, 0);"><span style="color: rgb(0, 102, 0);">The more important part of the exercise was to notice how the same type of problem changed over the different years. <a href="http://www.slideshare.net/dkuropatwa/ap-calculus-slides-april-17-2008?src=embed">Looking at them</a>, you can see that the graphs are the most noticeable things that change. They become less familiar in behaviour and less points are actually drawn and labeled. The graphs also begin to have a lot of corners which are discrepancies in the second derivative as the are undefined. It then asks if these points are points of inflection, which are usually calculated via the second derivative. That is probably the trickiest part of these questions, but with a little understanding of how the derivatives and their graphs relate to each other, it is easy to get around.<br />The questions also change fairly drastically, as they go from just understanding how to use the graph and the given functions, to looking past the given graph and fuctions and using it to find characteristics of the parent/derivative functions and their graphs. In the later examples, they begin to ask questions about behaviour of the functions over certain intervals and even ask to draw the graph of the parent function on the axes.<br /><span style="color: rgb(0, 153, 0);">The final example (from 2007) is quite different from the others as there is no graph given, but a table of values is. The table includes two functions (ƒ and g) and their derivatives. It then asks questions about the function h(x) = ƒ(g(x)) - 6 and how it behaves over certain intervals. It also asks a new question that we have not seen yet.<br />It asks us to find the derivative of the inverse of a function ( g^-1 ). It is a rule that Mr. K. has not yet taught us, but it was explained to me as "the derivative of an inverse of a function is equal to the inverse of its derivative" basically, ∂/∂x(g^-1) = 1/g'. From there just apply it as you would any derivative and solve the question.<br /><span style="color: rgb(0, 0, 153);"><br />Finally, Mr. K. said that this last example is probably the closest example of how it will be presented on our exam, so it would be a good idea to go over this some time in the next two weeks to make sure you have this under your belt, it should be easy marks now.<br /><br /><span style="color: rgb(0, 0, 0);">That is all for my scribe post, remember to start studying "fiendishly" (if you haven't already) there are only 18 days left. O_O</span><br /><br /><span style="color: rgb(0, 0, 0);">The next scribe is <span style="font-size:180%;"><span style="color: rgb(51, 51, 255);">Van</span></span> I guess...<br /><br />Enjoy the rest of your weekends!<br /><br /></span></span></span></span></span></span></span><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-9193559161297058758?l=apcalc07.blogspot.com' alt='' /></div>«Craig»http://www.blogger.com/profile/08860268378924842699noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-54060665456650247662008-04-17T13:40:00.003-05:002008-04-17T13:43:45.163-05:00<span style="color: rgb(255, 0, 0); font-weight: bold;font-size:150%;" >20 Days Until The Exam And Counting ...</span><br /><br />Here are the slides ...<br /><br /><center><br /><div style="width:425px;text-align:left" id="__ss_358949"><object style="margin:0px" width="425" height="355"><param name="movie" value="http://static.slideshare.net/swf/ssplayer2.swf?doc=apcalc07080417-1208457518131961-9"/><param name="allowFullScreen" value="true"/><param name="allowScriptAccess" value="always"/><embed src="http://static.slideshare.net/swf/ssplayer2.swf?doc=apcalc07080417-1208457518131961-9" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="355"></embed></object><div style="font-size:11px;font-family:tahoma,arial;height:26px;padding-top:2px;"><a href="http://www.slideshare.net/?src=embed"><img src="http://static.slideshare.net/swf/logo_embd.png" style="border:0px none;margin-bottom:-5px" alt="SlideShare"/></a> | <a href="http://www.slideshare.net/dkuropatwa/ap-calculus-slides-april-17-2008?src=embed" title="View 'AP Calculus Slides April 17, 2008' on SlideShare">View</a> | <a href="http://www.slideshare.net/upload?src=embed">Upload your own</a></div></div><br /></center><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-5406066545665024766?l=apcalc07.blogspot.com' alt='' /></div>dkuropatwahttp://www.blogger.com/profile/08462283847470560887noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-89248334366534411422008-04-15T22:42:00.013-05:002008-11-18T18:01:10.094-06:00<a href="http://1.bp.blogspot.com/_fvK_Af4KmnA/SAWHMowrxYI/AAAAAAAAAG8/7VIhHea9RzM/s1600-h/Post+%231_1.jpeg"><img id="BLOGGER_PHOTO_ID_5189702796784158082" style="CURSOR: hand" height="324" alt="" src="http://1.bp.blogspot.com/_fvK_Af4KmnA/SAWHMowrxYI/AAAAAAAAAG8/7VIhHea9RzM/s400/Post+%231_1.jpeg" width="439" border="0" /></a><a href="http://4.bp.blogspot.com/_fvK_Af4KmnA/SAWHQYwrxZI/AAAAAAAAAHE/tR3MloCd9bs/s1600-h/Post+%231_2.jpeg"><img id="BLOGGER_PHOTO_ID_5189702861208667538" style="CURSOR: hand" height="330" alt="" src="http://4.bp.blogspot.com/_fvK_Af4KmnA/SAWHQYwrxZI/AAAAAAAAAHE/tR3MloCd9bs/s400/Post+%231_2.jpeg" width="439" border="0" /></a><a href="http://3.bp.blogspot.com/_fvK_Af4KmnA/SAWHUIwrxaI/AAAAAAAAAHM/-Jl5dZ6yzB0/s1600-h/Post+%231_3.jpeg"><img id="BLOGGER_PHOTO_ID_5189702925633176994" style="CURSOR: hand" height="338" alt="" src="http://3.bp.blogspot.com/_fvK_Af4KmnA/SAWHUIwrxaI/AAAAAAAAAHM/-Jl5dZ6yzB0/s400/Post+%231_3.jpeg" width="439" border="0" /></a><a href="http://4.bp.blogspot.com/_fvK_Af4KmnA/SAWHaYwrxbI/AAAAAAAAAHU/UDs6e1m24fE/s1600-h/Post+%231_4.jpeg"><img id="BLOGGER_PHOTO_ID_5189703033007359410" style="CURSOR: hand" height="339" alt="" src="http://4.bp.blogspot.com/_fvK_Af4KmnA/SAWHaYwrxbI/AAAAAAAAAHU/UDs6e1m24fE/s400/Post+%231_4.jpeg" width="440" border="0" /></a><a href="http://2.bp.blogspot.com/_fvK_Af4KmnA/SAWHz4wrxhI/AAAAAAAAAIE/VXWZ-fL95NA/s1600-h/Post+%231_5.jpeg"><img id="BLOGGER_PHOTO_ID_5189703471094023698" style="CURSOR: hand" height="337" alt="" src="http://2.bp.blogspot.com/_fvK_Af4KmnA/SAWHz4wrxhI/AAAAAAAAAIE/VXWZ-fL95NA/s400/Post+%231_5.jpeg" width="441" border="0" /></a><br /><div><a href="http://1.bp.blogspot.com/_fvK_Af4KmnA/SAWHwowrxgI/AAAAAAAAAH8/aTYp1hCETpw/s1600-h/Post+%231_6.jpeg"><img id="BLOGGER_PHOTO_ID_5189703415259448834" style="WIDTH: 438px; CURSOR: hand; HEIGHT: 320px" height="312" alt="" src="http://1.bp.blogspot.com/_fvK_Af4KmnA/SAWHwowrxgI/AAAAAAAAAH8/aTYp1hCETpw/s400/Post+%231_6.jpeg" width="413" border="0" /></a><br /><br /><div><a href="http://2.bp.blogspot.com/_fvK_Af4KmnA/SAWHs4wrxfI/AAAAAAAAAH0/zRGbU1mvlBI/s1600-h/Post+%231_7.jpeg"><img id="BLOGGER_PHOTO_ID_5189703350834939378" style="CURSOR: hand" height="336" alt="" src="http://2.bp.blogspot.com/_fvK_Af4KmnA/SAWHs4wrxfI/AAAAAAAAAH0/zRGbU1mvlBI/s400/Post+%231_7.jpeg" width="439" border="0" /></a><br /><br /><br /><div><a href="http://4.bp.blogspot.com/_fvK_Af4KmnA/SAWHpYwrxeI/AAAAAAAAAHs/dqkMdTH5bHI/s1600-h/Post+%231_8.jpeg"><img id="BLOGGER_PHOTO_ID_5189703290705397218" style="CURSOR: hand" height="356" alt="" src="http://4.bp.blogspot.com/_fvK_Af4KmnA/SAWHpYwrxeI/AAAAAAAAAHs/dqkMdTH5bHI/s400/Post+%231_8.jpeg" width="441" border="0" /></a><br /><br /><br /><br /><div><a href="http://1.bp.blogspot.com/_fvK_Af4KmnA/SAWHlowrxdI/AAAAAAAAAHk/Y5BUFThwUI0/s1600-h/Post+%231_9.jpeg"><img id="BLOGGER_PHOTO_ID_5189703226280887762" style="WIDTH: 438px; CURSOR: hand; HEIGHT: 341px" height="333" alt="" src="http://1.bp.blogspot.com/_fvK_Af4KmnA/SAWHlowrxdI/AAAAAAAAAHk/Y5BUFThwUI0/s400/Post+%231_9.jpeg" width="449" border="0" /></a><br /><div><a href="http://3.bp.blogspot.com/_fvK_Af4KmnA/SAWHiIwrxcI/AAAAAAAAAHc/kcWTuDlIxMw/s1600-h/Post+%231_10.jpeg"><img id="BLOGGER_PHOTO_ID_5189703166151345602" style="CURSOR: hand" height="325" alt="" src="http://3.bp.blogspot.com/_fvK_Af4KmnA/SAWHiIwrxcI/AAAAAAAAAHc/kcWTuDlIxMw/s400/Post+%231_10.jpeg" width="438" border="0" /></a><br /><div><div><div><div> </div></div></div></div></div></div></div></div></div><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-8924833436653441142?l=apcalc07.blogspot.com' alt='' /></div>etimzhttp://www.blogger.com/profile/09626175119223517919noreply@blogger.com2tag:blogger.com,1999:blog-5788051255938485746.post-73922864390550290632008-04-15T12:49:00.002-05:002008-04-15T12:55:30.897-05:00Here they are ...<br /><br /><center><br /><div style="width:425px;text-align:left" id="__ss_354809"><object style="margin:0px" width="425" height="355"><param name="movie" value="http://static.slideshare.net/swf/ssplayer2.swf?doc=apcalc07080415-1208281867294246-9"/><param name="allowFullScreen" value="true"/><param name="allowScriptAccess" value="always"/><embed src="http://static.slideshare.net/swf/ssplayer2.swf?doc=apcalc07080415-1208281867294246-9" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="355"></embed></object><div style="font-size:11px;font-family:tahoma,arial;height:26px;padding-top:2px;"><a href="http://www.slideshare.net/?src=embed"><img src="http://static.slideshare.net/swf/logo_embd.png" style="border:0px none;margin-bottom:-5px" alt="SlideShare"/></a> | <a href="http://www.slideshare.net/dkuropatwa/ap-calculus-slides-april-15-2008?src=embed" title="View 'AP Calculus Slides April 15, 2008' on SlideShare">View</a> | <a href="http://www.slideshare.net/upload?src=embed">Upload your own</a></div></div><br /></center><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-7392286439055029063?l=apcalc07.blogspot.com' alt='' /></div>dkuropatwahttp://www.blogger.com/profile/08462283847470560887noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-81274623450715698662008-04-12T09:24:00.002-05:002008-11-18T18:01:10.217-06:00<a href="http://apcalc07.blogspot.com/2008/04/bob.html">Craig mentioned in his BOB</a>:<br /><blockquote>Anyway, this is the last unit and from now on it's all exam review.<br />It may seem like it's far away (a little less than a month), but there are ONLY 8 MORE CLASSES!!! So the intense studying should probably begin.... NOW!</blockquote><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://4.bp.blogspot.com/_HiqPzpd3O9Y/SADGrCVZZOI/AAAAAAAAAHQ/uUa_iRIaFkc/s1600-h/poster+good+seize+the+day.jpg"><img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;" src="http://4.bp.blogspot.com/_HiqPzpd3O9Y/SADGrCVZZOI/AAAAAAAAAHQ/uUa_iRIaFkc/s200/poster+good+seize+the+day.jpg" border="0" alt=""id="BLOGGER_PHOTO_ID_5188365213394101474" /></a><br />How's your hold on calculus?<br /><br /><span style="font-style:italic;"><a href="http://www.flickr.com/photos/acousticdad/164544542/">image source</a></span><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-8127462345071569866?l=apcalc07.blogspot.com' alt='' /></div>Lani Ritter Hallhttp://www.blogger.com/profile/16352862711544966770noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-15477945577525270022008-04-10T23:58:00.002-05:002008-04-11T00:05:16.154-05:00Woah! We had the test moved back two days and I still almost forgot.<br />But yes, this last unit has been okay mainly because it is just an elaboration on material we've already covered.<br />However, I believe the hardest part for me is that we started it quite a while ago. It's almost been a month since we began the unit.<br /> But, it's nothing a little studying won't fix.<br />I'm trying to think of the parts that are most challenging, but none seem to stick out. I think the place where I'll make my mistakes is on the tricky little curves Mr. K. enjoys throwing into the tests =P<br />Anyway, this is the last unit and from now on it's all exam review.<br />It may seem like it's far away (a little less than a month), but there are ONLY 8 MORE CLASSES!!! So the intense studying should probably begin.... NOW!<br /><br />Good Luck to all on the test tomorrow, and good luck to all on the upcoming exam.<br /><br />Keep On Pushing!<div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-1547794557752527002?l=apcalc07.blogspot.com' alt='' /></div>«Craig»http://www.blogger.com/profile/08860268378924842699noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-44045468135537063152008-04-09T00:52:00.002-05:002008-04-09T01:11:16.838-05:00Oh wow, I completely forgot about bobbing. Thankfully, I had to take a picture of the blog for a presentation, and at first glance the "Blogging on Blogging" title immediately jogged my memory. I also can't believe that there aren't that many BOB's up yet, especially considering the fact that I'm posting this at 1 A.M. Well anyways, on with my final BOB of the year (=.<br /><br />It's been a crazy year crammed full of information, and this final unit was certainly no exception to working hard. This unit, <span style="color: rgb(0, 0, 153);">differential equations</span>, didn't prove too complicated since it basically solidified the connection between differentiation and integration / antidifferentiation by drawing upon all of the units we've covered so far in the year.<br /><br />The most challenging idea in this unit is probably Euler's method, though I don't think that it will be that big of a deal on the test tomorrow. The other ideas included in this unit were<span style="color: rgb(255, 102, 0);"> slope fields</span>, <span style="color: rgb(204, 0, 0);">solving differential equations</span>, <span style="color: rgb(0, 153, 0);">separating the variables </span>(which greatly aids the solving process) and <span style="color: rgb(51, 102, 255);">Newton's Law of Cooling</span>. I don't think that comprehending each idea on it's own was too difficult, though in the whole scheme of differential equations some questions (especially Mr. K questions xD) might prove troublesome. For me, this is particularly due to my lack of effort in committing myself to complete homework and consistently study the ideas in this unit.<br /><br />I hope that I won't find too much difficulty with this test, and that I have enough time to study tonight and catch up on all that missed work over the past week or two. I hope that all of you will do the same, and <span style="color: rgb(255, 0, 0);">I wish everyone good luck on the test tomorrow!</span><span style="font-style: italic;"><span style="font-style: italic;"><span style="font-style: italic;"><span style="font-style: italic;"></span></span></span></span><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-4404546813553706315?l=apcalc07.blogspot.com' alt='' /></div>MrSiwWyhttp://www.blogger.com/profile/18072494079396275311noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-86726562605232585382008-04-08T21:08:00.002-05:002008-04-08T21:13:54.832-05:00Oh my, it's test time tomorrow! Well, here for another bob (throws bob post onto large stack of bobs). This time it's for the test on Differential Equations! How exciting? Well anyway, this unit, like most of the other units had its ups and downs. With regards to my muddiest point, I would have to say that I don't have a specific problem. All that I'm worried about, like most of the times is the battle against time, by trying to make everything click on the test. Thats the thing though, tests aren't as difficult. Well what I'm trying to say is, the solutions aren't as bad as we usually think, but the difficult thing is, figuring the key solution that would unlock insight to finding the final answer. Asides from the likes of that, theres also remembering to attempt to solve a question using as little time as possible (such as taking advantage of your calculator). Well, thats all i really have to say about this unit! Good luck everyone! I hope people remember to bob, cause I almost forgot =(<div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-8672656260523258538?l=apcalc07.blogspot.com' alt='' /></div>Tim_MATH_yhttp://www.blogger.com/profile/09929458936647070771noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-11176314073540381362008-04-08T20:07:00.003-05:002008-04-08T20:16:25.118-05:00Well, this unit was one of the small units that we had done although I still say the smaller the units the harder they are. Especially for this unit, I am still a little nervous going into this test as I still cannot wrap my head around Euler's method, it is very confusing to understand although I am still studying it. Another spot I am having a little bit of trouble is with the <span class="blsp-spelling-corrected" id="SPELLING_ERROR_0">anti differentiating</span> first-order and second-order equations although that also is a working progress which I should have figured out by <span class="blsp-spelling-corrected" id="SPELLING_ERROR_1">tomorrow</span>. <span style="color: rgb(255, 0, 0);">Well good luck everybody for </span><span style="color: rgb(255, 0, 0);" class="blsp-spelling-corrected" id="SPELLING_ERROR_2">tomorrow's</span><span style="color: rgb(255, 0, 0);"> test and good luck.</span><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-1117631407354038136?l=apcalc07.blogspot.com' alt='' /></div>Dinohttp://www.blogger.com/profile/16720903626604404687noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-90867333012123101322008-04-07T22:08:00.003-05:002008-04-07T22:22:09.435-05:00<div style="text-align: center;"><span style="font-weight: bold;font-family:courier new;font-size:180%;" ><span style="color: rgb(0, 0, 0);"><span style="color: rgb(255, 0, 0);">H</span><span style="color: rgb(255, 102, 0);">A</span><span style="color: rgb(255, 204, 0);">P</span><span style="color: rgb(0, 153, 0);">P</span><span style="color: rgb(51, 102, 255);">Y</span> <span style="color: rgb(0, 0, 153);">E</span><span style="color: rgb(102, 0, 204);">I</span><span style="color: rgb(255, 0, 0);">G</span><span style="color: rgb(255, 102, 0);">H</span><span style="color: rgb(255, 204, 0);">T</span><span style="color: rgb(0, 153, 0);">E</span><span style="color: rgb(51, 102, 255);">E</span><span style="color: rgb(0, 0, 153);">N</span><span style="color: rgb(102, 0, 204);">T</span><span style="color: rgb(255, 0, 0);">H</span> <span style="color: rgb(255, 102, 0);">B</span><span style="color: rgb(255, 204, 0);">I</span><span style="color: rgb(0, 153, 0);">R</span><span style="color: rgb(51, 102, 255);">T</span><span style="color: rgb(0, 0, 153);">H</span><span style="color: rgb(102, 0, 204);">D</span><span style="color: rgb(255, 0, 0);">A</span><span style="color: rgb(255, 102, 0);">Y</span><span style="color: rgb(255, 204, 0);">!<span style="color: rgb(0, 153, 0);">!</span></span><span style="color: rgb(51, 102, 255);">!</span></span></span><br /><span style="font-weight: bold;font-family:courier new;font-size:180%;" ><span style="color: rgb(0, 0, 0);"><span style="color: rgb(51, 102, 255);"></span></span></span></div><span style="font-weight: bold;font-family:courier new;font-size:180%;" ><span style="color: rgb(0, 0, 0);"><span style="color: rgb(51, 102, 255);"><br /></span></span></span><div style="text-align: center;"><span style="font-weight: bold;font-family:courier new;font-size:130%;" ><span style="color: rgb(0, 0, 0);"><span style="color: rgb(51, 102, 255);"><span style="font-family:times new roman;"><span style="color: rgb(0, 0, 0); font-style: italic;">***Craig-4th (belated)***</span></span></span></span></span><span style="font-size:130%;"><br /></span><span style="font-weight: bold;font-family:courier new;font-size:130%;" ><span style="color: rgb(0, 0, 0);"><span style="color: rgb(51, 102, 255);"><span style="font-family:times new roman;"><span style="color: rgb(0, 0, 0); font-style: italic;">***Graeme-7th***</span></span></span></span></span><span style="font-size:130%;"><br /></span><span style="font-weight: bold;font-family:courier new;font-size:130%;" ><span style="color: rgb(0, 0, 0);"><span style="color: rgb(51, 102, 255);"><span style="font-family:times new roman;"><span style="color: rgb(0, 0, 0); font-style: italic;">***Chris-13th (advanced)***</span></span></span></span></span><br /><span style="font-weight: bold;font-family:courier new;font-size:180%;" ><span style="color: rgb(0, 0, 0);"><span style="color: rgb(51, 102, 255);"><span style="font-family:times new roman;"><span style="font-size:130%;"><span style="color: rgb(0, 0, 0); font-style: italic;"></span></span></span></span></span></span></div><span style="font-weight: bold;font-family:courier new;font-size:180%;" ><span style="color: rgb(0, 0, 0);"><span style="color: rgb(51, 102, 255);"><span style="font-family:times new roman;"><span style="font-size:130%;"><span style="color: rgb(0, 0, 0); font-style: italic;"><br /></span></span></span></span></span></span><div style="text-align: center;"><span style=";font-family:courier new;font-size:100%;" ><span style="color: rgb(0, 0, 0);"><span style="color: rgb(51, 102, 255);"><span style="font-family:times new roman;"><span style="color: rgb(0, 0, 0);"><span style="font-family:arial;">Much love ! =) </span></span></span></span></span><br /><span style=";font-family:courier new;font-size:100%;" ><span style="color: rgb(0, 0, 0);"><span style="color: rgb(51, 102, 255);"><span style="font-family:times new roman;"><span style="color: rgb(0, 0, 0);"><span style="font-family:arial;">Good luck with your DEV =)<br /><br /><br />---<span style="color: rgb(0, 0, 102);">Aichelle ROCKS!</span></span></span></span></span></span></span><br /><span style=";font-family:courier new;font-size:100%;" ><span style="color: rgb(0, 0, 0);"><span style="color: rgb(51, 102, 255);"><span style="font-family:times new roman;"><span style="color: rgb(0, 0, 0);"><span style="font-family:arial;"></span></span></span></span></span></span><span style="font-weight: bold;font-family:courier new;font-size:180%;" ><span style="color: rgb(0, 0, 0);"><span style="color: rgb(51, 102, 255);"><span style="font-family:times new roman;"><span style="font-size:130%;"><span style="color: rgb(0, 0, 0); font-style: italic;"></span></span></span></span></span></span><br /><span style="font-weight: bold;font-family:courier new;font-size:180%;" ><span style="color: rgb(0, 0, 0);"><span style="color: rgb(51, 102, 255);"><span style="font-family:times new roman;"></span></span></span></span></span></div><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-9086733301212310132?l=apcalc07.blogspot.com' alt='' /></div>aichelle s.http://www.blogger.com/profile/17302966674461415089noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-85955111078513006582008-04-07T13:57:00.003-05:002008-04-07T14:06:04.966-05:00Here they are ...<br /><br /><center><br /><div style="width:425px;text-align:left" id="__ss_340670"><object style="margin:0px" width="425" height="355"><param name="movie" value="http://static.slideshare.net/swf/ssplayer2.swf?doc=apcalc07080407-1207594018132976-8"/><param name="allowFullScreen" value="true"/><param name="allowScriptAccess" value="always"/><embed src="http://static.slideshare.net/swf/ssplayer2.swf?doc=apcalc07080407-1207594018132976-8" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="355"></embed></object><div style="font-size:11px;font-family:tahoma,arial;height:26px;padding-top:2px;"><a href="http://www.slideshare.net/?src=embed"><img src="http://static.slideshare.net/swf/logo_embd.png" style="border:0px none;margin-bottom:-5px" alt="SlideShare"/></a> | <a href="http://www.slideshare.net/dkuropatwa/ap-calculus-slides-april-7-2008?src=embed" title="View 'AP Calculus Slides April 7, 2008' on SlideShare">View</a> | <a href="http://www.slideshare.net/upload?src=embed">Upload your own</a></div></div><br /></center><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-8595511107851300658?l=apcalc07.blogspot.com' alt='' /></div>dkuropatwahttp://www.blogger.com/profile/08462283847470560887noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-87864724117632105392008-03-27T15:49:00.001-05:002008-03-27T16:01:18.952-05:00Here they are ...<br /><br /><center><br /><div style="width:425px;text-align:left" id="__ss_324433"><object style="margin:0px" width="425" height="355"><param name="movie" value="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-march-27-2008-120665136252688-2"/><param name="allowFullScreen" value="true"/><param name="allowScriptAccess" value="always"/><embed src="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-march-27-2008-120665136252688-2" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="355"></embed></object><div style="font-size:11px;font-family:tahoma,arial;height:26px;padding-top:2px;"><a href="http://www.slideshare.net/?src=embed"><img src="http://static.slideshare.net/swf/logo_embd.png" style="border:0px none;margin-bottom:-5px" alt="SlideShare"/></a> | <a href="http://www.slideshare.net/dkuropatwa/ap-calculus-slides-march-27-2008?src=embed" title="View 'AP Calculus Slides March 27, 2008' on SlideShare">View</a> | <a href="http://www.slideshare.net/upload?src=embed">Upload your own</a></div></div><br /></center><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-8786472411763210539?l=apcalc07.blogspot.com' alt='' /></div>dkuropatwahttp://www.blogger.com/profile/08462283847470560887noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-31394118238454370072008-03-25T12:42:00.001-05:002008-03-25T13:21:00.018-05:00Here they are ...<br /><br /><center><br /><div style="width:425px;text-align:left" id="__ss_321153"><object style="margin:0px" width="425" height="355"><param name="movie" value="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-march-25-2008-1206468996189607-2"/><param name="allowFullScreen" value="true"/><param name="allowScriptAccess" value="always"/><embed src="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-march-25-2008-1206468996189607-2" 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/></div>dkuropatwahttp://www.blogger.com/profile/08462283847470560887noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-21483207716495818592008-03-23T20:16:00.008-05:002008-11-18T18:01:11.013-06:00Hello everyone! I'm known as <span style="color: rgb(204, 102, 0);">Tim-math-y</span> and I'm the <span style="color: rgb(204, 102, 0);">scribe for Thursday's lessons</span>. On Thursday, our topic was: Solving Differential Equations Symbolically and Newton's Law of Cooling.<br /><br /><span style="font-weight: bold;">Introduction:</span><br /><br /><span style="color: rgb(0, 0, 102);">Solving Differential Equations Symbolically and Newton's Law of Cooling:</span><br /><br />On Thursday, we learned more about differential equations, this time using the symbolic (algebraic) approach. This topic tied into <span style="color: rgb(0, 0, 153);">Newton's Law of Cooling</span>. According to this law, <span style="color: rgb(0, 0, 153);">a hot object cools at a rate proportional to the difference between its own temperature and that of its environment.</span><br /><br />By applying this concept, we solved questions that dealt with cooling.<br /><br /><span style="font-weight: bold;">Content and Lessons:</span><br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://3.bp.blogspot.com/_HLm_KkQlC58/R-cRu9-6pfI/AAAAAAAAAIU/4-xHvCneB6k/s1600-h/NLaw1.JPG"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://3.bp.blogspot.com/_HLm_KkQlC58/R-cRu9-6pfI/AAAAAAAAAIU/4-xHvCneB6k/s320/NLaw1.JPG" alt="" id="BLOGGER_PHOTO_ID_5181129394923152882" border="0" /></a>First we began by <span style="color: rgb(0, 0, 153);">deconstructing the equation</span> of "<span style="color: rgb(0, 0, 153);">dy/dx = ky</span>." Mr. K mentioned in the past that normally, you are not allowed to pull apart the differential operator. However, without going through the complexities of why it is possible, we pulled it apart.<br /><br />By seperating the two variables, including both '<span style="color: rgb(0, 0, 153);">dy</span>' and '<span style="color: rgb(0, 0, 153);">dx</span>', we found a familiar term that could be <span style="color: rgb(0, 0, 153);">antidifferentiated</span>.<br /><br />Through algebraic play, we found the equation that defined the parent function of cooling.<br /><br />*Note: the variable "c" and "C" are different in value but generally both represent a <span style="color: rgb(0, 0, 153);">constant</span>.<br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://3.bp.blogspot.com/_HLm_KkQlC58/R-cUk9-6phI/AAAAAAAAAIk/TidBJwoliyg/s1600-h/NL3.JPG"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://3.bp.blogspot.com/_HLm_KkQlC58/R-cUk9-6phI/AAAAAAAAAIk/TidBJwoliyg/s320/NL3.JPG" alt="" id="BLOGGER_PHOTO_ID_5181132521659344402" border="0" /></a>Next we looked at an example question.<br /><ul><li>First we assigned variables, <span style="color: rgb(204, 102, 0);">F = Temperature in Degrees of F</span> and <span style="color: rgb(204, 102, 0);">t = T time in Hours</span></li><li>Second we applied the variables to "<span style="color: rgb(0, 0, 153);">dy/dx=ky</span>" to produce "<span style="color: rgb(0, 0, 153);">dF/dt=k(F-20)</span>" where 2o is the lowest temperature the object can attain in an environment</li><li>By applying the same steps to seperate 'dF' and 'dt', we antidifferentiated and worked the out an <span style="color: rgb(0, 0, 153);">equation that could be used to find 'F', the temperature, at any given time. The only unknown left is the constant that we have yet to solve.</span><br /></li></ul><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://3.bp.blogspot.com/_HLm_KkQlC58/R-cW09-6piI/AAAAAAAAAIs/OZhYBNvmoIo/s1600-h/NL4.JPG"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://3.bp.blogspot.com/_HLm_KkQlC58/R-cW09-6piI/AAAAAAAAAIs/OZhYBNvmoIo/s320/NL4.JPG" alt="" id="BLOGGER_PHOTO_ID_5181134995560506914" border="0" /></a>By using the information given (two pieces of data that showed us the temperature at a specific point in time) we substituted the values in and solved for the final equation that represented the temperature at any given time of this situation.<br /><br />At this point, we considerred both questions solved.<br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://3.bp.blogspot.com/_HLm_KkQlC58/R-cT79-6pgI/AAAAAAAAAIc/-814XOjTJ8s/s1600-h/NL2.JPG"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://3.bp.blogspot.com/_HLm_KkQlC58/R-cT79-6pgI/AAAAAAAAAIc/-814XOjTJ8s/s320/NL2.JPG" alt="" id="BLOGGER_PHOTO_ID_5181131817284707842" border="0" /></a>Next, we looked at another example question (Don't be overwhelmed by the length of the question as most of it is just a background story that Mr. K conjured up: The information required is located in <span style="color: rgb(0, 0, 153);">paragraph 3</span>).<br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://3.bp.blogspot.com/_HLm_KkQlC58/R-cYB9-6pjI/AAAAAAAAAI0/tnDXLwW0UHA/s1600-h/NL5.JPG"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://3.bp.blogspot.com/_HLm_KkQlC58/R-cYB9-6pjI/AAAAAAAAAI0/tnDXLwW0UHA/s320/NL5.JPG" alt="" id="BLOGGER_PHOTO_ID_5181136318410434098" border="0" /></a>By applying the same steps from the previous question, we found the solution to the problem. Again, our variable assignments were the same. The only difference is that the coldest temperature in this situation was 25 degrees. <span style="color: rgb(153, 0, 0);">We found the equation that still included the constant value of C.</span><br /><br /><span style="color: rgb(0, 102, 0);">By taking this a step further, we inputted the coordinates given (temperatures at given times) and solved for the value of C to in turn, solve for the final equation.</span><br /><br /><span style="color: rgb(0, 0, 153);">Finally, by inputting the temperature of the cola at an unknown point in time, we solved for the uknown value of 't'.<br /><br /><span style="color: rgb(0, 0, 0);"><span style="font-weight: bold;">Conlusion:</span><br /><br />We went through two problems that dealt with Newton's Law of Cooling, solving differential equations symbolically.<br /><br />******<br /><br />Homework is on the end of the slides posted in the previous post! Answer is on the following slide.<br /><br /><span style="color: rgb(102, 0, 204);">Tomorrow's Scribe is... </span><span style="font-weight: bold; color: rgb(102, 0, 204);">Craig!</span><br /><br />Happy Easter? Haha, good night.<br /></span></span><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-2148320771649581859?l=apcalc07.blogspot.com' alt='' /></div>Tim_MATH_yhttp://www.blogger.com/profile/09929458936647070771noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-13484987505169986192008-03-20T12:46:00.001-05:002008-03-20T12:53:15.038-05:00Here they are ...<br /><br /><center><br /><div style="width:425px;text-align:left" id="__ss_315545"><object style="margin:0px" width="425" height="355"><param name="movie" value="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-march-20-2008-1206035081886250-3"/><param name="allowFullScreen" value="true"/><param name="allowScriptAccess" value="always"/><embed src="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-march-20-2008-1206035081886250-3" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="355"></embed></object><div style="font-size:11px;font-family:tahoma,arial;height:26px;padding-top:2px;"><a href="http://www.slideshare.net/?src=embed"><img src="http://static.slideshare.net/swf/logo_embd.png" style="border:0px none;margin-bottom:-5px" alt="SlideShare"/></a> | <a href="http://www.slideshare.net/dkuropatwa/ap-calculus-slides-march-20-2008?src=embed" title="View 'AP Calculus Slides March 20, 2008' on SlideShare">View</a> | <a href="http://www.slideshare.net/upload?src=embed">Upload your own</a></div></div><br /></center><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-1348498750516998619?l=apcalc07.blogspot.com' alt='' /></div>dkuropatwahttp://www.blogger.com/profile/08462283847470560887noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-39899330298331695582008-03-19T21:46:00.006-05:002008-11-18T18:01:11.691-06:00Hey everyone, it's MrSiwWy here as the scribe for March 18's class on Euler's method. I know I started this scribe post pretty late, but I just got home from the fashion show at school, which was definitely an enjoyable occasion. I really found this topic quite interesting, so I'll very much enjoy explaining his method as intricately as possible. Now with classes only every second day, I didn't think scribe duties would remain so frequent, though the class is indeed quite minuscule in comparison to the class size first semester. Oh yes, and before I begin, I think i must note that all explanations will be accompanied by an example furthering my explanation in a different text colour. Text in black will be initial explanations, <span style="color: rgb(0, 102, 0);">while text in green will pertain to explanations using an example for the first part of the post</span> and text in <span style="color: rgb(0, 0, 153);">blue will pertain to explanations using an example for the second part of the post</span>. Hope it works out well. Enjoy!<br /><br /><span style="font-weight: bold; font-style: italic;font-size:130%;" >Scribe</span><br />~~~~~~<br /><br />I can't remember exactly how class started, though I do recall it initiated with the usual abstract chatter and various technological utility exposures by Mr. K. But what I do remember, is beginning the actual lesson with a very fundamentally elegant and quite remarkable equation; <a href="http://en.wikipedia.org/wiki/Euler%27s_identity">Euler's identity</a>.<br /><br />Though we didn't go in depth into what Euler's identity is and how it was formulated, but we did vaguely discuss the implications of the identity. The sheer elegance of the equation derives from the fact that it contains and also intertwines the destiny of five of the most important constants in mathematics: 0, 1, (pi), <span style="font-style: italic;">i</span>, and <span style="font-style: italic;">e</span>. Euler's identity is as follows:<br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://1.bp.blogspot.com/_dvbjJeJb4LE/R-HuC6-mV7I/AAAAAAAAAPo/yatAeUNkdAI/s1600-h/pic1.JPG"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://1.bp.blogspot.com/_dvbjJeJb4LE/R-HuC6-mV7I/AAAAAAAAAPo/yatAeUNkdAI/s400/pic1.JPG" alt="" id="BLOGGER_PHOTO_ID_5179682780411746226" border="0" /></a><br />Now we transitioned into the basis of the day's topic: Euler's method. I think it's quite a useful technique, and is a very innovative technique with easy implementation (at our level at least). We started off with a demonstration of Euler's method without any formal introduction quite yet. This demonstration can be found <a href="http://www.math.ubc.ca/%7Efeldman/demos/demo1.html">here</a>. If you follow the link, it might ease the subsequent explanations tremendously.<br /><br />- First off, imagine that you are given an initial value problem. Recall that an initial value problem is a problem in which you are given a differential equation to solve as usual, but you are also given a point that exists on the parent function/solution to the differential equation.<br /><br /><span style="color: rgb(0, 153, 0);">*In this case, as used by the aforementioned demonstration, take the differential equation to be y'(t) = 1 - t + 4y(t) and the initial point to be y(0) = 1. This means that at t = 0, y(t) = 1. Following the above link will drastically clarify this example. Also, take note that we know the point (0,1) on the differential equation solution, and therefore have an exact solution and not just a general one. A general solution only represents the family of functions that could fit the solution for the differential equation. An exact one means that it's only specific function, and not a whole family.</span><br /><br />- Now, there is an important idea that I must stress in order for Euler's method to be utilized easily, so pay attention in case you missed it in class. We must find a way to get another point on the parent function, which can be easily determined by plugging in the initial value/point into the differential equation to solve for the slope of the line tangent to the function at that point. Using this slope we can use our classic definition of slope to yield the next point that exists on the function y(t) given a certain step.<br /><br /><span style="color: rgb(0, 153, 0);">*The point (0,1) is known to lie on the function y(t), but how are we possibly going to get another point on this function? Well, as I stated earlier, since we know the differential equation, we can plug in the point (0,1) into the differential equation (which is y'(t) = 1 - t + 4y(t) in this case) and determine </span><span style="font-weight: bold; color: rgb(0, 153, 0);">the slope of the line tangent to the function at that point</span><span style="color: rgb(0, 153, 0);">. This is because the differential equation will solve for y'(t), which is the derivative of y(t) or rather <span style="font-weight: bold;">the instantaneous rate of change of y(t) at a given t</span>. This is particularly useful sinc</span><span style="color: rgb(0, 153, 0);">e using our rather classic definition of "slope" (rise over run) will yield our next point. By plugging in the initial point into the differential equation as follows, we can determine the required slope:<br /></span><br /><span style="color: rgb(0, 153, 0);">y'(0) = 1 - (0) + 4(1)</span><br /><span style="color: rgb(0, 153, 0);">y'(0) = 5</span><br /><br /><span style="color: rgb(0, 153, 0);">Using this slope with the "old-school" rise over run definition of slope, we can easily determine the next point. But be careful, you must <span style="font-weight: bold;">pay careful attention to the steps or the scale of each axis</span> to correctly apply this definition. Since the steps we will be using in this case are 0.1, that means that instead of the function increasing upwards by 5 and increasing rightwards by 1, the function will increase upwards by 0.5 as it travels rightwards by 0.1. This means that the next point will be (0.1, 1.5). Using the next button on the demonstration page will automatically graph the next point on the function.</span><br /><br /><span style="color: rgb(0, 153, 0);"><span style="color: rgb(0, 0, 0);">- Now that we have two points on the graph, we can easily repeat the above process to determine any subsequent points and find an accurate portrayal of the solution. The only problem is that you must use a sufficient amount of steps by decreasing the amount of change along the x-axis (<span style="color: rgb(0, 153, 0);">or in this case t</span>) so that each point will be closer to each other and each subsequent derivative will be more accurate than with little points. I advise using <a href="http://www.ecs.fullerton.edu/%7Emathews/a2001/Animations/OrdinaryDE/Euler1/Euleraa.html">this website</a> or <a href="http://www.ugrad.math.ubc.ca/coursedoc/math100/notes/mordifeqs/euler.html">this website</a> to help grasp this concept. Review one of these sites to further your understanding of what I have just said. Also, concerning this idea, I tend to think of Euler's method as being quite similar to how an integral works. In an integral, you must let the <span style="font-style: italic;">dx </span>values (<span style="color: rgb(0, 153, 0);">or <span style="font-style: italic;">dt</span><span style="color: rgb(0, 0, 0);">) get as close to 0 as possible, so as to increase how accurate the solution truly is or how well the integral fits the actual shape of the function. This is exactly how Euler's method works.<br /><br /></span></span></span></span><span style="color: rgb(0, 153, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 153, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 153, 0);">*Each time that you press the next button on the above demonstration, the graph will further the use of Euler's method and graph the next point on the parent function, thereby creating an approximate graph of said function. But notice how choppy each segment looks, even though as a calculus student(and others =p) you should recognize the function to be curved.<br /></span></span></span></span></span><br /><span style="color: rgb(0, 153, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 153, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 153, 0);"><span style="color: rgb(0, 0, 0);">Once we were introduced to Euler's method using a bevy of demonstrations and tools to ease the idea of his method to us, we were asked to apply what we had just learned to a problem. This can be seen on the following slide:<br /><br /></span></span></span></span></span></span><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://2.bp.blogspot.com/_dvbjJeJb4LE/R-HuDK-mV8I/AAAAAAAAAPw/g4rcQlzGMQU/s1600-h/pic2.JPG"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://2.bp.blogspot.com/_dvbjJeJb4LE/R-HuDK-mV8I/AAAAAAAAAPw/g4rcQlzGMQU/s400/pic2.JPG" alt="" id="BLOGGER_PHOTO_ID_5179682784706713538" border="0" /></a><span style="color: rgb(0, 153, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 153, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 153, 0);"><span style="color: rgb(0, 0, 0);"><br />In the above slide, all that we really did was find a way (basically on our own) to efficiently repeat the process I detailed above. Though there were some alterations in this problem.<br /><br /></span></span></span></span></span></span><span style="color: rgb(0, 153, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 153, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 153, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 0, 153);">*Basically, when you were given the initial point, you could plug the point into the differential equation (<span style="color: rgb(0, 0, 153);">which is y' = y - x this time</span><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 0, 153);">) to find the derivative of the function at that point. This derivative will be </span><span style="font-style: italic; color: rgb(0, 0, 153);">m</span><span style="color: rgb(0, 0, 153);">, or the slope of the tangent line at that point, which can be used by</span></span></span></span></span></span></span></span></span><span style="color: rgb(0, 153, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 153, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 153, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 0, 153);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 0, 153);"> rearranging the <span style="font-style: italic;">m </span>=<span style="font-style: italic;"> Δy/Δx </span>equation into <span style="font-style: italic;">Δy </span>= <span style="font-style: italic;">m * Δx</span>. Since <span style="font-style: italic;">m </span>has just been calculated, and <span style="font-style: italic;">Δx</span> (which is the <span style="font-style: italic;">step </span>as I mentioned above) is given in the question as 0.25. Calculating <span style="font-style: italic;">Δy</span> will give us the change in y from the first point to the next point given a certain step, which means we'll know the next points x-coordinate (the first point x-coordinate plus the step) and y-coordinate (the first point's y-coordinate plus the <span style="font-style: italic;">Δy</span> calculated). Thus, this "next" point will be our "initial point" in the next part of the solution. If you use the above slide to help with understanding this idea, then <span style="font-weight: bold;">each row will be one iteration </span>of determining the solution to the differential</span></span></span></span></span></span></span></span></span><span style="color: rgb(0, 153, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 153, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 153, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 0, 153);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 0, 153);"> equation. An iteration means a complete round of doing something, such as the process that is being repeated within a loop (which can be taken as the process I detailed above in the first</span></span></span></span></span></span></span></span></span><span style="color: rgb(0, 153, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 153, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 153, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 0, 153);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 0, 153);"> detailed section of this scribe post).<br /><br /><span style="color: rgb(0, 0, 0);">But how can we make this process simpler and reduce the irksomeness of the entire process, especially for cases with a lot more repetitions? One way that we worked out during class involved using the store function in our calculators. This method can actually have several approaches, but basically you can utilize your calculator's store functions in many ways to achieve the same thing. Here's basically what you do:</span></span></span></span></span></span></span></span></span></span><br /><br /><span style="color: rgb(0, 153, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 153, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 153, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 0, 153);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 0, 153);"><span style="color: rgb(0, 0, 0);">- Store the initial y value into any variable in your calculator, say A.<br /><br /></span></span></span></span></span></span></span></span></span></span><span style="color: rgb(0, 153, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 153, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 153, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 0, 153);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 0, 153);"><span style="color: rgb(0, 0, 0);">- Input this variable into the differential equation on your calculator, and multiply this new expression by the step, and finally add A to this whole thing and store the new answer back into A. It might be hard at first to type this all in within one shot, but it is doable if you do it enough. Below is an example of how it should look in your calculator using the above question.<br /><br /><span style="color: rgb(0, 0, 153);">*Applying this step to the slide above, here's what it would look like in your calculator:<br />(((A - 0)0.25) + A) -> A<br /><br /></span></span></span></span></span></span></span></span></span></span></span><span style="color: rgb(0, 153, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 153, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 153, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 0, 153);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 0, 153);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 0, 153);"><span style="color: rgb(0, 0, 0);">- Simply repeat this line using the [2nd] then [enter] function in your calculator, each time changing the x value according to how the point has changed.<br /><br /></span></span></span></span></span></span></span></span></span></span></span></span><span style="color: rgb(0, 153, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 153, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 153, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 0, 153);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 0, 153);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 0, 153);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 0, 153);">*Again applying this step to the slide above, here's what the next line would look like:<br />(((A - 0.25)0.25) + A) -> A<br /><br /><span style="color: rgb(0, 0, 0);">Using this method yielded the answers shown in the above slide, which could be taken as solved since the last point on the interval asked for (which was [0,1]) was determined, the question was answered. Though, Mr. K wrote down the exact answer to the problem at the bottom of the slide, which wasn't that close to the answer we arrived at.<br /></span></span></span></span></span></span></span></span></span></span></span></span></span></span><br /><span style="color: rgb(0, 153, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 153, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 153, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 0, 153);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 0, 153);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 0, 153);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 0, 153);"><span style="color: rgb(0, 0, 0);">So then we tried using more segments to approximate a solution that is much closer than achieved above. We did this by using our <span class="blsp-spelling-corrected" id="SPELLING_ERROR_0">new found</span> method of numerically solving initial value problems with Euler's method, we attempted another problem. Though it may look quite empty, all of the work was quickly repeated on our calculators using a method roughly equivalent to the one above, and we all arrived at the same answers as shown in the following slide. I suggest trying it out on your own for some practice. (note: The question is basically the exact same as the first, but it uses far more segments and a much smaller step).<br /><br /></span></span></span></span></span></span></span></span></span></span></span></span></span></span><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://2.bp.blogspot.com/_dvbjJeJb4LE/R-HuDK-mV9I/AAAAAAAAAP4/dzZ93XOea8o/s1600-h/pic3.JPG"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://2.bp.blogspot.com/_dvbjJeJb4LE/R-HuDK-mV9I/AAAAAAAAAP4/dzZ93XOea8o/s400/pic3.JPG" alt="" id="BLOGGER_PHOTO_ID_5179682784706713554" border="0" /></a><span style="color: rgb(0, 153, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 153, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 153, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 0, 153);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 0, 153);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 0, 153);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 0, 153);"><span style="color: rgb(0, 0, 0);"><br />Just before class ended, Mr. K distributed the "EULER" calculator program for us to use to quickly solve problems involving Euler's method.<br /><br /></span></span></span></span></span></span></span></span></span></span></span></span></span></span><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://3.bp.blogspot.com/_dvbjJeJb4LE/R-HuDa-mV-I/AAAAAAAAAQA/xz34E17Cs18/s1600-h/pic4.JPG"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://3.bp.blogspot.com/_dvbjJeJb4LE/R-HuDa-mV-I/AAAAAAAAAQA/xz34E17Cs18/s400/pic4.JPG" alt="" id="BLOGGER_PHOTO_ID_5179682789001680866" border="0" /></a><span style="color: rgb(0, 153, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 153, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 153, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 0, 153);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 0, 153);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 0, 153);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 0, 153);"><span style="color: rgb(0, 0, 0);"><br />I will post the algorithms and full code for the program here when I get the full version again, since someone accidentally erased a line in the program and I can't remember what that line was.<br /><br />Okay, that's all for my scribe for now. I still have some stuff to edit, but it's getting pretty late now and I don't want to be late for Chemistry again tomorrow (I bet people in Chem would doubt that). Anyway, the next scribe will be:<br /><span style="font-size:130%;"><span style="color: rgb(204, 0, 0);">Tim-MATH-y!<span style="font-size:100%;"><span style="color: rgb(0, 0, 0);"><br />Well good night everyone, see you all in class tomorrow! Please just talk to me or comment if there are any questions, complaints, anxieties, confusions, etc. =p Good bye all!<br /></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span style="color: rgb(0, 153, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 153, 0);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(0, 153, 0);"></span></span></span></span></span><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-3989933029833169558?l=apcalc07.blogspot.com' alt='' /></div>MrSiwWyhttp://www.blogger.com/profile/18072494079396275311noreply@blogger.com1tag:blogger.com,1999:blog-5788051255938485746.post-90688987768027259002008-03-18T15:36:00.002-05:002008-03-18T15:39:52.377-05:00Here they are ...<br /><br /><center><br /><div style="width:425px;text-align:left" id="__ss_312582"><object style="margin:0px" width="425" height="355"><param name="movie" value="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-march-18-2008-1205872523940770-4"/><param name="allowFullScreen" value="true"/><param name="allowScriptAccess" value="always"/><embed src="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-march-18-2008-1205872523940770-4" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="355"></embed></object><div style="font-size:11px;font-family:tahoma,arial;height:26px;padding-top:2px;"><a href="http://www.slideshare.net/?src=embed"><img src="http://static.slideshare.net/swf/logo_embd.png" style="border:0px none;margin-bottom:-5px" alt="SlideShare"/></a> | <a href="http://www.slideshare.net/dkuropatwa/ap-calculus-slides-march-18-2008?src=embed" title="View 'AP Calculus Slides March 18, 2008' on SlideShare">View</a> | <a href="http://www.slideshare.net/upload?src=embed">Upload your own</a></div></div><br /></center><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-9068898776802725900?l=apcalc07.blogspot.com' alt='' /></div>dkuropatwahttp://www.blogger.com/profile/08462283847470560887noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-72428320941410481642008-03-17T23:16:00.001-05:002008-11-18T18:01:11.837-06:00If you'd like to leave a comment to Al Upton and the minilegends in Australia click on the picture below to get to their blog.<br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://alupton.edublogs.org/"><img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;" src="http://3.bp.blogspot.com/_tlF3qs3v9cw/R99AHlbTURI/AAAAAAAAAVQ/2zJ2q7-iB38/s320/minilegends.jpg" border="0" alt=""id="BLOGGER_PHOTO_ID_5178928595549245714" /></a><br /><br />You can also read more about Article 13 and the Convention on the Rights of the Child <a href="http://www.unicef.org/knowyourrights/convention_cartoons.php">here</a>. This movie illustrates what it's all about:<br /><br /><center><object width="425" height="355"><param name="movie" value="http://www.youtube.com/v/xKBu2B8xZYE&hl=en"></param><param name="wmode" value="transparent"></param><embed src="http://www.youtube.com/v/xKBu2B8xZYE&hl=en" type="application/x-shockwave-flash" wmode="transparent" width="425" height="355"></embed></object></center><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-7242832094141048164?l=apcalc07.blogspot.com' alt='' /></div>dkuropatwahttp://www.blogger.com/profile/08462283847470560887noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-28948094007830826732008-03-17T00:54:00.004-05:002008-03-17T00:59:42.351-05:00Well, a certainly enjoyable pi day it was. Lots of treats and Friday!. Doesn't get much better than that.<br /><br />With all the celebrating, we didn't do very much.<br /><br />We learned, the slope fields were the graphs of the parent function. Just moved all over the place.<br /><br />Due to my internet moving unbelievably slow, I will add more "pi" pictures when it starts to pick up. (Litterally, 3 minutes to load a single page)<br /><br />Next scribe.<br /><br />Mr.SiwWy-chan<div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-2894809400783082673?l=apcalc07.blogspot.com' alt='' /></div>Vanhttp://www.blogger.com/profile/07679810325423377354noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-66290816608584237632008-03-14T14:42:00.003-05:002008-03-14T14:46:33.634-05:00Here they are ...<br /><br /><center><br /><div style="width:425px;text-align:left" id="__ss_306738"><object style="margin:0px" width="425" height="355"><param name="movie" value="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-march-14-2008-1205523714472743-2"/><param name="allowFullScreen" value="true"/><param name="allowScriptAccess" value="always"/><embed src="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-march-14-2008-1205523714472743-2" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="355"></embed></object><div style="font-size:11px;font-family:tahoma,arial;height:26px;padding-top:2px;"><a href="http://www.slideshare.net/?src=embed"><img src="http://static.slideshare.net/swf/logo_embd.png" style="border:0px none;margin-bottom:-5px" alt="SlideShare"/></a> | <a href="http://www.slideshare.net/dkuropatwa/ap-calculus-slides-march-14-2008?src=embed" title="View 'AP Calculus Slides March 14, 2008' on SlideShare">View</a> | <a href="http://www.slideshare.net/upload?src=embed">Upload your own</a></div></div><br /></center><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-6629081660858423763?l=apcalc07.blogspot.com' alt='' /></div>dkuropatwahttp://www.blogger.com/profile/08462283847470560887noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-48449174080465971872008-03-14T08:55:00.001-05:002008-03-14T08:57:31.144-05:00<center>Happy <font size="+2">&pi;</font> Day Everyone!!<br /><br /><a href="http://flickr.com/search/show/?q=pi+pie">Have some <font size="+2">&pi;</font> Pie!</a></center><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-4844917408046597187?l=apcalc07.blogspot.com' alt='' /></div>dkuropatwahttp://www.blogger.com/profile/08462283847470560887noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-31123834896740371592008-03-14T00:36:00.013-05:002008-11-18T18:01:14.001-06:00WOW! Very, very sorry about the tardiness of this scribe... I worked both last and this evening and today was very busy.<br /><br />So Wednesday's class started with a very unusual period in which we were given our tests back and allowed 10 minutes to continue them since we didn't get a chance last Thursday.<br /><br />Then we briefly discussed the Developing Expert Voices projects as Mr. K. handed back the proposals.<br /><br />Following that was a little conversation about "π day" (which is today actually). I'm bringing either a Strawberry-Mango or Blueberry-Peach pie =D.<br /><br /><span style="color: rgb(0, 0, 153);">Finally we started the lesson for the day.</span> <span style="color: rgb(0, 0, 153);">The first thing we did was a quick review of Differential Problems.<br /><span style="color: rgb(255, 0, 0);">It dealt with a set amount of oil in a reserve, states that a function A(t) as the rate of consumption, and provides the derivative of the rate [ A'(t) ]. How long until the reserve is completely emptied.<br /><br />Basically, to solve it, find the antiderivative, A(t)+C, of the the given function and then plug in t=0 to find the constant (C).<br />From there it is as simple as finding the positive root of that function and that will be the answer.<br /><br /><span style="color: rgb(0, 0, 153);">Next we moved onto the new content.<br /><span style="color: rgb(0, 102, 0);">SLOPE FIELDS<br /><br />these are kind of tough to explain... I'll let some pictures speak some of my words.<br /></span></span></span></span><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://3.bp.blogspot.com/_n1XY9NEWU6I/R9oWa9r7FpI/AAAAAAAAAG4/hZiMhOwKJO4/s1600-h/Picture+3.png"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://3.bp.blogspot.com/_n1XY9NEWU6I/R9oWa9r7FpI/AAAAAAAAAG4/hZiMhOwKJO4/s400/Picture+3.png" alt="" id="BLOGGER_PHOTO_ID_5177475374107006610" border="0" /></a><span style="color: rgb(0, 102, 0);">This is the FIELD in which we plot the SLOPES. Easy enough right?<br />So we first tried it with the expression <span style="color: rgb(153, 51, 0);">( ∂y/∂x = y )<span style="color: rgb(0, 102, 0);">. This means that the derivative of any point in the function is the value of the function at that point (y-coordinate).<br />•Let's say we want to plot the point <span style="color: rgb(51, 51, 255);"><span style="color: rgb(51, 102, 255);">(1,1)</span><span style="color: rgb(0, 102, 0);"><span style="color: rgb(51, 102, 255);">.</span> The derivative (SLOPE) of the function at that point is <span style="color: rgb(51, 51, 255);"><span style="color: rgb(51, 102, 255);">1</span><span style="color: rgb(0, 102, 0);"><span style="color: rgb(51, 102, 255);">,</span> so we draw a small line with <span style="color: rgb(51, 51, 255);"><span style="color: rgb(51, 102, 255);">slope 1 at that point.</span><br /></span></span></span></span></span></span></span></span><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://4.bp.blogspot.com/_n1XY9NEWU6I/R9oYBNr7FqI/AAAAAAAAAHA/uDQdbyb5lvY/s1600-h/Picture+4.png"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://4.bp.blogspot.com/_n1XY9NEWU6I/R9oYBNr7FqI/AAAAAAAAAHA/uDQdbyb5lvY/s400/Picture+4.png" alt="" id="BLOGGER_PHOTO_ID_5177477130748630690" border="0" /></a><span style="color: rgb(0, 102, 0);">•Next, take the point <span style="color: rgb(51, 102, 255);">(3,2).<span style="color: rgb(0, 102, 0);"> The derivative at that point is <span style="color: rgb(51, 102, 255);">2<span style="color: rgb(0, 102, 0);">, so there's another small line, but with a <span style="color: rgb(51, 102, 255);">slope 2 at the point (3,2).<br /></span></span></span></span></span></span><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://3.bp.blogspot.com/_n1XY9NEWU6I/R9oYx9r7FrI/AAAAAAAAAHI/HFq1KFI1T7c/s1600-h/Picture+5.png"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://3.bp.blogspot.com/_n1XY9NEWU6I/R9oYx9r7FrI/AAAAAAAAAHI/HFq1KFI1T7c/s400/Picture+5.png" alt="" id="BLOGGER_PHOTO_ID_5177477968267253426" border="0" /></a><span style="color: rgb(0, 102, 0);">Now, you understand how they are constructed, this is how the SLOPE FIELD of the function(s) that have </span><span style="color: rgb(0, 102, 0);"><span style="color: rgb(153, 51, 0);">( ∂y/∂x = y ).<br /></span></span><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://4.bp.blogspot.com/_n1XY9NEWU6I/R9ocaNr7FuI/AAAAAAAAAHg/HpXR5Tp2dSc/s1600-h/Picture+8.png"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://4.bp.blogspot.com/_n1XY9NEWU6I/R9ocaNr7FuI/AAAAAAAAAHg/HpXR5Tp2dSc/s400/Picture+8.png" alt="" id="BLOGGER_PHOTO_ID_5177481958291871458" border="0" /></a><span style="color: rgb(0, 102, 0);">From here, you can solve for the point <span style="color: rgb(153, 51, 0);">(1,1).<br /></span></span><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://2.bp.blogspot.com/_n1XY9NEWU6I/R9ocltr7FvI/AAAAAAAAAHo/_8FV7SGSM-c/s1600-h/Picture+9.png"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://2.bp.blogspot.com/_n1XY9NEWU6I/R9ocltr7FvI/AAAAAAAAAHo/_8FV7SGSM-c/s400/Picture+9.png" alt="" id="BLOGGER_PHOTO_ID_5177482155860367090" border="0" /></a><span style="color: rgb(153, 51, 0);">To do this, draw a curve that follows the slope of the line in either direction until the next whole number is reached. At that point, change the curve's slope to that of the new poitn and continue until the next whole number and so on... WHAT FUNCTION DOES THE CURVE LOOK LIKE???<br /></span><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://2.bp.blogspot.com/_n1XY9NEWU6I/R9odEtr7FwI/AAAAAAAAAHw/e28xCIQyk_c/s1600-h/Picture+10.png"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://2.bp.blogspot.com/_n1XY9NEWU6I/R9odEtr7FwI/AAAAAAAAAHw/e28xCIQyk_c/s400/Picture+10.png" alt="" id="BLOGGER_PHOTO_ID_5177482688436311810" border="0" /></a><span style="color: rgb(0, 102, 0);">Correct! It is easy to see that the <span style="color: rgb(153, 51, 0);">function that has a derivative equal to the value of the function</span> and <span style="color: rgb(153, 51, 0);">passes through the point (1,1)</span> is <span style="color: rgb(153, 51, 0);">y=e^x.<br /><span style="color: rgb(0, 102, 0);">However, you can also see that there are many other functions with a similar shape that could have been drawn, depending on the starting point.<br />This is because the SLOPE FIELD shows the entire family of functions [ ∫ƒ(x)∂x + C ].<br />So, by solving for a point, you can find the exact one function from that family.<br /><br /><span style="color: rgb(0, 0, 153);"><span style="color: rgb(0, 102, 0);">We also did SLOPE FIELDS for <span style="color: rgb(255, 0, 0);">∂y/∂x = 2x <span style="color: rgb(0, 102, 0);">and</span> ∂y/∂x = -x/y<span style="color: rgb(0, 102, 0);">... can you guess what functions they are???<br /></span></span></span></span></span></span></span><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://4.bp.blogspot.com/_n1XY9NEWU6I/R9ogLNr7FxI/AAAAAAAAAH4/Vyfrw17oSE0/s1600-h/Picture+11.png"><img style="margin: 0pt 10px 10px 0pt; float: left; cursor: pointer;" src="http://4.bp.blogspot.com/_n1XY9NEWU6I/R9ogLNr7FxI/AAAAAAAAAH4/Vyfrw17oSE0/s400/Picture+11.png" alt="" id="BLOGGER_PHOTO_ID_5177486098640344850" border="0" /></a><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://4.bp.blogspot.com/_n1XY9NEWU6I/R9ogVNr7FyI/AAAAAAAAAIA/188zuDQbDPI/s1600-h/Picture+12.png"><img style="margin: 0pt 10px 10px 0pt; float: left; cursor: pointer;" src="http://4.bp.blogspot.com/_n1XY9NEWU6I/R9ogVNr7FyI/AAAAAAAAAIA/188zuDQbDPI/s400/Picture+12.png" alt="" id="BLOGGER_PHOTO_ID_5177486270439036706" border="0" /></a><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><span style="color: rgb(0, 0, 153);">This was the end of the lesson as we realized that any function can be determined by plotting it's derivative in a SLOPE FIELD and we can find which function out of a family given a point on the specific function.<br /></span><br /><span style="color: rgb(0, 102, 0);"><span style="color: rgb(153, 51, 0);"><span style="color: rgb(0, 102, 0);"><span style="color: rgb(0, 0, 153);"><span style="color: rgb(0, 0, 0);"><br />This concludes my scribe post, once again, sorry it was so late.<br /><br />The scribe for PI (π) Day will be...<br /><br /><span style="color: rgb(102, 0, 204);font-size:180%;" >VAN!!!!!!!!!!</span><br /></span></span></span></span></span><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-3112383489674037159?l=apcalc07.blogspot.com' alt='' /></div>«Craig»http://www.blogger.com/profile/08860268378924842699noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-53957421280184078522008-03-13T08:10:00.000-05:002008-11-18T18:01:14.098-06:00<div style="text-align: center;"><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://2.bp.blogspot.com/_HiqPzpd3O9Y/R9knopn8vDI/AAAAAAAAAG8/F5_2_zP7PpA/s1600-h/pi+day.gif"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://2.bp.blogspot.com/_HiqPzpd3O9Y/R9knopn8vDI/AAAAAAAAAG8/F5_2_zP7PpA/s200/pi+day.gif" alt="" id="BLOGGER_PHOTO_ID_5177212825960758322" border="0" /></a><a href="http://www.123greetings.com/view/MJ30313060109599">Click here to view your e card! </a><br /></div><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-5395742128018407852?l=apcalc07.blogspot.com' alt='' /></div>Lani Ritter Hallhttp://www.blogger.com/profile/16352862711544966770noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-59850663303083059302008-03-12T16:31:00.001-05:002008-03-12T16:40:11.591-05:00Here they are ...<br /><br /><center><br /><div style="width:425px;text-align:left" id="__ss_304261"><object style="margin:0px" width="425" height="355"><param name="movie" value="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-march-12-2008-1205357848601217-2"/><param name="allowFullScreen" value="true"/><param name="allowScriptAccess" value="always"/><embed src="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-march-12-2008-1205357848601217-2" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="355"></embed></object><div style="font-size:11px;font-family:tahoma,arial;height:26px;padding-top:2px;"><a href="http://www.slideshare.net/?src=embed"><img src="http://static.slideshare.net/swf/logo_embd.png" style="border:0px none;margin-bottom:-5px" alt="SlideShare"/></a> | <a href="http://www.slideshare.net/dkuropatwa/ap-calculus-slides-march-12-2008?src=embed" title="View 'AP Calculus Slides March 12, 2008' on SlideShare">View</a> | <a href="http://www.slideshare.net/upload?src=embed">Upload your own</a></div></div><br /></center><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-5985066330308305930?l=apcalc07.blogspot.com' alt='' /></div>dkuropatwahttp://www.blogger.com/profile/08462283847470560887noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-26458443991438304992008-03-11T23:56:00.014-05:002008-11-18T18:01:15.966-06:00Grrrr.. Can't do slideshow<br /><div><a href="http://2.bp.blogspot.com/_fvK_Af4KmnA/R9djnJGUSNI/AAAAAAAAAFc/or4HfRsdYjc/s1600-h/scribe+5_1.jpeg"><img id="BLOGGER_PHOTO_ID_5176715820794136786" style="CURSOR: hand" height="341" alt="" src="http://2.bp.blogspot.com/_fvK_Af4KmnA/R9djnJGUSNI/AAAAAAAAAFc/or4HfRsdYjc/s400/scribe+5_1.jpeg" width="440" border="0" /></a><br /><br /><a href="http://4.bp.blogspot.com/_fvK_Af4KmnA/R9djzpGUSPI/AAAAAAAAAFs/t9gUPtFJn0g/s1600-h/scribe+5_2.jpeg"><img id="BLOGGER_PHOTO_ID_5176716035542501618" style="CURSOR: hand" height="323" alt="" src="http://4.bp.blogspot.com/_fvK_Af4KmnA/R9djzpGUSPI/AAAAAAAAAFs/t9gUPtFJn0g/s400/scribe+5_2.jpeg" width="435" border="0" /></a><br /><a href="http://1.bp.blogspot.com/_fvK_Af4KmnA/R9dj45GUSQI/AAAAAAAAAF0/Yljc87amvcQ/s1600-h/scribe+5_3.jpeg"><img id="BLOGGER_PHOTO_ID_5176716125736814850" style="CURSOR: hand" height="333" alt="" src="http://1.bp.blogspot.com/_fvK_Af4KmnA/R9dj45GUSQI/AAAAAAAAAF0/Yljc87amvcQ/s400/scribe+5_3.jpeg" 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border="0" /></a><br /><br /><a href="http://1.bp.blogspot.com/_fvK_Af4KmnA/R9dkn5GUSWI/AAAAAAAAAGk/cevMopytnYQ/s1600-h/scribe+5_10.jpeg"><img id="BLOGGER_PHOTO_ID_5176716933190666594" style="CURSOR: hand" height="339" alt="" src="http://1.bp.blogspot.com/_fvK_Af4KmnA/R9dkn5GUSWI/AAAAAAAAAGk/cevMopytnYQ/s400/scribe+5_10.jpeg" width="435" border="0" /></a><br /><br /><a href="http://1.bp.blogspot.com/_fvK_Af4KmnA/R9dkw5GUSXI/AAAAAAAAAGs/jtteTMiTy9g/s1600-h/scribe+5_11.jpeg"><img id="BLOGGER_PHOTO_ID_5176717087809489266" style="CURSOR: hand" height="338" alt="" src="http://1.bp.blogspot.com/_fvK_Af4KmnA/R9dkw5GUSXI/AAAAAAAAAGs/jtteTMiTy9g/s400/scribe+5_11.jpeg" width="437" border="0" /></a> <a href="http://4.bp.blogspot.com/_fvK_Af4KmnA/R9dk2pGUSYI/AAAAAAAAAG0/3S_olBPSiCQ/s1600-h/scribe+5_12.jpeg"><img id="BLOGGER_PHOTO_ID_5176717186593737090" style="CURSOR: hand" height="330" alt="" src="http://4.bp.blogspot.com/_fvK_Af4KmnA/R9dk2pGUSYI/AAAAAAAAAG0/3S_olBPSiCQ/s400/scribe+5_12.jpeg" width="435" border="0" /></a><br /><br /><div></div></div><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-2645844399143830499?l=apcalc07.blogspot.com' alt='' /></div>etimzhttp://www.blogger.com/profile/09626175119223517919noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-3796778512284106882008-03-10T12:33:00.000-05:002008-03-10T12:34:03.924-05:00Here they are ...<br /><br /><center><br /><div style="width:425px;text-align:left" id="__ss_300537"><object style="margin:0px" width="425" height="355"><param name="movie" value="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-march-10-2008-1205170223168194-3"/><param name="allowFullScreen" value="true"/><param name="allowScriptAccess" value="always"/><embed src="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-march-10-2008-1205170223168194-3" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="355"></embed></object><div style="font-size:11px;font-family:tahoma,arial;height:26px;padding-top:2px;"><a href="http://www.slideshare.net/?src=embed"><img src="http://static.slideshare.net/swf/logo_embd.png" style="border:0px none;margin-bottom:-5px" alt="SlideShare"/></a> | <a href="http://www.slideshare.net/dkuropatwa/ap-calculus-slides-march-10-2008?src=embed" title="View 'AP Calculus Slides March 10, 2008' on SlideShare">View</a> | <a href="http://www.slideshare.net/upload?src=embed">Upload your own</a></div></div><br /></center><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-379677851228410688?l=apcalc07.blogspot.com' alt='' /></div>dkuropatwahttp://www.blogger.com/profile/08462283847470560887noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-53353171334895138482008-03-05T23:28:00.002-06:002008-03-05T23:32:03.105-06:00this unit was just the same as the rest. im still trying to catch back up. thanks to GreyM I think i will be able to do just that. Im putting more effort into cathing backup now so i hope i do good.<div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-5335317133489513848?l=apcalc07.blogspot.com' alt='' /></div>etimzhttp://www.blogger.com/profile/09626175119223517919noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-15092964208404685182008-03-05T21:10:00.003-06:002008-03-05T21:13:34.092-06:00I have to say I feel more confident going into this test then I did going into the other test earlier today. I get how these problems work most of the time, just using the unit logic and the idea of taking one of the parameters and chopping it up into little bits. I grok the average value of a function idea. I waver a bit on the solids but really they are not terribly hard. I think I will do fairly well tomorrow. G'night.<div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-1509296420840468518?l=apcalc07.blogspot.com' alt='' /></div>Grey-Mhttp://www.blogger.com/profile/10237241798100727258noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-23269963391119427912008-03-05T19:59:00.004-06:002008-03-05T20:27:32.413-06:00Well, I went on to check out the calculus blog to BOB and surprisingly enough, there aren't many <span class="blsp-spelling-error" id="SPELLING_ERROR_0">BOB's</span> up for this test yet, especially considering what time is it already. I hope that people didn't forget =/.<br /><br />Anyways, this unit, applications of integrals, was definitely not a cake walk. I think there's quite a bit of material left for me to review, mostly due to the fact that I seem to be lost in a haze since we only have class every second day now and I'm usually too busy with other things to finish all of the homework. But all in all, a lot of the applications of integrals we have covered during this unit are quite similar. Such as the similarity between evaluating the volumes of various solids revolved about the x or y-axis or a particular line like y = -1 and using density functions and such to find a mass or population (the last section of this unit).<br /><br />Since I mentioned the last section of this unit anyway, I think that's probably the only place where I might run into some trouble on the test tomorrow. The main reason for this is because the process of solving such problems aren't straight forward in the least, though once you repeat the logic undertaken each time you attempt a problem of this sort (where unit analysis <span class="blsp-spelling-corrected" id="SPELLING_ERROR_1">incredibly</span> simplifies the matter) the approach for each question becomes more apparent each time.<br /><br />Besides the aforementioned troublesome questions, I don't think that there is anything else I might find excruciatingly difficult on the test. The only things we really have to know for tomorrow are:<br /><br /><span style="color: rgb(0, 0, 153);">- The difference between</span><span style="font-weight: bold; color: rgb(0, 0, 153);"> displacement</span><span style="color: rgb(0, 0, 153);"> and </span><span style="font-weight: bold; color: rgb(0, 0, 153);">distance</span><span style="color: rgb(0, 0, 153);"> when integrating a velocity/speed function.</span><br /><span style="color: rgb(0, 0, 153);">- Determining definite integrals to represent the <span style="font-weight: bold;">volume of a solid</span>, and evaluating Riemann sums and these self-generated definite integrals.</span><br /><span style="color: rgb(0, 0, 153);">- How to use integrals to determine the <span style="font-weight: bold;">average value of a function</span>. (Just think of finding the average normally, ex. adding up all your marks (an integral) over the total amount of marks (the size of the interval)).</span><br /><span style="color: rgb(0, 0, 153);">- How to use integrals for differing scenarios, such as <span style="font-weight: bold;">oil density</span> or <span style="font-weight: bold;">population density</span> of cities. I suggest reviewing chapter 8.5 in the textbook for elaboration and some practice on this topic.</span><br /><br />Well that's it, I haven't completed such a long BOB in quite a while. Well, I hope everyone does great on the test tomorrow, don't forget to go over all your notes and practice some problems tonight. <span style="color: rgb(0, 153, 0);">Study hard everyone!</span> I know I will.<br /><br />I wish everyone <span style="color: rgb(204, 0, 0);">good luck</span> on tomorrow's test and a good night!<div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-2326996339111942791?l=apcalc07.blogspot.com' alt='' /></div>MrSiwWyhttp://www.blogger.com/profile/18072494079396275311noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-356556247210780052008-03-05T19:38:00.014-06:002008-11-18T18:01:17.505-06:00what do you know I made it back for class :) <div><div><div><div><div><div><div><div><div><br /><div>Sorry but i forgot about scribe until tonight, but all Im going over is the Pre-test. Nothing else really happened...I believe.</div><br /><div>Personally i needed the pre-test because they was somethings I didnt even realize I didnt know and now I have the chance to look over it again.</div><br /><div>Now on to the pre-test.</div><br /><div>It was made up of 6 questions. Ill be going over each of the questions. </div><br /><div>Heres the questions. All the work is done on the pages itself.</div><br /><div>Most of these are for me to rework them so sorry if something seems really obvious.</div><br /><div><a href="http://3.bp.blogspot.com/_fvK_Af4KmnA/R8988MH9JqI/AAAAAAAAAEM/ZPNJLQHLVUU/s1600-h/ethan+math_1.jpeg"><img id="BLOGGER_PHOTO_ID_5174491870360577698" style="WIDTH: 408px; CURSOR: hand; HEIGHT: 295px" height="325" alt="" src="http://3.bp.blogspot.com/_fvK_Af4KmnA/R8988MH9JqI/AAAAAAAAAEM/ZPNJLQHLVUU/s400/ethan+math_1.jpeg" width="519" border="0" /></a><a href="http://3.bp.blogspot.com/_fvK_Af4KmnA/R899HMH9JrI/AAAAAAAAAEU/KdE5ItIO1jA/s1600-h/ethan+math_2.jpeg"><img id="BLOGGER_PHOTO_ID_5174492059339138738" style="WIDTH: 403px; CURSOR: hand; HEIGHT: 362px" height="310" alt="" src="http://3.bp.blogspot.com/_fvK_Af4KmnA/R899HMH9JrI/AAAAAAAAAEU/KdE5ItIO1jA/s400/ethan+math_2.jpeg" width="444" border="0" /></a><a href="http://4.bp.blogspot.com/_fvK_Af4KmnA/R899ccH9JsI/AAAAAAAAAEc/yqz1GQI8irk/s1600-h/ethan+math_3.jpeg"><img id="BLOGGER_PHOTO_ID_5174492424411358914" style="WIDTH: 411px; CURSOR: hand; HEIGHT: 249px" height="310" alt="" src="http://4.bp.blogspot.com/_fvK_Af4KmnA/R899ccH9JsI/AAAAAAAAAEc/yqz1GQI8irk/s400/ethan+math_3.jpeg" width="637" border="0" /></a><a href="http://4.bp.blogspot.com/_fvK_Af4KmnA/R89-McH9JtI/AAAAAAAAAEk/Spc2Bopg__U/s1600-h/ethan+math_4.jpeg"><img id="BLOGGER_PHOTO_ID_5174493249045079762" style="WIDTH: 445px; CURSOR: hand; HEIGHT: 207px" height="322" alt="" src="http://4.bp.blogspot.com/_fvK_Af4KmnA/R89-McH9JtI/AAAAAAAAAEk/Spc2Bopg__U/s400/ethan+math_4.jpeg" width="682" border="0" /></a><a href="http://2.bp.blogspot.com/_fvK_Af4KmnA/R89-U8H9JuI/AAAAAAAAAEs/t3jj-ZJwLEM/s1600-h/ethan+math_5.jpeg"><img id="BLOGGER_PHOTO_ID_5174493395073967842" style="WIDTH: 410px; CURSOR: hand; HEIGHT: 320px" height="359" alt="" src="http://2.bp.blogspot.com/_fvK_Af4KmnA/R89-U8H9JuI/AAAAAAAAAEs/t3jj-ZJwLEM/s400/ethan+math_5.jpeg" width="683" border="0" /></a><a href="http://3.bp.blogspot.com/_fvK_Af4KmnA/R89-cMH9JvI/AAAAAAAAAE0/_XRSJJH2AhQ/s1600-h/ethan+math_6.jpeg"><img id="BLOGGER_PHOTO_ID_5174493519628019442" style="WIDTH: 408px; CURSOR: hand; HEIGHT: 329px" height="364" alt="" src="http://3.bp.blogspot.com/_fvK_Af4KmnA/R89-cMH9JvI/AAAAAAAAAE0/_XRSJJH2AhQ/s400/ethan+math_6.jpeg" width="680" border="0" /></a><a href="http://1.bp.blogspot.com/_fvK_Af4KmnA/R89-msH9JwI/AAAAAAAAAE8/OhHBMXqv2_g/s1600-h/ethan+math_7.jpeg"><img id="BLOGGER_PHOTO_ID_5174493700016645890" style="WIDTH: 400px; CURSOR: hand; HEIGHT: 328px" height="387" alt="" src="http://1.bp.blogspot.com/_fvK_Af4KmnA/R89-msH9JwI/AAAAAAAAAE8/OhHBMXqv2_g/s400/ethan+math_7.jpeg" width="670" border="0" /></a><a href="http://1.bp.blogspot.com/_fvK_Af4KmnA/R89-5sH9JxI/AAAAAAAAAFE/kPqN0LtlZmA/s1600-h/ethan+math_8.jpeg"><img id="BLOGGER_PHOTO_ID_5174494026434160402" style="WIDTH: 405px; CURSOR: hand; HEIGHT: 376px" height="396" alt="" src="http://1.bp.blogspot.com/_fvK_Af4KmnA/R89-5sH9JxI/AAAAAAAAAFE/kPqN0LtlZmA/s400/ethan+math_8.jpeg" width="675" border="0" /></a><a href="http://4.bp.blogspot.com/_fvK_Af4KmnA/R89_JcH9JyI/AAAAAAAAAFM/EzLeBxB-fQ8/s1600-h/ethan+math_9.jpeg"><img id="BLOGGER_PHOTO_ID_5174494297017100066" style="WIDTH: 416px; CURSOR: hand; HEIGHT: 417px" height="422" alt="" src="http://4.bp.blogspot.com/_fvK_Af4KmnA/R89_JcH9JyI/AAAAAAAAAFM/EzLeBxB-fQ8/s400/ethan+math_9.jpeg" width="686" border="0" /></a><a href="http://2.bp.blogspot.com/_fvK_Af4KmnA/R89_R8H9JzI/AAAAAAAAAFU/Azii5P2H4hk/s1600-h/ethan+math_10.jpeg"><img id="BLOGGER_PHOTO_ID_5174494443045988146" style="WIDTH: 410px; CURSOR: hand; HEIGHT: 281px" height="318" alt="" src="http://2.bp.blogspot.com/_fvK_Af4KmnA/R89_R8H9JzI/AAAAAAAAAFU/Azii5P2H4hk/s400/ethan+math_10.jpeg" width="682" border="0" /></a></div></div></div></div></div></div></div></div></div></div><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-35655624721078005?l=apcalc07.blogspot.com' alt='' /></div>etimzhttp://www.blogger.com/profile/09626175119223517919noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-40597923964336103312008-03-04T20:02:00.003-06:002008-03-04T22:45:47.420-06:00Ah! Once again there's an upcoming test! I am confident in saying that I am prepared, but not quite.<br /><br />There are still a couple of things that I'm worried about. However I can't seem to place it. It's just that feeling in the gut. Hopefully some more practice will clear out any doubts.<br /><br />Confidence may be my only hope! Good luck to everyone else on the test :)<div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-4059792396433610331?l=apcalc07.blogspot.com' alt='' /></div>Tim_MATH_yhttp://www.blogger.com/profile/09929458936647070771noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-27321379113546874522008-03-04T13:09:00.002-06:002008-03-04T13:19:16.863-06:00Here they are ...<br /><br /><center><br /><div style="width:425px;text-align:left" id="__ss_292431"><object style="margin:0px" width="425" height="355"><param name="movie" value="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-march-4-2008-1204658056891921-4"/><param name="allowFullScreen" value="true"/><param name="allowScriptAccess" value="always"/><embed src="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-march-4-2008-1204658056891921-4" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="355"></embed></object><div style="font-size:11px;font-family:tahoma,arial;height:26px;padding-top:2px;"><a href="http://www.slideshare.net/?src=embed"><img src="http://static.slideshare.net/swf/logo_embd.png" style="border:0px none;margin-bottom:-5px" alt="SlideShare"/></a> | <a href="http://www.slideshare.net/dkuropatwa/ap-calculus-slides-march-4-2008?src=embed" title="View 'AP Calculus Slides March 4, 2008' on SlideShare">View</a> | <a href="http://www.slideshare.net/upload?src=embed">Upload your own</a></div></div><br /></center><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-2732137911354687452?l=apcalc07.blogspot.com' alt='' /></div>dkuropatwahttp://www.blogger.com/profile/08462283847470560887noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-40342643385506344312008-03-03T23:47:00.003-06:002008-03-04T00:04:29.420-06:00Well, just here to do a quick little B.O.B. The unit of Applications of Integrals has been a pretty interesting one because of how close it gets to real time problems. I have enjoyed solving these problems and even though at times they seemed complicated, once you leave all the words behind and put it into mathematical equations it became a lot easier.<br /><br />As well, it gave a bit more practice for solving integrals a bit quicker and allowed us to use a lot of previous techniques/skills to do them.<br /><br /><br />I would have to say that the hardest part of the unit was the last bit with the radial density because of how much translation from real circumstances to mathematical equations and diagrams. However, once again as soon as it was translated, it became quite easy to solve.<br /><br />So after the pre-test, we will see what kinds of questions to expect on the test, and what areas we need to brush up on a bit.<br /><br /><br />See you all tomorrow and good luck on the pre-test and test!<div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-4034264338550634431?l=apcalc07.blogspot.com' alt='' /></div>«Craig»http://www.blogger.com/profile/08860268378924842699noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-45189004416047410392008-02-29T13:01:00.003-06:002008-03-02T12:40:52.007-06:00Here they are ...<br /><br /><center><br /><div style="width:425px;text-align:left" id="__ss_287282"><object style="margin:0px" width="425" height="355"><param name="movie" value="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-march-1-2008-1204312142635882-2"/><param name="allowFullScreen" value="true"/><param name="allowScriptAccess" value="always"/><embed src="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-march-1-2008-1204312142635882-2" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="355"></embed></object><div style="font-size:11px;font-family:tahoma,arial;height:26px;padding-top:2px;"><a href="http://www.slideshare.net/?src=embed"><img src="http://static.slideshare.net/swf/logo_embd.png" style="border:0px none;margin-bottom:-5px" alt="SlideShare"/></a> | <a href="http://www.slideshare.net/dkuropatwa/ap-calculus-slides-march-1-2008?src=embed" title="View 'AP Calculus Slides March 1, 2008' on SlideShare">View</a> | <a href="http://www.slideshare.net/upload?src=embed">Upload your own</a></div></div><br /></center><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-4518900441604741039?l=apcalc07.blogspot.com' alt='' /></div>dkuropatwahttp://www.blogger.com/profile/08462283847470560887noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-49561591716399034392008-02-27T18:28:00.021-06:002008-11-18T18:01:18.583-06:00Hello everyone, MrSiwWy here today for the scribe substituting for Ethan, since he was chosen yesterday but he wasn't in class today. So I guess after my scribe today, if Ethan attends class he'll be the scribe for next class. Though, I think I'll designate another replacement scribe for next class just in case Ethan ends up missing class once again.<br /><br />Now, I know we didn't really cover a lot of material today but were presented with a rather new type of problem and also a quick quiz at the beginning of class. For the problem, we have in fact encountered somewhat similar problems, such as the oil slick problem, but the main difference is that we are now dealing with a numeric representation of the density function, rather than a symbolic one. Remember that on the exam, we could be given a varied amount of different representations, ideally numeric, symbolic and graphical ones. So try to practice not only questions that convey a specified density function in the question, but also practice those that have a numerical representation (say a table of values such as today's question) or an accompanying graph. I believe that this idea was the basis of today's problem and overall lesson.<br /><br />Before I continue onward with the detailed progression of today's problem, I'll first digress and explain the quiz that preceded the challenging question. Though we went over the answers to the quiz afterwards, we did not detail each questions solutions. I felt obligated (not entirely sure why =p) to do so now for each question.<br /><br /><div align="center"><span style="color:#cc0000;">Question #1:</span></div><div align="left"><span style="color:#cc0000;"><img id="BLOGGER_PHOTO_ID_5171861421091026834" style="DISPLAY: block; MARGIN: 0px auto 10px; CURSOR: hand; TEXT-ALIGN: center" alt="" src="http://1.bp.blogspot.com/_dvbjJeJb4LE/R8Ykj4-td5I/AAAAAAAAAOI/cZ9mpNLRHD8/s400/quizquestion1.PNG" border="0" /></div></span><div align="left"><span style="color:#cc0000;">First off, it's obvious that all there really is to this question is to analyze the graph. Still, make sure that you read the question carefully, because it's asking for a LOCAL minimum, which brought up some silly bewilderment today when some of us haphazardly chose their answer without making sure they read the question dutifully enough (this group includes me =/). Now, if you look closely at the graph, we know that since the function in the graph is the derivative of <em>f</em>, in order to determine where <em>f</em> has a local maximum or a local minimum in this case is to analyze the roots of <em>f'</em>.<br /><br />Knowing the behaviour of the derivative of a function and it's relation to the parent function helps incredibly in this analysis. You must know that not all roots of <em>f'</em> are maximums or minimums, such at x = 2. This is because since there is no sign change, what happens at x = 2 is that there exists a plateau or flat portion in the parent function, thus the horizontal slope at x = 2. Also, looking at x = 4 as another candidate for a local minimum or maximum yields a sign change from positive to negative. This change indicates that x = 4 yields a maximum on the parent function, because since <em>f'</em> is positive to the left of x = 4 and negative to the right of x = 4 the parent function is increasing until it hits x = 4 then turns downwards, therefore it is a local maximum. Now there's only one thing left to analyze since the zero's are now out of the question.<br /><br />Don't forget about the endpoints of the function. At the endpoints of <em>f</em>', since <em>f</em> is increasing on (0, 4) and then <em>f</em> starts to decrease once again on (4, 5), they turn out to be LOCAL minima on the parent function. This is because since the function start out low at x = 0 and then increases from that point on until x = 4, it is a local minimum. After the function reaches x = 4, the function begins to decrease and therefore reaches another low point at the end of the function. Don't confuse the question with finding the global minimum, which is actually the mistake that I made when I chose my answer. You must recognize that local minima are only considered minima on a small interval central to that point, not over the entire function. Thus both endpoints qualify as LOCAL minima.</span></div><br /><div align="left"></div><div align="left"></div><div align="left"><span style="color:#000000;">Answer: There are local minima at x = 0 and x = 5.</span></div><br /><div align="center"><span style="color:#cc0000;"></span></div><div align="left"><span style="color:#cc0000;"></span></div><div align="center"><span style="color:#ff6600;">Question #2:</span></div><div align="center"><span style="color:#cc0000;"><img id="BLOGGER_PHOTO_ID_5171861425385994146" style="DISPLAY: block; MARGIN: 0px auto 10px; CURSOR: hand; TEXT-ALIGN: center" alt="" src="http://2.bp.blogspot.com/_dvbjJeJb4LE/R8YkkI-td6I/AAAAAAAAAOQ/KK5wuKyVb6U/s400/quizquestion2.PNG" border="0" /><span style="color:#ff6600;"></span> </div></span><div align="left"><span style="color:#ff6600;">For question number 2, here comes that same graph of <em>f</em> again. But this time we are asked to determine the points of inflection of the parent function <em>f</em>. Though the previous question relied on the endpoints of the function, we cannot do that this time since it's impossible to determine the derivative of the endpoints of a function since there's no one specific derivative at that point. If you imagine trying to draw a tangent line at x = 0 or x = 5 on <em>f</em>', there's no way you can draw just one definite tangent line, thus the illegal procedure of differentiating this function. You might be asking as to why I'm mentioning differentiating this function when it already is the derivative. Recall that we're looking for the inflection point of <em>f</em>, which is a change in the concavity of the function, and that <em>f''</em> represents the concavity of the parent function, thus the reason for differentiating <em>f'</em>.<br /><br />We basically are now looking for the zero's of <em>f'', </em>which occur at the local maxima and minima of <em>f'</em>. We can quickly see that there is a maximum present at x = 3 and a minimum at x =2, therefore these are also the points of inflection of <em>f</em>. Don't forget, however, to analyze the shape of these points to determine whether they are in fact points of inflection or not.<br /><br /><span style="color:#000000;">Answer: There are points of inflection at x = 2 and x = 3.</span></span></div><div align="left"><br /></div><span style="color:#cc0000;"><p align="center"></span></p><div align="center"><span style="color:#cc0000;"><span style="color:#000099;">Question #3: </span></span><br /></div><img id="BLOGGER_PHOTO_ID_5171870620910975042" style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 465px; CURSOR: hand; HEIGHT: 118px; TEXT-ALIGN: center" height="118" alt="" src="http://3.bp.blogspot.com/_dvbjJeJb4LE/R8Ys7Y-teEI/AAAAAAAAAPg/KrMYiHxZAS8/s400/quizquestion3.PNG" width="452" border="0" /> <p align="left"><span style="color:#cc0000;"><span style="color:#000099;">In this question, the level of comprehension required for this question does not really bypass the simple rules of antidifferentiating polynomial functions and knowing the connection between acceleration, velocity, and position. This required connection involves the fundamental theorem of calculus, which implicates the process of integration as the reverse of differentiation. Knowing this, and seeing that velocity is defined as a change in position over a given time interval, it can also be perceived as the time derivative of position, which is also followed by the relationship between acceleration and velocity. This indicates that antidifferentiating the acceleration function given gives us the velocity function. But don't forget that the arbitrary constant of integration, C, must be determined using the fact that v(0) = 1. Using this, we can easily solve for C by substituting 0 in for t and setting this equal to 1. Now, since velocity and position are also related similarly, we can antidifferentiate this new velocity function to determine the position function. Don't forget once again, however, to use the fact that s(0) = 3 in order to determine the constants value and finish off the final function.<br /><br /><span style="color:#000000;">Answer: s(t) = t³ + t + 3</span> </span></span></p><p align="center"><span style="color:#cc0000;"><span style="color:#009900;">Question #4:</span> </span><br /></p><div align="center"><span style="color:#cc0000;"></div></span><span style="color:#cc0000;"><p></span></p><img id="BLOGGER_PHOTO_ID_5171861433975928770" style="DISPLAY: block; MARGIN: 0px auto 10px; CURSOR: hand; TEXT-ALIGN: center" height="297" alt="" src="http://4.bp.blogspot.com/_dvbjJeJb4LE/R8Ykko-td8I/AAAAAAAAAOg/M1jXB9FTVe0/s400/quizquestion4.PNG" width="413" border="0" /> <span style="color:#009900;">This question is a truly interesting one, and also sparked a discussion as Mr. K was pulling up the answers to the quiz. It was really just a discussion concerning the negative sign of the cos(x) function, though this quarrel was quickly settled as Mr. K began detailing the solution to the problem. Now on with a reiteration of Mr. K's solution, though I will tend to elaborate wherever I can.</span><br /><span style="color:#009900;"></span><br /><span style="color:#009900;">Since we're finding the area under two separate function on the interval x = 0 to x = k, we know that we're going to be finding the value of two separate integrals with the intervals lower and upper limits. Also, just looking at these functions yields should yield a sigh of relief since these functions aren't complicated at all to antidifferentiate each function, so it would be quite easy to continue on that path now. First, as it shows in Mr. K's work, we must set these two integrals equal to each other on the same interval, 0 to k. Then, all you must do is antidifferentiate each function, giving you an equality as shown in the first line in black print in Mr. K's solution above. Don't forget that the antiderivative of sin(x) is -cos(x) and not just cos(x). Remember this is so because the derivative of cos(x) is -sin(x), so just sin(x) is just missing that negative sign when going backwards. Also, don't neglect substituting the 0 into the trigonometric function -cos(x) when integrating the left side, since cos(0) yields a 1. Watch the negative signs though, and try not to get mixed up by the negatives in the function.</span><br /><span style="color:#009900;"></span><br /><span style="color:#009900;">Now once you have determined the final equality after antidifferentiating and inserting k and 0, you could solve this in two different ways. You could either leave the two functions equal to each other, and find points of intersection between the two graphs. Or you could find move everything to one side and find the zero's of the function. Either way, it'll require the use of your calculator but yet still yield the same answer.</span><br /><span style="color:#009900;"></span><br /><span style="color:#000000;">Answer: 1.300</span><br /><span style="color:#000000;">____________________________________________________________</span><br /><span style="color:#000000;"></span><br />And now, on to the problem for today's class, the Boston By The Sea" problem. First off, Mr. K quickly told us the origin of Boston as a harbour that slowly grew outwards into the ever-growing city it is now. He really only mentioned that since the question modeled the city using a semi-circle. Here is the question. <p><br /></p><p><img id="BLOGGER_PHOTO_ID_5171861433975928786" style="DISPLAY: block; MARGIN: 0px auto 10px; CURSOR: hand; TEXT-ALIGN: center" alt="" src="http://4.bp.blogspot.com/_dvbjJeJb4LE/R8Ykko-td9I/AAAAAAAAAOo/5eEXcsMfdwk/s400/problem1.PNG" border="0" /><span style="color:#cc0000;"></span></p><p><span style="color:#cc0000;">In class, we were only able to cover question (a), though our solution never came to fruition and is therefore homework for Friday's class. Now on to our discussion concerning the first portion of this question.<br /><br />Now you must first identify what your first task is while approaching this question. If you analyze the question thoroughly, as you should do for long answer questions similar to this, you should notice that the density function gives a value in units of population (in thousands)/miles².<br />You now must be able to try and find a way to multiply the thousands of people by a unit that will reduce with the miles² to only give the answer a final unit of population (in thousands of people). This can be done by multiplying the density function by the areas of various semi-circles with varying radii. All of our work on this problem can be seen on the following slide<br /><br /></p></span><img id="BLOGGER_PHOTO_ID_5171862099695859698" style="DISPLAY: block; MARGIN: 0px auto 10px; CURSOR: hand; TEXT-ALIGN: center" alt="" src="http://3.bp.blogspot.com/_dvbjJeJb4LE/R8YlLY-td_I/AAAAAAAAAO4/Fk7K3xT1RUo/s400/problem2.PNG" border="0" /><span style="color:#cc0000;">The function in the top left of the above slide depicts the integral Craig proposed to solve the problem part (a). Through much discussion, however, it was decided that this integral would not work for one main reason. The reason is that if we were to use this integral, we would be finding the area of successive semi-circles with changing radii, the further out you go from the center, the larger radii, but this poses a problem because each following area will include the area of the previous semi-circle, meaning you'd be adding the area of each semi-circle several times using this integral. Now we're also going to have to use only the interval given in the data, to find a Riemann sum, but we can only use the intervals of. </span><span style="color:#cc0000;">Therefore, estimating the area underneath the curve (which is the integral we're looking for) would require us to know the values between 0 and 1, 1 and 2, 2 and 3, etc.,but we don't know these values so we can't use left and right-hand sums unless we only use the given data.</span><br /><span style="color:#cc0000;"></span><br /><span style="color:#cc0000;">Now, follow the yellow section of the figure to the left, and the yellow highlighted function just to the right of the figure as they correspond to each other. Looking at the data, we can see that there is no change in density in the very first semi-circle of radius 1. Thus, we can create a simple expression to show the amount of people in this section by multiplying the density function by the area of this semi-circle. This very expression is what is highlighted in yellow.</span><br /><span style="color:#cc0000;"></span><br /><span style="color:#cc0000;">But now we must try and model the rest of the semi-circles, and we can do this by breaking each semi-circle into rings between each circumference. One of these rings is represented by the green section in the diagram. This ring can be taken out and straightened out to then be represented by a rectangle, which is also shown by the green highlighted section. The length of this rectangle will be equivalent to the circumference of the semi-circle, and will have a width equivalent to the change in the radius, which is consistently one in this case. Therefore, we now have a model to determine the actual amount of people in the entire section between r = 0 and r = 8. Adding the successive parts of each interval to the expression highlighted in yellow will yield us our final answer for part (a).</span><br /><span style="color:#cc0000;"></span><br /><span style="color:#cc0000;">Now, our homework for today's class was to finish up this problem and calculate the amount of people in the 8 mile radius. Don't forget to complete your homework for Friday's class everyone.</span><br /><span style="color:#cc0000;"></span><br /><span style="color:#cc0000;">I think that's all for my scribe post today. As I mentioned in the beginning of my scribe post, the scribe for tomorrow's class will be <span style="color:#000000;">ETHAN</span>, that is unless he can't make it to class again, in which case the scribe will be </span><span style="color:#000000;">VAN</span><span style="color:#cc0000;">. Goodnight everyone and have a pleasant tomorrow! ( I can't remember where that's from, but oh well! =D) I hope my scribe post helped anyone who was in seek of some elaboration upon today's class, or was yearning for some explanation on the subject of applications of integrals. Have a good one everyone!</span><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-4956159171639903439?l=apcalc07.blogspot.com' alt='' /></div>MrSiwWyhttp://www.blogger.com/profile/18072494079396275311noreply@blogger.com2tag:blogger.com,1999:blog-5788051255938485746.post-13797078410197564112008-02-27T15:39:00.001-06:002008-02-27T15:43:56.613-06:00Here they are ...<br /><br /><center><br /><div style="width:425px;text-align:left" id="__ss_284480"><object style="margin:0px" width="425" height="355"><param name="movie" value="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-february-27-2008-1204148267536093-2"/><param name="allowFullScreen" value="true"/><param name="allowScriptAccess" value="always"/><embed src="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-february-27-2008-1204148267536093-2" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="355"></embed></object><div style="font-size:11px;font-family:tahoma,arial;height:26px;padding-top:2px;"><a href="http://www.slideshare.net/?src=embed"><img src="http://static.slideshare.net/swf/logo_embd.png" style="border:0px none;margin-bottom:-5px" alt="SlideShare"/></a> | <a href="http://www.slideshare.net/dkuropatwa/ap-calculus-slides-february-27-2008?src=embed" title="View 'AP Calculus Slides February 27, 2008' on SlideShare">View</a> | <a href="http://www.slideshare.net/upload?src=embed">Upload your own</a></div></div><br /></center><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-1379707841019756411?l=apcalc07.blogspot.com' alt='' /></div>dkuropatwahttp://www.blogger.com/profile/08462283847470560887noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-17153590094641567662008-02-25T23:50:00.009-06:002008-11-18T18:01:19.393-06:00Sorry to all, this scribe is very late due to the fact that I worked till 11 both nights and I just woke up from falling asleep at the computer... o_O<br /><br />Okay, so Monday's class started for most people with walking into room 66 and getting ready to learn. Mine however started in the library as I was getting my Grad Pictures taken.<br /><br />So, when I walked into class, a couple minutes had already gone by and Mr. K. was quickly going over the Exercises on Volumes of Revolution from Thursday's class. Now, over the weekend Grey-M did a pretty decent job posting the answers in his Scribe, but happened to make a couple of mistakes.<br /><span style="color: rgb(255, 0, 0);"><br />To make sure you don't make the same ones:<br />•RECOGNIZE THE DIFFERENCE BETWEEN VERTICAL AND HORIZONTAL FUNCTIONS (of "y" and of "x"), IDENTIFY WHICH TERMS THE INTEGRAL IS IN (x or y), AND REMEMBER THE DIRECTION OF REVOLUTION.<span style="color: rgb(0, 0, 0);"><br /><br />In depth corrections may be seen in the <a href="http://www.slideshare.net/dkuropatwa/ap-calculus-slides-january-25-2008?src=embed">slides from today</a> .<br /><br />We then returned to the "Oil Leak Question" from Tuesday's class...<br /><br />If you don't remember, the question was:<br /></span></span><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://3.bp.blogspot.com/_n1XY9NEWU6I/R8UDSJX_6qI/AAAAAAAAAGI/EGdy4awrmpQ/s1600-h/Picture+1.png"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 386px; height: 215px;" src="http://3.bp.blogspot.com/_n1XY9NEWU6I/R8UDSJX_6qI/AAAAAAAAAGI/EGdy4awrmpQ/s320/Picture+1.png" alt="" id="BLOGGER_PHOTO_ID_5171543357393988258" border="0" /></a>Now, to explain it briefly... the function given is in <span style="color: rgb(0, 153, 0);">Gallons/Hour</span> and the value we wish to obtain is in <span style="color: rgb(0, 153, 0);">Gallons</span>. So, you do the math... (we have to multiply by <span style="color: rgb(0, 153, 0);">Hours</span>). Now the amount of hours it takes to spill can also be represented as ∂t (change in time) which, when multiplied by the function, results in a neat looking integral which can then be solved from 0 to 10.<br /><br />The next question we looked at was the "Density Question".<br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://1.bp.blogspot.com/_n1XY9NEWU6I/R8UEupX_6rI/AAAAAAAAAGQ/Zh3UrukuR2o/s1600-h/Picture+2.png"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://1.bp.blogspot.com/_n1XY9NEWU6I/R8UEupX_6rI/AAAAAAAAAGQ/Zh3UrukuR2o/s400/Picture+2.png" alt="" id="BLOGGER_PHOTO_ID_5171544946531887794" border="0" /></a>For this question, first visualize that there is a circular oil slick with a changing density (it is more dense in the center than it is towards the circumference). We are then told the density at any radius is given by a function. Once again looking at units, we see the function is in <span style="color: rgb(255, 102, 0);">KG/m^2</span> and we need a value with units <span style="color: rgb(255, 102, 0);">KG</span>. So we need to multiply the function by the area of the slick and the area in terms of r is <span style="color: rgb(255, 0, 0);">2πr•∂r. <span style="color: rgb(0, 0, 0);">So multiplying that onto the function once again provides an integral (thanks to the ∂r) and may be evaluated from 0 to 1000 to get the answer.<br /><br />Part B of this question asks to find the smallest radius that contains 75% of the slick's mass.<br /></span></span><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://3.bp.blogspot.com/_n1XY9NEWU6I/R8UHwJX_6sI/AAAAAAAAAGY/Pcmzjes1vR8/s1600-h/Picture+3.png"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://3.bp.blogspot.com/_n1XY9NEWU6I/R8UHwJX_6sI/AAAAAAAAAGY/Pcmzjes1vR8/s400/Picture+3.png" alt="" id="BLOGGER_PHOTO_ID_5171548270836574914" border="0" /></a>So for this one, we take the integral to find mass used in the previous question and use "o to r" as the interval. As well, set it equal to 75% of the total mass (3255). By doing this, we have set the end of the interval as the variable and therefor the first value of "r" from 0 that equals 75% of the mass will be the result.<br />Once this is set up, <span style="color: rgb(0, 153, 0);">integrate by parts with u=1+r^2</span> to solve it... the rest is grunt work =P<br /><br /><br /><br />Then we moved on to a new question dealing with density of cars along a length of street.<br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://3.bp.blogspot.com/_n1XY9NEWU6I/R8UJcJX_6tI/AAAAAAAAAGg/RdtjUbGZ2t4/s1600-h/Picture+4.png"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://3.bp.blogspot.com/_n1XY9NEWU6I/R8UJcJX_6tI/AAAAAAAAAGg/RdtjUbGZ2t4/s400/Picture+4.png" alt="" id="BLOGGER_PHOTO_ID_5171550126262446802" border="0" /></a>This is very similar to the first question in terms of how we solve it. It says the function gives value with the units <span style="color: rgb(51, 51, 255);">Number of Cars/KM</span> and we need just the <span style="color: rgb(51, 102, 255);">Number of Cars</span>. So, we somehow need to multiply by KM. It says that x=KM so the distance from zero (∂x) can be multiplied by the function to once again CREATE AN INTEGRAL!!! (Surprise!)<br />To find how many cars are along a stretch of street of given length, just plug the length into the equation (x=length) and solve.<br /><br /><br /><br /><br />I believe that is all for this scribe... don't forget to do some homework!<br /><br /><br />Next Scribe is.........<br /><br /><br /> ETHAN!!! Because I think he's the only one who hasn't done 7... I may be wrong, but yeah. You're scribe<br />XD<div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-1715359009464156766?l=apcalc07.blogspot.com' alt='' /></div>«Craig»http://www.blogger.com/profile/08860268378924842699noreply@blogger.com1tag:blogger.com,1999:blog-5788051255938485746.post-67620264258708928202008-02-25T13:26:00.001-06:002008-02-25T13:28:00.770-06:00Here they are ...<br /><br /><center><br /><div style="width:425px;text-align:left" id="__ss_281218"><object style="margin:0px" width="425" height="355"><param name="movie" value="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-january-25-2008-1203967324480667-4"/><param name="allowFullScreen" value="true"/><param name="allowScriptAccess" value="always"/><embed src="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-january-25-2008-1203967324480667-4" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="355"></embed></object><div style="font-size:11px;font-family:tahoma,arial;height:26px;padding-top:2px;"><a href="http://www.slideshare.net/?src=embed"><img src="http://static.slideshare.net/swf/logo_embd.png" style="border:0px none;margin-bottom:-5px" alt="SlideShare"/></a> | <a href="http://www.slideshare.net/dkuropatwa/ap-calculus-slides-january-25-2008?src=embed" title="View 'AP Calculus Slides January 25, 2008' on SlideShare">View</a> | <a href="http://www.slideshare.net/upload?src=embed">Upload your own</a></div></div><br /></center><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-6762026425870892820?l=apcalc07.blogspot.com' alt='' /></div>dkuropatwahttp://www.blogger.com/profile/08462283847470560887noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-63775839372050482712008-02-24T17:14:00.002-06:002008-02-24T17:18:17.176-06:00Here is my scribe! Supposed to be spinning solids but I think it made my head spin more than anything. Got 4 out of the 7 wrong according to the sheet but I worked hard and if I had time I would go through them with a fine tooth comb but cannot do that today (Plus number 7 was just mean).<br /><br /><br /><div style="width:425px;text-align:left" id="__ss_280294"><object style="margin:0px" width="425" height="355"><param name="movie" value="http://static.slideshare.net/swf/ssplayer2.swf?doc=scribe-6-1203894823860416-5"/><param name="allowFullScreen" value="true"/><param name="allowScriptAccess" value="always"/><embed src="http://static.slideshare.net/swf/ssplayer2.swf?doc=scribe-6-1203894823860416-5" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="355"></embed></object><div style="font-size:11px;font-family:tahoma,arial;height:26px;padding-top:2px;"><a href="http://www.slideshare.net/?src=embed"><img src="http://static.slideshare.net/swf/logo_embd.png" style="border:0px none;margin-bottom:-5px" alt="SlideShare"/></a> | <a href="http://www.slideshare.net/GreyM/scribe-6?src=embed" title="View 'Scribe 6' on SlideShare">View</a> | <a href="http://www.slideshare.net/upload?src=embed">Upload your own</a></div></div><br /><br /><br /><br /><br />The next scribe will be... Craig?<div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-6377583937205048271?l=apcalc07.blogspot.com' alt='' /></div>Grey-Mhttp://www.blogger.com/profile/10237241798100727258noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-65875261557276819012008-02-24T08:20:00.002-06:002008-02-24T08:21:48.708-06:00I wondered if this might apply to you? I know I should be paying attention--<br /><br /><object width="480" height="360"><param name="movie" value="http://voicethread.com/book.swf?b=61691"></param><param name="wmode" value="transparent"></param><embed src="http://voicethread.com/book.swf?b=61691" type="application/x-shockwave-flash" wmode="transparent" width="480" height="360"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-6587526155727681901?l=apcalc07.blogspot.com' alt='' /></div>Lani Ritter Hallhttp://www.blogger.com/profile/16352862711544966770noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-71352388155561348842008-02-21T04:31:00.001-06:002008-02-23T07:55:06.226-06:00The same applies here... PDF/Slideshare issues.<div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-7135238815556134884?l=apcalc07.blogspot.com' alt='' /></div>Vanhttp://www.blogger.com/profile/07679810325423377354noreply@blogger.com3tag:blogger.com,1999:blog-5788051255938485746.post-78557650550010425292008-02-19T13:23:00.002-06:002008-02-19T13:25:46.121-06:00Here they are ...<br /><br /><center><br /><div style="width:425px;text-align:left" id="__ss_272166"><object style="margin:0px" width="425" height="355"><param name="movie" value="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-february-19-2008-1203448905786541-2"/><param name="allowFullScreen" value="true"/><param name="allowScriptAccess" value="always"/><embed src="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-february-19-2008-1203448905786541-2" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="355"></embed></object><div style="font-size:11px;font-family:tahoma,arial;height:26px;padding-top:2px;"><a href="http://www.slideshare.net/?src=embed"><img src="http://static.slideshare.net/swf/logo_embd.png" style="border:0px none;margin-bottom:-5px" alt="SlideShare"/></a> | <a href="http://www.slideshare.net/dkuropatwa/ap-calculus-slides-february-19-2008?src=embed" title="View 'AP Calculus Slides February 19, 2008' on SlideShare">View</a> | <a href="http://www.slideshare.net/upload?src=embed">Upload your own</a></div></div><br /></center><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-7855765055001042529?l=apcalc07.blogspot.com' alt='' /></div>dkuropatwahttp://www.blogger.com/profile/08462283847470560887noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-64742533799774733722008-02-19T03:38:00.005-06:002008-03-17T00:54:28.943-05:00It's works. *jumps for joy* Finally.<br /><br /><div style="width: 425px; text-align: left;" id="__ss_282147"><a style="left: 0px ! important; top: 0px ! important;" title="Click here to block this object with Adblock Plus" class="abp-objtab-0352130266780758 visible" href="http://static.slideshare.net/swf/ssplayer2.swf?doc=super-late-scribe-1204026381701317-5"></a><object style="margin: 0px;" height="355" width="425"><param name="movie" value="http://static.slideshare.net/swf/ssplayer2.swf?doc=super-late-scribe-1204026381701317-5"><param name="allowFullScreen" value="true"><param name="allowScriptAccess" value="always"><embed src="http://static.slideshare.net/swf/ssplayer2.swf?doc=super-late-scribe-1204026381701317-5" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" height="355" width="425"></embed></object><div style="font-size: 11px; font-family: tahoma,arial; height: 26px; padding-top: 2px;"><a href="http://www.slideshare.net/?src=embed"><img src="http://static.slideshare.net/swf/logo_embd.png" style="border: 0px none ; margin-bottom: -5px;" alt="SlideShare" /></a> | <a href="http://www.slideshare.net/Navcan/super-late-scribe?src=embed" title="View 'Super Late Scribe' on SlideShare">View</a> | <a href="http://www.slideshare.net/upload?src=embed">Upload your own</a></div></div><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-6474253379977473372?l=apcalc07.blogspot.com' alt='' /></div>Vanhttp://www.blogger.com/profile/07679810325423377354noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-46193109518300507812008-02-14T11:12:00.002-06:002008-02-14T11:15:36.381-06:00Here they are ...<br /><br /><center><br /><div style="width:425px;text-align:left" id="__ss_265966"><object style="margin:0px" width="425" height="355"><param name="movie" value="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-february-14-2008-1203009288341487-2"/><param name="allowFullScreen" value="true"/><param name="allowScriptAccess" value="always"/><embed src="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-february-14-2008-1203009288341487-2" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="355"></embed></object><div style="font-size:11px;font-family:tahoma,arial;height:26px;padding-top:2px;"><a href="http://www.slideshare.net/?src=embed"><img src="http://static.slideshare.net/swf/logo_embd.png" style="border:0px none;margin-bottom:-5px" alt="SlideShare"/></a> | <a href="http://www.slideshare.net/dkuropatwa/ap-calculus-slides-february-14-2008?src=embed" title="View 'AP Calculus Slides February 14, 2008' on SlideShare">View</a> | <a href="http://www.slideshare.net/upload?src=embed">Upload your own</a></div></div><br /></center><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-4619310951830050781?l=apcalc07.blogspot.com' alt='' /></div>dkuropatwahttp://www.blogger.com/profile/08462283847470560887noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-39189434703158529172008-02-13T21:15:00.007-06:002008-02-14T10:13:51.022-06:00Hello back again for another scribe, Tuesday's class was not very difficult although very important. As we began the day with a simple question, which was "Find the volume generated between the x-axis and the graph f(x)=x^2, from x=0 to x=2. Here is the graph of x^2.<br /><br /><a href="http://fooplot.com/index.php?q0=x%5e2">http://fooplot.com/x^2</a><br /><br /><span style="font-size:100%;">So we begin to solve this problem by setting up the equation (0,2) <span style="font-size: 16pt; font-family: Arial;" lang="EN-CA">∫ </span><span style="font-size: 16pt; font-family: &quot;Times New Roman&quot;;" lang="EN-CA">&pi;r^2 dx<br /><br /> = (0,2) </span><span style="font-size: 16pt; font-family: Arial;" lang="EN-CA">∫</span><span style="font-size: 16pt; font-family: Arial;" lang="EN-CA"></span><span style="font-size: 16pt; font-family: &quot;Times New Roman&quot;;" lang="EN-CA"> Π(x^2)^2 dx<br /> = (0,2) </span><span style="font-size: 16pt; font-family: Arial;" lang="EN-CA">∫ </span><span style="font-size: 16pt; font-family: &quot;Times New Roman&quot;;" lang="EN-CA">Πx^4 dx<br /> = (0,2) </span><span style="font-size: 16pt; font-family: Arial;" lang="EN-CA">∫ </span><span style="font-size: 16pt; font-family: &quot;Times New Roman&quot;;" lang="EN-CA">Π(x^5/5) dx<br /></span><span style="font-size: 16pt; font-family: &quot;Times New Roman&quot;;" lang="EN-CA"> </span><span style="font-size: 16pt; font-family: Arial;" lang="EN-CA"> </span><span style="font-size: 16pt; font-family: &quot;Times New Roman&quot;;" lang="EN-CA"> = Π(2^5/5) - </span><span style="font-size: 16pt; font-family: Arial;" lang="EN-CA"> </span><span style="font-size: 16pt; font-family: &quot;Times New Roman&quot;;" lang="EN-CA">Π(0^5/5)<br /> = 32</span><span style="font-size: 16pt; font-family: &quot;Times New Roman&quot;;" lang="EN-CA">Π/5<br /> <span style="font-style: italic;"> Solved.<br /><span style="font-style: italic;"><br /><span style="font-family: arial;">Then Mr.K began showing us a new form in which solids of revolution can be formed, which was by rotating the function about the y-axis. The equation that represents a function about the y-axis is: V= </span></span></span></span></span><span style="font-size:100%;"><span style="font-size: 16pt; font-family: &quot;Times New Roman&quot;;" lang="EN-CA"></span><span style="font-size: 16pt; font-family: Arial;" lang="EN-CA">∫2</span><span style="font-size: 16pt; font-family: &quot;Times New Roman&quot;;" lang="EN-CA"><span style="font-style: italic;"><span style="font-style: italic;"><span style="font-family: arial;"> </span></span></span></span><span style="font-size: 16pt; font-family: &quot;Times New Roman&quot;;" lang="EN-CA">Πrf(x) dx</span><span style="font-size: 16pt; font-family: &quot;Times New Roman&quot;;" lang="EN-CA"><span style="font-style: italic;"><span style="font-style: italic;"><span style="font-family: arial;"> Here is a problem that was worked on in class using this method.<br /><br /><br />Region S is bonded between two functions, f(x) &amp; g(x), find the volume of the solid generating around the y-axis.<br /><br /> F(x)= 0.5x^2-2x+4 G(x)= 4+4x-x^2 <br /><br />Using the volume equation of a solid of revolution around the y-axis we get, just solve algebraically, do not solve completely:<br /><br /> = </span></span></span></span><span style="font-size: 16pt; font-family: &quot;Times New Roman&quot;;" lang="EN-CA"><span style="font-style: italic;"><span style="font-style: italic;"><span style="font-family: arial;"></span></span></span></span></span><span style="font-size:100%;"><span style="font-size: 16pt; font-family: &quot;Times New Roman&quot;;" lang="EN-CA"></span><span style="font-size: 16pt; font-family: Arial;" lang="EN-CA">2</span><span style="font-size: 16pt; font-family: &quot;Times New Roman&quot;;" lang="EN-CA"><span style="font-style: italic;"><span style="font-style: italic;"><span style="font-family: arial;"> </span></span></span></span><span style="font-size: 16pt; font-family: &quot;Times New Roman&quot;;" lang="EN-CA">Π (0,4)</span></span><span style="font-size:100%;"><span style="font-size: 16pt; font-family: Arial;" lang="EN-CA">∫ </span></span><span style="font-size:100%;"><span style="font-size: 16pt; font-family: &quot;Times New Roman&quot;;" lang="EN-CA">x [g(x)-f(x)] dx<br /><br />All in all this class was very important as the equation for a solid of revolution about the y-axis was given and explained. <span style="color: rgb(255, 0, 0);">Remember all the rest of 8.3 was for homework. The next scribe is VAN.</span><br /><br /></span><span style="font-size: 16pt; font-family: &quot;Times New Roman&quot;;" lang="EN-CA"><span style="font-style: italic;"><span style="font-style: italic;"><span style="font-family: arial;"><br /><br /> </span><br /></span></span></span></span><span style="font-size: 16pt; font-family: &quot;Times New Roman&quot;;" lang="EN-CA"><br /></span><span style="font-size: 16pt; font-family: Arial;" lang="EN-CA"></span><span style="font-size: 16pt; font-family: &quot;Times New Roman&quot;;" lang="EN-CA"></span><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-3918943470315852917?l=apcalc07.blogspot.com' alt='' /></div>Dinohttp://www.blogger.com/profile/16720903626604404687noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-82562348503031857202008-02-12T17:16:00.006-06:002008-02-13T22:31:29.142-06:00Hello back again for another scribe post. Sorry in advance to everyone who was waiting for the Friday's post, i was not able to get it up because of my internet connection to the blog was timing out. Well lets get into some math. Friday's class was all about rotating functions around the X-axis. Although this functions were not lines, they were functions like the area between two parabolas, or a function that had never been seen before. It started with a simple solids of revolution function, Find the volume of a solid revolution obtained by rotating the function x^2, bounded by the lines x=2 and x=1 around the x-axis. So here is a picture of that function revolved around the x-axis.<br /><a href="http://fooplot.com/index.php?q0=x%5e2">http://fooplot.com/index.php?q0=x%5e2</a><br /><br /> So between 1 and 2 we want to find the volume of the solid of revolution. So we find that once we make a cut and pull out a piece from the solid of revolution it looks like a circle with a hole in the middle. Where x^2 is the radius of the circle So the area of the circle is A(x)=<span lang="EN-CA">Πr^2= </span><span lang="EN-CA">Π(X^2)^2 =<br /></span><span lang="EN-CA">Π(x^4).<br /><br /> So the volume of the solid of revolution is V<span style="font-size:100%;">=1,2</span></span><span style="font-size:100%;"><span lang="EN-CA" style="font-family:Arial;"> ∫</span></span><span style="" lang="EN-CA">Π(x^4) dx = </span><span style="" lang="EN-CA">Π [x^5/5] 1,2 = [32</span><span style="" lang="EN-CA">Π/</span><span style="" lang="EN-CA">5]</span><span style="" lang="EN-CA"> - [1</span><span style="" lang="EN-CA">Π/5] =<span style="font-weight: bold;"> 31</span></span><span style="" lang="EN-CA">Π/5.</span><span style="" lang="EN-CA"><br /><span style="font-size:100%;"><br /></span><span style="font-weight: bold;font-size:100%;" >After this question we took up questions in our homework from the previous night that we were not able to complete. </span><span style="color: rgb(255, 0, 0); font-weight: bold;font-size:100%;" >The next scribe is going to be Dino once again.</span><br /></span><span style="" lang="EN-CA"><span style="font-size:180%;"><span style="font-size:100%;"><br /></span></span></span> <p class="MsoNormal"><span lang="EN-CA"><br /></span></p><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-8256234850303185720?l=apcalc07.blogspot.com' alt='' /></div>Dinohttp://www.blogger.com/profile/16720903626604404687noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-47816888437753963932008-02-12T12:43:00.000-06:002008-02-12T12:51:18.095-06:00Here they are ...<br /><br /><center><br /><div style="width:425px;text-align:left" id="__ss_263206"><object style="margin:0px" width="425" height="355"><param name="movie" value="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-february-12-2008-120284144824645-4"/><param name="allowFullScreen" value="true"/><param name="allowScriptAccess" value="always"/><embed src="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-february-12-2008-120284144824645-4" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="355"></embed></object><div style="font-size:11px;font-family:tahoma,arial;height:26px;padding-top:2px;"><a href="http://www.slideshare.net/?src=embed"><img src="http://static.slideshare.net/swf/logo_embd.png" style="border:0px none;margin-bottom:-5px" alt="SlideShare"/></a> | <a href="http://www.slideshare.net/dkuropatwa/ap-calculus-slides-february-12-2008?src=embed" title="View 'AP Calculus Slides February 12, 2008' on SlideShare">View</a> | <a href="http://www.slideshare.net/upload?src=embed">Upload your own</a></div></div><br /></center><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-4781688843775396393?l=apcalc07.blogspot.com' alt='' /></div>dkuropatwahttp://www.blogger.com/profile/08462283847470560887noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-11134631970220491882008-02-08T13:36:00.000-06:002008-02-08T13:37:42.151-06:00Here they are ...<br /><br /><center><br /><div style="width:425px;text-align:left" id="__ss_258253"><object style="margin:0px" width="425" height="355"><param name="movie" value="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-february-8-2008-1202499346164235-2"/><param name="allowFullScreen" value="true"/><param name="allowScriptAccess" value="always"/><embed src="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-february-8-2008-1202499346164235-2" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="355"></embed></object><div style="font-size:11px;font-family:tahoma,arial;height:26px;padding-top:2px;"><a href="http://www.slideshare.net/?src=embed"><img src="http://static.slideshare.net/swf/logo_embd.png" style="border:0px none;margin-bottom:-5px" alt="SlideShare"/></a> | <a href="http://www.slideshare.net/dkuropatwa/ap-calculus-slides-february-8-2008?src=embed" title="View 'AP Calculus Slides February 8, 2008' on SlideShare">View</a> | <a href="http://www.slideshare.net/upload?src=embed">Upload your own</a></div></div><br /></center><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-1113463197022049188?l=apcalc07.blogspot.com' alt='' /></div>dkuropatwahttp://www.blogger.com/profile/08462283847470560887noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-59079247851659780562008-02-06T16:06:00.001-06:002008-02-06T16:07:43.687-06:00Here they are ...<br /><br /><center><br /><div style="width:425px;text-align:left" id="__ss_255485"><object style="margin:0px" width="425" height="355"><param name="movie" value="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-february-6-2008-1202335356735332-4"/><param name="allowFullScreen" value="true"/><param name="allowScriptAccess" value="always"/><embed src="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-february-6-2008-1202335356735332-4" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="355"></embed></object><div style="font-size:11px;font-family:tahoma,arial;height:26px;padding-top:2px;"><a href="http://www.slideshare.net/?src=embed"><img src="http://static.slideshare.net/swf/logo_embd.png" style="border:0px none;margin-bottom:-5px" alt="SlideShare"/></a> | <a href="http://www.slideshare.net/dkuropatwa/ap-calculus-slides-february-6-2008?src=embed" title="View 'AP Calculus Slides February 6, 2008' on SlideShare">View</a> | <a href="http://www.slideshare.net/upload?src=embed">Upload your own</a></div></div><br /></center><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-5907924785165978056?l=apcalc07.blogspot.com' alt='' /></div>dkuropatwahttp://www.blogger.com/profile/08462283847470560887noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-62623435423280241592008-02-06T15:57:00.000-06:002008-11-18T18:01:21.124-06:00<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://flickr.com/photos/jimwinstead/70623170/"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://4.bp.blogspot.com/_HLm_KkQlC58/R6ozlYSW-4I/AAAAAAAAAGA/YPFovhAazYM/s320/Bread.JPG" alt="" id="BLOGGER_PHOTO_ID_5163996640001325954" border="0" /></a><span style="color: rgb(51, 51, 51);"><span style="color: rgb(51, 51, 51);"><br />Hello, I am known as <span style="color: rgb(204, 102, 0);">Tim-math-y</span> on our blog, and I will be the Scribe for today's lessons.<br /><br /><span style="font-weight: bold;">The Overview:</span><br /><br />Today, we discussed the concept of "How to Build a Solid." By rotating certain functions about the x-axis, three dimensional solids were formed, upon which we could use calculus to solve for the unknown volume. We found that the solution to calculating such values, especially with irregular shapes, involved the idea of Bread Slices. These concepts will be thoroughly discussed throughout this scribe post.<br /><br /><span style="font-weight: bold;">The Introduction:</span><br /><br />To begin the class, we started off with a concept we were already familiar with: <span style="color: rgb(204, 102, 0);">areas between curves</span>. We were given three questions that would lead us into our main lesson of today. They all involved similar steps, and did not present any new difficulties. Calculators were not permitted to solving the first two, but for the third one, they were permitted as the algebraic calculations could become disorganized and difficult.<br /><br /></span></span><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://1.bp.blogspot.com/_HLm_KkQlC58/R6o4_oSW-5I/AAAAAAAAAGI/E46-hruKvLU/s1600-h/SA1.JPG"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://1.bp.blogspot.com/_HLm_KkQlC58/R6o4_oSW-5I/AAAAAAAAAGI/E46-hruKvLU/s320/SA1.JPG" alt="" id="BLOGGER_PHOTO_ID_5164002588531030930" border="0" /></a><span style="color: rgb(51, 51, 51);">This was our first question. As one may see, it involves a set of three major steps.<br /><br /><span style="color: rgb(0, 0, 153);">1.) First, we set up an equation in hopes of solving for the points of intersections, shared by the two functions. We set each function on opposite sides of the equation and algebraically solve for the value of 'x'.<br /><br /><span style="color: rgb(255, 102, 0);">2.) Second, we attempt to find which function is 'on top of the other'. This may be done by plotting both functions onto a coordinate plane or using a number line.<br /><br /><span style="color: rgb(0, 0, 0);">3.) Third, we subtract the two areas underneath each curve (area1 - area2, where area1 maintains the upper position). To find the areas underneat each curve, bounded by the x-axis, we used integration, from 'a' to 'b', where a = 0, and b = 1 in this question.<br /><br />By solving, we calculated the area bounded by the pair of functions.<br /></span></span></span></span><span style="color: rgb(51, 51, 51);"><span style="color: rgb(51, 51, 51);"><br /><span style="color: rgb(51, 51, 51);">The Next two questions given included the same ideas and concepts for finding the solution. What is more important however, is how this connected to the concept that we learned today.<br /><br /><span style="font-weight: bold;">The Focus: Volumes by Slicing </span><br /><span style="font-style: italic;">Included are exerpts from the textbook (Calculus; Concepts and Calculators), hopefully to provide insights<br /><br /></span></span></span></span><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://3.bp.blogspot.com/_HLm_KkQlC58/R6pClISW-6I/AAAAAAAAAGQ/57csQEeP3OA/s1600-h/SA2.JPG"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://3.bp.blogspot.com/_HLm_KkQlC58/R6pClISW-6I/AAAAAAAAAGQ/57csQEeP3OA/s320/SA2.JPG" alt="" id="BLOGGER_PHOTO_ID_5164013128380775330" border="0" /></a><span style="color: rgb(51, 51, 51);"><span style="color: rgb(51, 51, 51);"><span style="color: rgb(51, 51, 51);"><span style="color: rgb(0, 0, 0);">The method we use to compute the volume of certain solids revolves about using the <span style="color: rgb(204, 102, 0);">definite integral</span>. By dividing the solid into small pieces whose volume we can easily approximate, we can compute the volume. As the number of terms in the sum grows larger and larger, the approximation improves and the limiting values is a definite integral.<br /><br />For any solid that has a constant cross-section its volume is the <span style="color: rgb(204, 102, 0);">product of its cross-section area 'A' and its thickness '(delta) x'</span>. Most solids do not have regular shapes and the task of computing their volumes requires calculus methods.<br /><br />Imagine a loaf of bread lying along the x-axis in the xy-plane between x = a and x = b.<br /><br />The loaf can be divided into several slices by making cuts <span style="color: rgb(204, 102, 0);">perpendicular to the x-axis</span>. The volume of the loaf is the <span style="color: rgb(204, 102, 0);">sum of the volumes of all of the individual slices</span>. In general, since the shape of the loaf varies, different slices have different volumes even if they all have the same thickness.<br /><br />For any 'x' between 'a' and 'b', let <span style="color: rgb(204, 102, 0);">A(x) represent the cross-sectional area</span> created by the cut at x. Suppose we divide the loaf into n slices of thickness <span style="color: rgb(204, 102, 0);">(delta) x = (b - a)/ n</span>. The cross-sectional area of a typical slice is not constant. However, if we replace it with a slice of the same thickness and constant cross-sectional area given by one face of the slive, then the volume of this slice is:<br /><br />Volume of one slice = A(x) * (delta) x<br /><br />To estimate the volume of the whole loaf, divide it into n pieces of equal thickness: </span></span></span></span><span style="color: rgb(51, 51, 51);"><span style="color: rgb(51, 51, 51);"><span style="color: rgb(51, 51, 51);"><span style="color: rgb(0, 0, 0);"><span style="color: rgb(204, 102, 0);">(delta) x = (b - a)/ n</span> , by making cuts at a = x0, x1, x2 ... xn = b.Using the cross-sectional areas A(x1), A(x2), ... , A(xn) for the individual slices leads to an estimate for total volume.<br /><br />Total Volume = A(x1)(delta)x + A(x2)(delta)x + ... + A(xn)(delta)x<br /><br />This is a Reimann sum for the cross-sectional area function A(x) on the interval [a,<span style="color: rgb(204, 102, 0);"><span style="color: rgb(0, 0, 0);">b].</span> Thus, the limiting value as n grows larger and larger is the definite integral concluded below: <xn using="" areas="" individual="" slices="" leads="" to="" an="" estimate="" total="" volume="A(x1)(delta)x" x="" this="" a="" riemann="" sum="" for="" sectional="" area="" function="" on="" interval="" limiting="" value="" as="" n="" grows="" larger="" is="" the="" definite="" integral="" shown=""></xn></span></span></span></span><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://3.bp.blogspot.com/_HLm_KkQlC58/R6pG1ISW-7I/AAAAAAAAAGY/hnzLMARIB_0/s1600-h/SA3.JPG"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://3.bp.blogspot.com/_HLm_KkQlC58/R6pG1ISW-7I/AAAAAAAAAGY/hnzLMARIB_0/s320/SA3.JPG" alt="" id="BLOGGER_PHOTO_ID_5164017801305193394" border="0" /></a><span style="color: rgb(51, 51, 51);">Coming back to our class lesson, we looked at an example question:<br /><br /></span><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://4.bp.blogspot.com/_HLm_KkQlC58/R6pIJYSW-8I/AAAAAAAAAGg/sbRfYYuxzOw/s1600-h/SA4.JPG"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://4.bp.blogspot.com/_HLm_KkQlC58/R6pIJYSW-8I/AAAAAAAAAGg/sbRfYYuxzOw/s320/SA4.JPG" alt="" id="BLOGGER_PHOTO_ID_5164019248709172162" border="0" /></a><span style="color: rgb(51, 51, 51);">In this situation, the linear function, within the interval of [0,8], was 'spun' around the x-axis to for a coned solid. We found that, by slicing the solid into smaller pieces, we could find the volume of the given piece. In this case, each piece took the shape of a cylinder, where its volume could be expressed as: <span style="color: rgb(204, 102, 0);">area x height</span>, where area is <span style="color: rgb(204, 102, 0);">(pi)r^2</span>, and height is the subinterval of <span style="color: rgb(204, 102, 0);">(delta)x</span>.<br /><br />As the amount of slices grew to <span style="color: rgb(204, 102, 0);">infinity</span>, or in other words, let (delta)x become infinitely small, the integral of the area would compute the volume of the solid.<br /></span><br /><span style="color: rgb(51, 51, 51);">We then continued our calculations to solve for the volume:</span><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://4.bp.blogspot.com/_HLm_KkQlC58/R6pJ9YSW-9I/AAAAAAAAAGo/rDGXYGwcJPI/s1600-h/SA5.JPG"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://4.bp.blogspot.com/_HLm_KkQlC58/R6pJ9YSW-9I/AAAAAAAAAGo/rDGXYGwcJPI/s320/SA5.JPG" alt="" id="BLOGGER_PHOTO_ID_5164021241573997522" border="0" /></a><span style="color: rgb(51, 51, 51);">Here, we found the volume of the coned solid to be 128(pi)/3. Referring back to the original equation for solving a volume of a cone, we checked whether or not our solution was correct.</span><br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://2.bp.blogspot.com/_HLm_KkQlC58/R6pKg4SW--I/AAAAAAAAAGw/Th5lV_hfxeg/s1600-h/SA6.JPG"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://2.bp.blogspot.com/_HLm_KkQlC58/R6pKg4SW--I/AAAAAAAAAGw/Th5lV_hfxeg/s320/SA6.JPG" alt="" id="BLOGGER_PHOTO_ID_5164021851459353570" border="0" /></a><span style="color: rgb(51, 51, 51);">By inputting the given values of 'r' (4) and 'h' (8), we found that our calculations were indeed correct.<br /><br /></span><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://3.bp.blogspot.com/_HLm_KkQlC58/R6pLHISW-_I/AAAAAAAAAG4/R6hDheHG8e0/s1600-h/SA7.JPG"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://3.bp.blogspot.com/_HLm_KkQlC58/R6pLHISW-_I/AAAAAAAAAG4/R6hDheHG8e0/s320/SA7.JPG" alt="" id="BLOGGER_PHOTO_ID_5164022508589349874" border="0" /></a><br /><span style="color: rgb(51, 51, 51);">Finally, there was one final concept that Mr. K parted to us. It was a brief explanation as to how to solve these type of questions:<br /><br /></span><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://1.bp.blogspot.com/_HLm_KkQlC58/R6pMMoSW_AI/AAAAAAAAAHA/z5K6z9NdQzc/s1600-h/SA8.JPG"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://1.bp.blogspot.com/_HLm_KkQlC58/R6pMMoSW_AI/AAAAAAAAAHA/z5K6z9NdQzc/s320/SA8.JPG" alt="" id="BLOGGER_PHOTO_ID_5164023702590258178" border="0" /></a><br /><span style="color: rgb(51, 51, 51);">The solution is as simple as the slide. To solve for the volume of a given solid, such as this, we utilize the same techniques. To solve this question, we find the area of the shaded portion. Simply put, we find the area by using subtraction: <span style="color: rgb(204, 102, 0);">A2 - A1</span>.<br /><br />That concludes our lesson today. I hope that any of the readers found this somewhat helpful, if required. I just thought I'd add a slight more effort in today's scribe post.<br /><br />I found a link to a site that provided me with some animations, to help anyone who wishes to visualize such shapes to a greater extent:<br /><br /><a href="http://curvebank.calstatela.edu/volrev/volrev.htm">http://curvebank.calstatela.edu/volrev/volrev.htm</a><br /><br />I'm not sure if we were assigned homework. However, I'm guessing<span style="font-weight: bold;"> exercise 8.2</span>. We will continue this topic the following day, said Mr. K.<br /><br />That's all everybody! Tomorrow's scribe will be: <span style="font-weight: bold;">Dino</span>, the first name that popped into my head. Good night everyone!<br /></span></span><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-6262343542328024159?l=apcalc07.blogspot.com' alt='' /></div>Tim_MATH_yhttp://www.blogger.com/profile/09929458936647070771noreply@blogger.com4tag:blogger.com,1999:blog-5788051255938485746.post-82797749656753992232008-02-04T20:41:00.000-06:002008-02-04T20:44:33.445-06:00<div style="width:425px;text-align:left" id="__ss_252599"><object style="margin:0px" width="425" height="355"><param name="movie" value="http://static.slideshare.net/swf/ssplayer2.swf?doc=scribefeb4-1202179414913430-2"><param name="allowFullScreen" value="true"><param name="allowScriptAccess" value="always"><embed src="http://static.slideshare.net/swf/ssplayer2.swf?doc=scribefeb4-1202179414913430-2" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="355"></embed></object><div style="font-size:11px;font-family:tahoma,arial;height:26px;padding-top:2px;"><a href="http://www.slideshare.net/?src=embed"><img src="http://static.slideshare.net/swf/logo_embd.png" style="border:0px none;margin-bottom:-5px" alt="SlideShare" /></a> | <a href="http://www.slideshare.net/aichelle/scribefeb4?src=embed" title="View 'Scribefeb4' on SlideShare">View</a> | <a href="http://www.slideshare.net/upload?src=embed">Upload your own</a></div></div><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-8279774965675399223?l=apcalc07.blogspot.com' alt='' /></div>aichelle s.http://www.blogger.com/profile/17302966674461415089noreply@blogger.com2tag:blogger.com,1999:blog-5788051255938485746.post-73039319782842664512008-02-04T12:59:00.000-06:002008-02-04T13:01:03.100-06:00Here they are ...<br /><br /><center><br /><div style="width:425px;text-align:left" id="__ss_252168"><object style="margin:0px" width="425" height="355"><param name="movie" value="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-february-4-2008-120215150848752-4"/><param name="allowFullScreen" value="true"/><param name="allowScriptAccess" value="always"/><embed src="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-february-4-2008-120215150848752-4" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="355"></embed></object><div style="font-size:11px;font-family:tahoma,arial;height:26px;padding-top:2px;"><a href="http://www.slideshare.net/?src=embed"><img src="http://static.slideshare.net/swf/logo_embd.png" style="border:0px none;margin-bottom:-5px" alt="SlideShare"/></a> | <a href="http://www.slideshare.net/dkuropatwa/ap-calculus-slides-february-4-2008?src=embed" title="View 'AP Calculus Slides February 4, 2008' on SlideShare">View</a> | <a href="http://www.slideshare.net/upload?src=embed">Upload your own</a></div></div><br /></center><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-7303931978284266451?l=apcalc07.blogspot.com' alt='' /></div>dkuropatwahttp://www.blogger.com/profile/08462283847470560887noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-34747627896919748352008-01-23T11:42:00.000-06:002008-01-23T11:47:14.289-06:00Hey †here, wow this was definitely a close one, MrSiwWy here with a near-miss bob post. Well antidifferentiation isn't too difficult once you are accustomed to manipulating equations and successfully applying the techniques such as u-substitution and integration by parts. I found that the most intricate method of preparation for this test is probably just practicing and honing your ability to apply these techniques to various equations, and practicing the process of identifying which scenarios require special approaches and such. I don't think that there were any troubled moments for me this unit, and I think that this test (assuming that it's similar to the difficulty of the pre-test) won't be too difficult as well. Hopefully, everyone studied and will perform at their maximum capacity on today's test. GOOD LUCK everyone!<div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-3474762789691974835?l=apcalc07.blogspot.com' alt='' /></div>MrSiwWyhttp://www.blogger.com/profile/18072494079396275311noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-18626644688920198452008-01-23T00:21:00.000-06:002008-01-23T00:25:48.981-06:00I found this unit a bit of work to do. I missed 2 classes so I had to do catch up work. I found this unit a bit of a problem because the problems dont click in right away. I know the steps but cant carry through with the first step required to solve it,if that makes any sense. anyways good luck to everyone writing tomorrow.<div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-1862664468892019845?l=apcalc07.blogspot.com' alt='' /></div>etimzhttp://www.blogger.com/profile/09626175119223517919noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-51022329377712952212008-01-22T22:12:00.000-06:002008-01-22T22:16:34.920-06:00Well I'm looking forward to this test! I think that this one is definitely aceable. I feel fairly comfortable with all the work that we have done except for maybe that brutal sheet of antiderivatives but we were assured that things like those wouldn't appear on the test so it's all good.<br /><br />Good luck every1!<div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-5102232937771295221?l=apcalc07.blogspot.com' alt='' /></div>Grey-Mhttp://www.blogger.com/profile/10237241798100727258noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-7282862023690800282008-01-22T22:04:00.000-06:002008-11-18T18:01:21.413-06:00<span style="font-family:trebuchet ms;"><span style="color: rgb(255, 102, 0);"><span style="font-weight: bold;">Well yeeeeaah Craig is BOBing again. This unit has been quite interesting, I have enjoyed it thoroughly. It seems that this unit has pretty good for, most has just come natural (a good thing because of all the content I have missed due to sleeping... Sorry Mr. K!!!) But after this pre-test I fully understand the application of all the content of the unit. It has been a great unit with plenty of memorable classes (thanks to the SMARTBoard and computer being at its best) But, I should wrap up this Beee - Ohhh - Beee, because I have plenty of Physics to finish tonight...<br /><br /></span></span></span><div style="text-align: center;"><span style="font-family:trebuchet ms;"><span style="color: rgb(255, 102, 0);"><span style="font-weight: bold;">Good Luck to all!!! =D</span></span></span><br /><br /><span style="font-family:trebuchet ms;"><span style="color: rgb(255, 102, 0);"><span style="font-weight: bold;">Keep On Pushing</span></span></span><br /></div><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://4.bp.blogspot.com/_n1XY9NEWU6I/R5bAXZRGD5I/AAAAAAAAAGA/1M7AOjXDdm8/s1600-h/sisyphus.jpg"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://4.bp.blogspot.com/_n1XY9NEWU6I/R5bAXZRGD5I/AAAAAAAAAGA/1M7AOjXDdm8/s400/sisyphus.jpg" alt="" id="BLOGGER_PHOTO_ID_5158521931351592850" border="0" /></a><br /><span style="font-family:trebuchet ms;"><span style="color: rgb(255, 102, 0);"><span style="font-weight: bold;">-Craig<br /></span></span></span><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-728286202369080028?l=apcalc07.blogspot.com' alt='' /></div>«Craig»http://www.blogger.com/profile/08860268378924842699noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-53342073413092801122008-01-22T21:59:00.000-06:002008-01-22T22:02:24.080-06:00My slideshare scribe... it seems more squished than it's supposed to be, but I guess that's okay. I gotta edit it over anyways. Enjoy!<br /><br /><div style="width: 425px; text-align: left;" id="__ss_237912"><object style="margin: 0px;" height="355" width="425"><param name="movie" value="http://static.slideshare.net/swf/ssplayer2.swf?doc=vans-scribe-1201060477642569-3"><param name="allowFullScreen" value="true"><param name="allowScriptAccess" value="always"><embed src="http://static.slideshare.net/swf/ssplayer2.swf?doc=vans-scribe-1201060477642569-3" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" height="355" width="425"></embed></object><div style="font-size: 11px; font-family: tahoma,arial; height: 26px; padding-top: 2px;"><a href="http://www.slideshare.net/?src=embed"><img src="http://static.slideshare.net/swf/logo_embd.png" style="border: 0px none ; margin-bottom: -5px;" alt="SlideShare" /></a> | <a href="http://www.slideshare.net/Navcan/vans-scribe" title="View 'Vans Scribe' on SlideShare">View</a> | <a href="http://www.slideshare.net/upload">Upload your own</a></div></div><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-5334207341309280112?l=apcalc07.blogspot.com' alt='' /></div>Vanhttp://www.blogger.com/profile/07679810325423377354noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-73349155814660134672008-01-22T21:05:00.000-06:002008-01-22T22:18:55.052-06:00Hello everyone! We started off todays class with a quick talk about the software out there to solve integrals and how we are better. A program called Maxima can find the answer to solvable integrals but they will be messy answers usually and we can do better! So we did as shown below... (Sorry for the terseness of the slides, it's not my style but I really don't have the time for all that tonight as it got busy real fast)<br /><br /><div style="width:425px;text-align:left" id="__ss_237922"><object style="margin:0px" width="425" height="355"><param name="movie" value="http://static.slideshare.net/swf/ssplayer2.swf?doc=scribe-5-1201061398688262-2"/><param name="allowFullScreen" value="true"/><param name="allowScriptAccess" value="always"/><embed src="http://static.slideshare.net/swf/ssplayer2.swf?doc=scribe-5-1201061398688262-2" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="355"></embed></object><div style="font-size:11px;font-family:tahoma,arial;height:26px;padding-top:2px;"><a href="http://www.slideshare.net/?src=embed"><img src="http://static.slideshare.net/swf/logo_embd.png" style="border:0px none;margin-bottom:-5px" alt="SlideShare"/></a> | <a href="http://www.slideshare.net/GreyM/scribe-5" title="View 'Scribe 5' on SlideShare">View</a> | <a href="http://www.slideshare.net/upload">Upload your own</a></div></div><br /><br />The next scribe will be..... <br /><br />Aichelle!<div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-7334915581466013467?l=apcalc07.blogspot.com' alt='' /></div>Grey-Mhttp://www.blogger.com/profile/10237241798100727258noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-57478862659419354422008-01-22T20:47:00.000-06:002008-01-22T20:55:29.173-06:00In my opinion I found this unit really challenging. I had a lot of difficulty on many of the problems and I still don't know the answers to many of them. I have trouble sometimes deciding on which method to use when I start to antidifferentiate. It takes me longer than others to get to the answer. On the other hand I found this unit interesting when I found out how to antidifferentiate inverse trig functions. Good luck to everyone tomorrow and the best of luck!<div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-5747886265941935442?l=apcalc07.blogspot.com' alt='' /></div>Robert Phttp://www.blogger.com/profile/03046070808724541650noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-31033912155151410802008-01-22T20:00:00.000-06:002008-01-22T20:05:45.694-06:00Personally, I found this unit to be one of the most difficult units. A large portion of the questions were undeniably difficult to solve, such as those where "algebraic massaging" or "adding/subtracting zero or multiplying/dividing by one" in various forms were required. I still have many questions left unanswered that I hope to be assauged tomorrow during my spare and at lunch. I know much of the mechanics, but much still remains clueless for myself. Thats my reflection on this unit. The pretest wasn't too bad either - lets hope that the test will be quite similar.<br /><br />Good luck to everyone on the Test tomorrow! I'm glad that I rememberred to bob.<div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-3103391215515141080?l=apcalc07.blogspot.com' alt='' /></div>Tim_MATH_yhttp://www.blogger.com/profile/09929458936647070771noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-40683672081701110682008-01-22T19:11:00.000-06:002008-01-22T19:29:56.639-06:00<span style="font-family: trebuchet ms; color: rgb(102, 0, 204);">This unit was definitely a harder unit for me to understand. I like how things go together but I have a lot of trouble figuring out what I have to do first. Sometimes I can start solving a problem but I don't know what to do next. I find this unit really challenging and hope that I can do well on the test. The pretest was all right; I answered most of the multiple choice questions and I had trouble with the long answer question. Although we spent a lot of time on this unit I still have a lot of trouble with it but hopefully I can do better on the test. Good luck everyone and don't forget to BOB or study!</span><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-4068367208170111068?l=apcalc07.blogspot.com' alt='' /></div>aichelle s.http://www.blogger.com/profile/17302966674461415089noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-11432940859720517952008-01-22T15:46:00.001-06:002008-01-22T15:55:08.312-06:00Here they are ...<br /><br /><center><br /><div style="width:425px;text-align:left" id="__ss_237610"><object style="margin:0px" width="425" height="355"><param name="movie" value="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-january-22-2008-1201038735545323-5"/><param name="allowFullScreen" value="true"/><param name="allowScriptAccess" value="always"/><embed src="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-january-22-2008-1201038735545323-5" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="355"></embed></object><div style="font-size:11px;font-family:tahoma,arial;height:26px;padding-top:2px;"><a href="http://www.slideshare.net/?src=embed"><img src="http://static.slideshare.net/swf/logo_embd.png" style="border:0px none;margin-bottom:-5px" alt="SlideShare"/></a> | <a href="http://www.slideshare.net/dkuropatwa/ap-calculus-slides-january-22-2008" title="View 'AP Calculus Slides January 22, 2008' on SlideShare">View</a> | <a href="http://www.slideshare.net/upload">Upload your own</a></div></div><br /></center><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-1143294085972051795?l=apcalc07.blogspot.com' alt='' /></div>dkuropatwahttp://www.blogger.com/profile/08462283847470560887noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-39207741553059076892008-01-21T16:36:00.000-06:002008-01-21T16:37:28.338-06:00Here they are ...<br /><br /><center><br /><div style="width:425px;text-align:left" id="__ss_236247"><object style="margin:0px" width="425" height="355"><param name="movie" value="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-january-21-2008-1200954815601047-3"/><param name="allowFullScreen" value="true"/><param name="allowScriptAccess" value="always"/><embed src="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-january-21-2008-1200954815601047-3" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="355"></embed></object><div style="font-size:11px;font-family:tahoma,arial;height:26px;padding-top:2px;"><a href="http://www.slideshare.net/?src=embed"><img src="http://static.slideshare.net/swf/logo_embd.png" style="border:0px none;margin-bottom:-5px" alt="SlideShare"/></a> | <a href="http://www.slideshare.net/dkuropatwa/ap-calculus-slides-january-21-2008" title="View 'AP Calculus Slides January 21, 2008' on SlideShare">View</a> | <a href="http://www.slideshare.net/upload">Upload your own</a></div></div><br /></center><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-3920774155305907689?l=apcalc07.blogspot.com' alt='' /></div>dkuropatwahttp://www.blogger.com/profile/08462283847470560887noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-2503975430900950652008-01-19T20:46:00.000-06:002008-11-18T18:01:23.118-06:00<div><div>Hello everyone, it's MrSiwWy here again for yet another scribe (since I chose myself for the beginning of cycle 7) to explore (along the lines of the title of this scribe) just how approximate something can be. Now for this post, I'd most definitely enjoy starting with the same image that Mr. K used to introduce this subtopic on Friday's class (though I'm sure he quickly showed it to us some time before).<br /><br /><p><img id="BLOGGER_PHOTO_ID_5157450055970549010" style="DISPLAY: block; MARGIN: 0px auto 10px; CURSOR: hand; TEXT-ALIGN: center" alt="" src="http://2.bp.blogspot.com/_dvbjJeJb4LE/R5LxgCTxbRI/AAAAAAAAAMY/QrmaWCkVRkk/s400/pic0.PNG" border="0" /> </p><p>So now that we've explored the many techniques associated with antidifferentiating a function or group or functions, we can now apply them calculate any complex definite integrals we might encounter. But how would you calculate a definite integral which we don't necessarily know the antiderivative for? Such as:<img id="BLOGGER_PHOTO_ID_5157450060265516322" style="DISPLAY: block; MARGIN: 0px auto 10px; CURSOR: hand; TEXT-ALIGN: center" alt="" src="http://3.bp.blogspot.com/_dvbjJeJb4LE/R5LxgSTxbSI/AAAAAAAAAMg/EVMPzMjsPno/s400/pic1.PNG" border="0" /></p><p>If you stop and think for a second, there is actually another way to solve such a definite integral without applying the techniques of antidifferentiation (since determining the antiderivative of the above function has so far proved quite impossible). As we were first introduced to the concept of the integral, we focused mainly on the geometric interpretation of various functions. This can easily be applied to the above function, by using Riemannian sums on the above equation, a definite value can in fact be calculated. The main concern is determining just how close we can get to this value in the simplest manner or using the simplest techniques.<br /></p><br /><p>Digress from the above analysis on non-antidifferentiable functions and consider the following functions.<img id="BLOGGER_PHOTO_ID_5157450060265516338" style="DISPLAY: block; MARGIN: 0px auto 10px; CURSOR: hand; TEXT-ALIGN: center" alt="" src="http://3.bp.blogspot.com/_dvbjJeJb4LE/R5LxgSTxbTI/AAAAAAAAAMo/v0F8y_nMAYg/s400/pic2.PNG" border="0" /></p><p>Now though these functions aren't difficult to antidifferentiate at all, they will be quite an aid for analyzing the various methods we have been introduced to when calculating definite integrals. The first of which, concerns the left and right-hand sums of the given function.<br /><span style="color:#000099;"></span></p><br /><p><span style="color:#000000;"></span></p><p><span style="color:#000000;">***Before determining the left and right hand sums for each function, recall the significance of monotonicity. What is meant by a monotonic increasing function? What is meant by a monotonic decreasing function? How do they affect approximations?</span></p><p><br /><span style="color:#000000;">If you can't quite remember what a monotonic increasing/decreasing function represents, the underlying or basic idea of such a concept is not too difficult to grasp, though it has been a while since this concept has been applied in any way.<br /><br />-- If a function <em>f</em> on the interval [a,b] is increasing throughout the interval, then that function is said to be monotonic increasing.<br />-- If a function <em>f</em> on the interval [a,b] is decreasing throughout the interval, then that function is said to be monotonic decreasing.<br /><br />You may also take note of the fact that if a function <em>f</em> is increasing on a given interval [a,b] (monotonic increasing), then the function's derivative <em>f' </em>is > 0 throughout that interval. Likewise for a monotonic decreasing function where it's derivative is negative throughout.</span><br /><br /><span style="color:#000099;"><u><span style="font-size:130%;">Left and Right hand sums</span></u><br /><br />Now back to the main concept at hand; to left and right hand sums. If you are not familiar with the process of calculating left and right hand sums or you have forgotten but have the RSUM / RIESUM program on your graphing calculator, here's a breakdown on how to use the program.<br /><br />- Hit the [y=] button, then insert the given function in the Y1 slot.<br />- Hit [prgm] and then choose the RSUM / RIESUM program.<br />- It will prompt you to insert a "LEFT" value, this is the lower limit of the integral.<br />- It will then prompt you to insert a "RIGHT" value, which is now the upper limit of the integral.<br />- Next will be a prompt for you to insert a "X CHOICE" value, which indicates the type of sum desired. For a left sum, X CHOICE = 0, for a midpoint sum X CHOICE = 0.5, for a right sum X CHOICE = 1. Try not to forget this.<br />- It will finally ask for a "NUMBER" value, this is the number of desired <span class="blsp-spelling-corrected" id="SPELLING_ERROR_3">sub intervals</span>.<br />- Hit enter, and it will output the value of the calculated sum, whether it be left, right or mid.<br /><br />The following graphs show how the left and right hand sums compare to the actual graphs and also to each other. </p><img id="BLOGGER_PHOTO_ID_5158064425272438242" style="DISPLAY: block; MARGIN: 0px auto 10px; CURSOR: hand; TEXT-ALIGN: center" alt="" src="http://1.bp.blogspot.com/_dvbjJeJb4LE/R5UgRCTxbeI/AAAAAAAAAOA/JeGoHVMx788/s400/pic3.PNG" border="0" /></span><span style="color:#000099;">Do you notice any patterns lying within the above graphs? One important factor to notice is that one function is monotonic increasing, while the other is monotonic decreasing. This does in fact play an important role in the accuracy and orientation of a particular left or right hand sum with regard to the main function. Here's a breakdown on this connection.<br /><br /><u>For a monotonic increasing function:</u><br /><br />-- The left hand sum will always be an underestimate. This means that using a left hand sum will always be slightly less than the actual value of the integral over that interval.<br />-- The right hand sum will always be an overestimate. This means that using a right hand sum will always be slightly greater than the actual value of the integral over that interval.<br /><br /><u>For a monotonic decreasing function:</u><br /><br />-- The left hand sum will always be an overestimate.<br />-- The right hand sum will always be an underestimate.<br /><br />At this point you might be asking yourself which is better, a left hand sum or a right hand sum. But do these answers have any correlation with the <span class="blsp-spelling-corrected" id="SPELLING_ERROR_4">tonicity</span> of the function in question?<br /><br />Well if you analyze the left and right hand sums for both monotonic increasing and monotonic decreasing functions above, they appear as though both sums produce approximately the same amount of error. But if you take into account the fact that a trapezoid sum (discussed further below) is the average between these two sums (left and right), then you can actually determine which sum truly retains a greater error. This can be achieved by analyzing where a trapezoid sum is an under or an overestimate, now since this sum is the average between a left and a right sum then this under or overestimate in the trapezoid sum directly reflects which sum has a greater error for a given function. This basically means that if the error of the trapezoid sum is known to be an underestimate, then whichever sum yielded an underestimate therefore had the greater error, and likewise for a trapezoid sum that is an overestimate. Now, I don't think that it's important to memorize this fact, but just know the basic concepts of analyzing the geometric aspect of functions and you should be fine.<br /><br />One might argue that this could be compensated for by introduced a greater amount of <span class="blsp-spelling-corrected" id="SPELLING_ERROR_5">sub intervals</span>, just as Riemann had showed the world. But how could you compensate for such a deficit with a fixed number of <span class="blsp-spelling-corrected" id="SPELLING_ERROR_6">sub intervals</span>? Basically using varied width for each <span class="blsp-spelling-corrected" id="SPELLING_ERROR_7">sub interval</span> could work. Since Riemann had basically surmised that it does not matter how wide each <span class="blsp-spelling-corrected" id="SPELLING_ERROR_8">sub interval</span> is, just as long as you go to infinity, that's what matters. So now that the idea of possibly using varied <span class="blsp-spelling-corrected" id="SPELLING_ERROR_9">sub interval</span> widths has quickly implanted itself into your mind and has just as quickly departed, beware that it's possible that a question involving such varied <span class="blsp-spelling-corrected" id="SPELLING_ERROR_10">sub intervals</span> might pop up on the AP exam. Remember, using Riemann sums and determining an integral by hand relies solely on your interpretation of the geometry of the function in question. Remember that.<br /></span><br /><span style="color:#ff6600;"><u><span style="font-size:130%;">Trapezoid Sums</span></u><br /><u><span style="font-size:130%;"></span></u><br />Recall another sum that was quite important in calculating definite integrals was the trapezoid sum. We were both informally introduced to this concept as being the mean of the left and right hand sums for the function and also formally introduced to the fact that averaging the left and right hand sums creates a series of trapezoid in order to approximate the area under the curve. This technique is sufficiently more approximate than just a left or a right hand sum, but is it right on or is there some inaccuracy associated with a trapezoid sum? Of course there is inaccuracy when using any type of sum that we have encountered so far, but which sum has the greatest accuracy?<br /><br />Here is an example depicting the idea behind a trapezoid sum:<br /></span><br /><img id="BLOGGER_PHOTO_ID_5157469056905866658" style="DISPLAY: block; MARGIN: 0px auto 10px; CURSOR: hand; TEXT-ALIGN: center" alt="" src="http://3.bp.blogspot.com/_dvbjJeJb4LE/R5MCyCTxbaI/AAAAAAAAANg/uQ910soAWcU/s400/pic4.PNG" border="0" /></div><div><p><span style="color:#ff6600;">Now we must find the trapezoid sums for the above functions in order to aptly contrast each process.</span><img id="BLOGGER_PHOTO_ID_5157466922307120498" style="DISPLAY: block; MARGIN: 0px auto 10px; CURSOR: hand; TEXT-ALIGN: center" alt="" src="http://2.bp.blogspot.com/_dvbjJeJb4LE/R5MA1yTxbXI/AAAAAAAAANI/zR2tyA7L1WI/s400/pic5.PNG" border="0" /><span style="color:#ff6600;">Notice by the above graphs that for an monotonic increasing function, the trapezoid sum is a slight overestimate, and that for a monotonic decreasing function the trapezoid sum is a slight underestimate. By looking at the graphs, it would appear that the trapezoid sum is overall the best sum by far. But what about the final type of sum that we have encountered throughout our study of integration. This sum is of course the midpoint sum.<br /></span><span style="font-size:130%;"><u><br /><span style="color:#ff0000;">Midpoint Sums<br /></span></u></span><br /><span style="color:#ff0000;">Do you remember what the purpose or the main idea behind a midpoint sum is? Basically, instead of using rectangles that use the left or right hand values of each <span class="blsp-spelling-corrected" id="SPELLING_ERROR_11">sub interval</span> as their height, the rectangles take upon themselves the value of the midpoint of each <span class="blsp-spelling-corrected" id="SPELLING_ERROR_12">sub interval</span> as their height. Below are two example of how a midpoint sum works, where the second example uses a tangent line instead of a straight line.<br /></span><br /><img id="BLOGGER_PHOTO_ID_5157466926602087810" style="DISPLAY: block; MARGIN: 0px auto 10px; CURSOR: hand; TEXT-ALIGN: center" alt="" src="http://3.bp.blogspot.com/_dvbjJeJb4LE/R5MA2CTxbYI/AAAAAAAAANQ/UStHJVu8iCE/s400/pic6.PNG" border="0" /> <span style="color:#ff0000;">Also, as with the above summing techniques, applying them to the aforementioned functions is required to better receive a deeper understanding of how these processes compare to each other. </span><img id="BLOGGER_PHOTO_ID_5157466926602087826" style="DISPLAY: block; MARGIN: 0px auto 10px; CURSOR: hand; TEXT-ALIGN: center" alt="" src="http://3.bp.blogspot.com/_dvbjJeJb4LE/R5MA2CTxbZI/AAAAAAAAANY/Mvr_4_7HMXM/s400/pic7.PNG" border="0" /></p><p><span style="color:#ff0000;">Now take note that inaccuracy attained by using a midpoint sum apparently rivals that of a trapezoid sum. This is true since a midpoint sum for a monotonic increasing function is always an underestimate, conversely a midpoint sum for a monotonic decreasing function yields an overestimate. Now we've covered the main sum techniques encountered so far this year, it's time to go a step further.</span><br /><span style="font-size:130%;"><u></u></span><br /><span style="color:#006600;"><span style="font-size:130%;"><u>The Best Approximation?</u></span><br /><br />So which technique appears to be the best? So far, it's either going to be the midpoint sum technique or the trapezoid technique. A possible way to determine which is truly more precise, apply them both to an integral with a known value. Setting up a table for different amounts of <span class="blsp-spelling-corrected" id="SPELLING_ERROR_13">sub intervals</span> and also with positions to record the given error of each respective summing technique will be essential in conducting this analysis. For the integral below, the value is known to be exactly 1/4, so by subtracting the received value from each calculation from 1/4 you can acquire the amount of error accompanying the calculation. Take note that it's now possible to receive both positive and negative errors, but let positive error values represent overestimates and negative error values represent underestimates. </span></p><img id="BLOGGER_PHOTO_ID_5158063750962572754" style="DISPLAY: block; MARGIN: 0px auto 10px; CURSOR: hand; TEXT-ALIGN: center" alt="" src="http://4.bp.blogspot.com/_dvbjJeJb4LE/R5UfpyTxbdI/AAAAAAAAAN4/VtL_PbA_-us/s400/pic8.PNG" border="0" /><span style="color:#006600;">Under close inspection of the error of using each sum technique, it appears as though a midpoint sum is twice as accurate as a trapezoid sum. So now we know how we can find the best approximation possible when <span class="blsp-spelling-corrected" id="SPELLING_ERROR_14">antidifferentiation</span> is impossible at our level. Since a midpoint sum yields half the error, and is therefore twice as better, is appears as though midpoint sums are the ideal candidates for numerical integration. Though there must be better techniques, such as a technique that will NOT be on the AP exam, but will be prominent in higher level calculus classes. This technique is known as the Simpson sum.</span><br /><br /><span style="color:#33cc00;"><span style="color:#993300;">The idea behind a Simpson sum is simple. I must note, however, that it must be understood that the minimum amount of points required to configure a straight line is two, while the minimum amount of points required to configure a parabolic curve is three, this being an important idea behind a Simpson sum. Now, since using a midpoint sum yields such a great accuracy using a line that connects two points, what if you were to use curves with three points to attempt to fit the curve? Basically, instead of having a straight line connecting two points within a <span class="blsp-spelling-corrected" id="SPELLING_ERROR_15">sub interval</span>, you can use three points connecting two lines within that <span class="blsp-spelling-corrected" id="SPELLING_ERROR_16">sub interval</span>. Now since a trapezoid sum's inaccuracy is always the opposite the inaccuracy of the midpoint sum, but twice the value, if you add the trapezoid sum to twice the midpoint sum and divide that value by three, the error would approach 0 significantly faster than with a trapezoid or midpoint sum thus producing remarkable accuracy.</span><br /><br /></span>Now, formal introductions to the idea of trapezoid and midpoint sum error calculations are necessary. These error calculations are as follows:<br /><br /><img id="BLOGGER_PHOTO_ID_5157469056905866674" style="DISPLAY: block; MARGIN: 0px auto 10px; CURSOR: hand; TEXT-ALIGN: center" alt="" src="http://3.bp.blogspot.com/_dvbjJeJb4LE/R5MCyCTxbbI/AAAAAAAAANo/TRJRtV8BQho/s400/pic9.PNG" border="0" /> ______________________________________________________________<br /><br />Wow, I'm finally finished scribing. I didn't think that this post would be so long, and I really wanted to complete this scribe using notebook software or something else, but I was having a great amount of tribulation when trying to export, publish, upload, etc. Well anyway, this is pretty much complete now, all that's left is to state who the next scribe will be! Well goodnight and goodbye everyone! Have a great weekend.<br />Ah yes, the next scribe will be :<br /><br /><strong><span style="font-size:130%;color:#ff0000;"><span class="blsp-spelling-error" id="SPELLING_ERROR_17">GreyM</span></span></strong></div></div><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-250397543090095065?l=apcalc07.blogspot.com' alt='' /></div>MrSiwWyhttp://www.blogger.com/profile/18072494079396275311noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-25093281430800448822008-01-19T17:14:00.000-06:002008-01-20T00:19:30.905-06:00Hello everyone! I'm MrSiwWy and I'm the scribe for Thursday's class. There wasn't anything that we learned during this class, though we did go over some of the tougher questions from the sheet we received the previous class. This sheet consisted of many questions, thus the party of antiderivatives in the title. The questions we went over in class were numbers 5, 6, and 10 on the sheet. Here are a compilation of a closer look the process used to solve each problem along with a little text to accompany each step. I decided to capitalize on Graeme's scribe post which was highlighted in class by Mr. K which involves explaining the process of integrating step-by-step using colour each time as an indicator (a very brilliant yet quite simplistic idea). Kudos to Graeme for the idea. Well here's the scribe for Thursday.<br /><br /><br /><center><div id="__ss_234452" style="WIDTH: 425px; TEXT-ALIGN: left"><object style="MARGIN: 0px" height="355" width="425"><param name="movie" value="http://static.slideshare.net/swf/ssplayer2.swf?doc=scribe-post-6-1200808817454444-2"><param name="allowFullScreen" value="true"><param name="allowScriptAccess" value="always"><embed src="http://static.slideshare.net/swf/ssplayer2.swf?doc=scribe-post-6-1200808817454444-2" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="355"></embed></object><div style="FONT-SIZE: 11px; PADDING-TOP: 2px; FONT-FAMILY: tahoma,arial; HEIGHT: 26px"><a href="http://www.slideshare.net/?src=embed"><img style="BORDER-RIGHT: 0px; BORDER-TOP: 0px; MARGIN-BOTTOM: -5px; BORDER-LEFT: 0px; BORDER-BOTTOM: 0px" alt="SlideShare" src="http://static.slideshare.net/swf/logo_embd.png" /></a> <a title="View 'Scribe Post 6' on SlideShare" href="http://www.slideshare.net/MrSiwWy/scribe-post-6">View</a> <a href="http://www.slideshare.net/upload">Upload your own</a></div></div></center><div align="left"><br />Well that's it folks! Goodnight and goodbye everyone! </div><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-2509328143080044882?l=apcalc07.blogspot.com' alt='' /></div>MrSiwWyhttp://www.blogger.com/profile/18072494079396275311noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-74116401841063766762008-01-18T15:37:00.000-06:002008-01-18T16:05:59.477-06:00Here they are ...<br /><br /><center><br /><div style="width:425px;text-align:left" id="__ss_233423"><object style="margin:0px" width="425" height="355"><param name="movie" value="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-january-18-2008-1200692550669301-4"/><param name="allowFullScreen" value="true"/><param name="allowScriptAccess" value="always"/><embed src="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-january-18-2008-1200692550669301-4" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="355"></embed></object><div style="font-size:11px;font-family:tahoma,arial;height:26px;padding-top:2px;"><a href="http://www.slideshare.net/?src=embed"><img src="http://static.slideshare.net/swf/logo_embd.png" style="border:0px none;margin-bottom:-5px" alt="SlideShare"/></a> | <a href="http://www.slideshare.net/dkuropatwa/ap-calculus-slides-january-18-2008" title="View 'AP Calculus Slides January 18, 2008' on SlideShare">View</a> | <a href="http://www.slideshare.net/upload">Upload your own</a></div></div><br /></center><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-7411640184106376676?l=apcalc07.blogspot.com' alt='' /></div>dkuropatwahttp://www.blogger.com/profile/08462283847470560887noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-29467899658614945672008-01-17T15:34:00.000-06:002008-01-17T15:38:33.885-06:00Here they are ...<br /><br /><center><br /><div style="width:425px;text-align:left" id="__ss_232089"><object style="margin:0px" width="425" height="355"><param name="movie" value="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-january-17-2008-1200605742364819-2"/><param name="allowFullScreen" value="true"/><param name="allowScriptAccess" value="always"/><embed src="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-january-17-2008-1200605742364819-2" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="355"></embed></object><div style="font-size:11px;font-family:tahoma,arial;height:26px;padding-top:2px;"><a href="http://www.slideshare.net/?src=embed"><img src="http://static.slideshare.net/swf/logo_embd.png" style="border:0px none;margin-bottom:-5px" alt="SlideShare"/></a> | <a href="http://www.slideshare.net/dkuropatwa/ap-calculus-slides-january-17-2008" title="View 'AP Calculus Slides January 17, 2008' on SlideShare">View</a> | <a href="http://www.slideshare.net/upload">Upload your own</a></div></div><br /></center><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-2946789965861494567?l=apcalc07.blogspot.com' alt='' /></div>dkuropatwahttp://www.blogger.com/profile/08462283847470560887noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-43403759912097866852008-01-16T22:13:00.000-06:002008-01-16T22:27:56.948-06:00Hello everyone, Im <span style="color: rgb(0, 0, 153);">Tim-Math-Y</span> your scribe for todays session. I do apologize for the late scribe.<br /><br /><div style="width: 425px; text-align: left;" id="__ss_230796"><object style="margin: 0px;" height="355" width="425"><param name="movie" value="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-january-16-2008-1200520331413151-5"><param name="allowFullScreen" value="true"><param name="allowScriptAccess" value="always"><embed src="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-january-16-2008-1200520331413151-5" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" height="355" width="425"></embed></object><div style="font-size: 11px; font-family: tahoma,arial; height: 26px; padding-top: 2px;"><a href="http://www.slideshare.net/?src=embed"><img src="http://static.slideshare.net/swf/logo_embd.png" style="border: 0px none ; margin-bottom: -5px;" alt="SlideShare" /></a> | <a href="http://www.slideshare.net/dkuropatwa/ap-calculus-slides-january-16-2008" title="View 'AP Calculus Slides January 16, 2008' on SlideShare">View</a> | <a href="http://www.slideshare.net/upload">Upload your own</a></div></div><br /><br />Today, we applied the lessons that we've learned during the previous day. We found the derivatives of <span style="color: rgb(0, 0, 153);">arcsinx</span>, <span style="color: rgb(0, 0, 153);">arccosx</span>, and <span style="color: rgb(0, 0, 153);">arctanx</span>. With this new knowledge, we have discovered that underlying, are the rules for antidifferentiating these functions. Truly, we only need to remember the derivative of arcsinx, as the derivatives of arccosx and arctanx are only slightly changed forms of the former.<br /><br />We started out by taking a look at multiple questions where we were instructed to solve them freely; use any method you would think is the least difficult. These include:<span style="color: rgb(0, 0, 153);"> substitution, antidifferentiating by parts and the new method</span> that we have discovered.<br /><br />As we made progress, we found that the questions became more complicated. These questions involved <span style="color: rgb(0, 0, 153);">algebraic massage</span>, where we have to work the questions to find the solution. By looking at the expressions, we found that most of the times, the questions were near perfect from a simple antidifferentiation. By algebraicly massaging, we can add the value of 0, multiply by 1 or divide by 1 in various ways to solve each question.<br /><br />Truly it comes down to that.<br /><br />To end off the class, we were given a handout that included 16 questions. It should be completed for tomorrow, as it would greatly aid in your practice experience.<br /><br />Tomorrow's scribe is: <span style="color: rgb(0, 102, 0);">MrSiwwy</span>?? I don't know who else to pick =)<br /><br />Again, I apologize for the late scribe and, the lack of detail incorporated in the explanations.<br /><br />Have a great night everyone, see you guys tomorrow.<div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-4340375991209786685?l=apcalc07.blogspot.com' alt='' /></div>Tim_MATH_yhttp://www.blogger.com/profile/09929458936647070771noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-57578804479833870352008-01-16T17:31:00.000-06:002008-11-18T18:01:23.192-06:00<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://3.bp.blogspot.com/_HiqPzpd3O9Y/R46Uwdp1uJI/AAAAAAAAAGE/B1kqNRvPALc/s1600-h/window.jpg"><img style="margin: 0pt 10px 10px 0pt; float: left; cursor: pointer;" src="http://3.bp.blogspot.com/_HiqPzpd3O9Y/R46Uwdp1uJI/AAAAAAAAAGE/B1kqNRvPALc/s200/window.jpg" alt="" id="BLOGGER_PHOTO_ID_5156222183700543634" border="0" /></a><br />Did you know I wrote this post last year and now I get to write it again?<br /><br />Did you know I can see your classroom from two windows?!<br /><br />My first window is your blog. I am excited by what I see and hear! I never cease to be amazed by the quality and sophistication of your scribes; you constantly achieve new heights in illustrating and annotating your scribes. More than that I am so impressed when you celebrate each others’ learning, when you are creative, and when you critically reflect upon your learning in your BOBs.<br /><br />Did you know Mr. K’s blog is my second window? I admire and respect what I see and hear here too! Did you know that Mr. K celebrates your learning on his blog? that he reflects upon what best helped you to learn and why? that he unselfishly shares all he knows with those who read his blog? that he learns from the conversations on his blog? that he writes with passion and is creative? and that he commits many random acts of kindness by honoring other teachers’ accomplishments in his posts? Did you know all he expects of you, he shares those same expectations for himself?<br /><br />Did you know that because of all that and more, Mr. K.’s blog has been nominated for “Best Education Blog” on the <a href="http://cdnba.wordpress.com/"><span style="text-decoration: underline;"><span style="font-weight: bold;">Canadian Blog Awards 2007 website</span></span></a>?<br /><br />I just happen to think that no one deserves this honor more than Mr. K. <br /><br />What about you?<div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-5757880447983387035?l=apcalc07.blogspot.com' alt='' /></div>Lani Ritter Hallhttp://www.blogger.com/profile/16352862711544966770noreply@blogger.com2tag:blogger.com,1999:blog-5788051255938485746.post-48060912096312679402008-01-16T15:55:00.000-06:002008-01-16T15:57:57.991-06:00Here they are ...<br /><br /><center><br /><div style="width:425px;text-align:left" id="__ss_230796"><object style="margin:0px" width="425" height="355"><param name="movie" value="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-january-16-2008-1200520331413151-5"/><param name="allowFullScreen" value="true"/><param name="allowScriptAccess" value="always"/><embed src="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-january-16-2008-1200520331413151-5" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="355"></embed></object><div style="font-size:11px;font-family:tahoma,arial;height:26px;padding-top:2px;"><a href="http://www.slideshare.net/?src=embed"><img src="http://static.slideshare.net/swf/logo_embd.png" style="border:0px none;margin-bottom:-5px" alt="SlideShare"/></a> | <a href="http://www.slideshare.net/dkuropatwa/ap-calculus-slides-january-16-2008" title="View 'AP Calculus Slides January 16, 2008' on SlideShare">View</a> | <a href="http://www.slideshare.net/upload">Upload your own</a></div></div><br /></center><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-4806091209631267940?l=apcalc07.blogspot.com' alt='' /></div>dkuropatwahttp://www.blogger.com/profile/08462283847470560887noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-83649403301232099442008-01-15T23:06:00.000-06:002008-01-15T23:52:39.922-06:00<span style="font-size:100%;">Well back once again for another scribe. Well class started off very cold, the classroom itself was freezing. Although class started as usually and we began with a couple inverse trig functions. They started off being on the easier side, like <span style="color: rgb(255, 0, 0);">sin^-1(sin(pi/3)</span>) in which case sin was in the perfect quadrant, so arcsin was the original value of sin, <span style="color: rgb(255, 0, 0);">(pi/3).<br /><br /><span style="color: rgb(0, 0, 0);">Then they became a little harder like, <span style="color: rgb(0, 0, 153);">arcsin(sin(5pi/4))</span> in which case <span style="color: rgb(0, 0, 153);">5pi/4</span> is <span style="color: rgb(51, 51, 153);">-2^(.5)/2</span>. So we know that Sin lives in quadrants 1&amp;4, so <span style="color: rgb(0, 0, 153);">arcsin(-2^(.5)/2)</span> would be in quadrants 3&amp;4, so quadrant four will be picked in which case the <span style="color: rgb(0, 0, 153);">arcsin(-2^(.5)/2)</span> in quadrant 4 is<span style="color: rgb(0, 0, 153);"> (7pi/4 )</span>.<br /><br /><span style="color: rgb(0, 0, 0);">Next we went in the opposite direction, which was by taking a length and finding its arc value, then in which case the trig value wanting to be determined will be found. An example is : <span style="color: rgb(0, 102, 0);">cos(arctan 1),</span> in which case we work backwards. We first find the <span style="color: rgb(0, 102, 0);">arctan of 1</span>, which is <span style="color: rgb(0, 102, 0);">pi/4</span>. Then we want to find the trig value, <span style="color: rgb(0, 102, 0);">cos of pi/4</span>. The value of cos<span style="color: rgb(0, 102, 0);">(pi/4)</span> is <span style="color: rgb(0, 102, 0);">2^(.5)/2</span>. So the answer is <span style="color: rgb(0, 102, 0);">2^(.5)/2</span>.<br /><br />Next we went to something completely new, and that was to find a length that was not common and find its arc value. Here is an example: cot(arcsin(2/3)), the first thing we do is solve as a function by using x. So it becomes sin(arcsin(2/3)) = sin x ----> (2/3) = sin x. So after we find this we solve using Pythagoras's theorem by using the length and the trig function given. opp=2, hyp=3, and by solving adj= 5^(.5). So now bring in the other trig value of the equation, cot which is adj/opp it solves as 5^(.5)/2, which is the final answer to that question.<br /><br />Just before the end of class Mr.K asked the class to draw the graphs of arccos(x), arcsin(x), and arctan(x). So just something to remember, the graph of arc trig functions are its Cap trig graphs reflected over the function y=x. due to the fact that if you do not use the Cap trig graphs reflected over the function y=x, the arc trig function would not be a function as it would fail the vertical line test. So remember that piece of advice, as Mr. K said in class today with the use of Star Trek.<br /><br />Mr.K did not assign any homework today, so we have the night off. <span style="color: rgb(102, 51, 102);">Tomorrow's scribe is going to be Tim_Math_y.</span><br /></span></span></span></span><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-8364940330123209944?l=apcalc07.blogspot.com' alt='' /></div>Dinohttp://www.blogger.com/profile/16720903626604404687noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-13111478387143793252008-01-15T16:49:00.000-06:002008-01-15T16:56:36.178-06:00Here they are ...<br /><br /><center><br /><div style="width:425px;text-align:left" id="__ss_229331"><object style="margin:0px" width="425" height="355"><param name="movie" value="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-january-15-2008-1200437732442322-5"/><param name="allowFullScreen" value="true"/><param name="allowScriptAccess" value="always"/><embed src="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-january-15-2008-1200437732442322-5" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="355"></embed></object><div style="font-size:11px;font-family:tahoma,arial;height:26px;padding-top:2px;"><a href="http://www.slideshare.net/?src=embed"><img src="http://static.slideshare.net/swf/logo_embd.png" style="border:0px none;margin-bottom:-5px" alt="SlideShare"/></a> | <a href="http://www.slideshare.net/dkuropatwa/ap-calculus-slides-january-15-2008" title="View 'AP Calculus Slides January 15, 2008' on SlideShare">View</a> | <a href="http://www.slideshare.net/upload">Upload your own</a></div></div><br /></center><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-1311147838714379325?l=apcalc07.blogspot.com' alt='' /></div>dkuropatwahttp://www.blogger.com/profile/08462283847470560887noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-37568534885050480562008-01-14T18:29:00.000-06:002008-01-14T19:13:37.506-06:00Well it's that time to scribe again and today is the day that Mr. K doesn't put up the slides.<br /><br />Today's class was a review on what we have been learning so far and it was also an introduction into inverse trig functions. We divided up into groups and had a workshop class.<br /><br />We took a look at a couple of questions and then we moved onto some inverse trig functions.<br /><br />Homework for tonight is to be posted on the blog (the rest of the slides).<br /><br />Dino you are up for the next scribe!<div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-3756853488505048056?l=apcalc07.blogspot.com' alt='' /></div>Robert Phttp://www.blogger.com/profile/03046070808724541650noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-53697018057177823742008-01-14T16:41:00.000-06:002008-01-15T16:49:52.385-06:00Here they are ...<br /><br /><center><br /><div style="width:425px;text-align:left" id="__ss_228160"><object style="margin:0px" width="425" height="355"><param name="movie" value="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-january-14-2008-1200349403553557-3"/><param name="allowFullScreen" value="true"/><param name="allowScriptAccess" value="always"/><embed src="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-january-14-2008-1200349403553557-3" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="355"></embed></object><div style="font-size:11px;font-family:tahoma,arial;height:26px;padding-top:2px;"><a href="http://www.slideshare.net/?src=embed"><img src="http://static.slideshare.net/swf/logo_embd.png" style="border:0px none;margin-bottom:-5px" alt="SlideShare"/></a> | <a href="http://www.slideshare.net/dkuropatwa/ap-calculus-slides-january-14-2008" title="View 'AP Calculus Slides January 14, 2008' on SlideShare">View</a> | <a href="http://www.slideshare.net/upload">Upload your own</a></div></div><br /></center><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-5369701805717782374?l=apcalc07.blogspot.com' alt='' /></div>dkuropatwahttp://www.blogger.com/profile/08462283847470560887noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-57107897351187681792008-01-13T22:16:00.000-06:002008-01-13T22:21:58.413-06:00<div style="width: 425px; text-align: left;" id="__ss_227148"><object style="margin: 0px;" height="355" width="425"><param name="movie" value="http://static.slideshare.net/swf/ssplayer2.swf?doc=scribe-1200284044683378-2"><param name="allowFullScreen" value="true"><param name="allowScriptAccess" value="always"><embed src="http://static.slideshare.net/swf/ssplayer2.swf?doc=scribe-1200284044683378-2" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" height="355" width="425"></embed></object><div style="font-size: 11px; font-family: tahoma,arial; height: 26px; padding-top: 2px;"><a href="http://www.slideshare.net/?src=embed"><img src="http://static.slideshare.net/swf/logo_embd.png" style="border: 0px none ; margin-bottom: -5px;" alt="SlideShare" /></a> | <a href="http://www.slideshare.net/Rakiztah/scribe-227148" title="View 'scribe' on SlideShare">View</a> | <a href="http://www.slideshare.net/upload">Upload your own</a></div></div><br /><br /><br /><br />The next scribe will be...... Robert. I have to do a lot of digging up just to find who hasn't scribe for the sixth cycle. Our homework is exercise 7.4 all odds and number 22. Have a good night everyone!<div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-5710789735118768179?l=apcalc07.blogspot.com' alt='' /></div>m@rkhttp://www.blogger.com/profile/00509544958003356512noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-72387410421271999852008-01-11T15:44:00.000-06:002008-01-11T15:48:16.077-06:00Here they are ...<br /><br /><center><br /><div style="width:425px;text-align:left" id="__ss_225165"><object style="margin:0px" width="425" height="355"><param name="movie" value="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-january-11-2008-120008772224532-3"/><param name="allowFullScreen" value="true"/><param name="allowScriptAccess" value="always"/><embed src="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-january-11-2008-120008772224532-3" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="355"></embed></object><div style="font-size:11px;font-family:tahoma,arial;height:26px;padding-top:2px;"><a href="http://www.slideshare.net/?src=embed"><img src="http://static.slideshare.net/swf/logo_embd.png" style="border:0px none;margin-bottom:-5px" alt="SlideShare"/></a> | <a href="http://www.slideshare.net/dkuropatwa/ap-calculus-slides-january-11-2008" title="View 'AP Calculus Slides January 11, 2008' on SlideShare">View</a> | <a href="http://www.slideshare.net/upload">Upload your own</a></div></div><br /></center><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-7238741042127199985?l=apcalc07.blogspot.com' alt='' /></div>dkuropatwahttp://www.blogger.com/profile/08462283847470560887noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-54124158310776200672008-01-10T18:05:00.000-06:002008-01-10T19:54:22.874-06:00Hi this is my scribe (despite the odd title)!<br /><br /><br /><div style="width:425px;text-align:left" id="__ss_224321"><object style="margin:0px" width="425" height="355"><param name="movie" value="http://static.slideshare.net/swf/ssplayer2.swf?doc=scribe-4-1200016300702476-4"/><param name="allowFullScreen" value="true"/><param name="allowScriptAccess" value="always"/><embed src="http://static.slideshare.net/swf/ssplayer2.swf?doc=scribe-4-1200016300702476-4" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="355"></embed></object><div style="font-size:11px;font-family:tahoma,arial;height:26px;padding-top:2px;"><a href="http://www.slideshare.net/?src=embed"><img src="http://static.slideshare.net/swf/logo_embd.png" style="border:0px none;margin-bottom:-5px" alt="SlideShare"/></a> | <a href="http://www.slideshare.net/GreyM/scribe-4" title="View 'Scribe 4' on SlideShare">View</a> | <a href="http://www.slideshare.net/upload">Upload your own</a></div></div><br /><br />Hope you enjoyed!<br />The next scribe will be....<br /><br /><br /><br />Mark!<div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-5412415831077620067?l=apcalc07.blogspot.com' alt='' /></div>Grey-Mhttp://www.blogger.com/profile/10237241798100727258noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-7788757651668772372008-01-10T15:39:00.001-06:002008-01-10T15:51:47.267-06:00Here they are ...<br /><br /><center><br /><div style="width:425px;text-align:left" id="__ss_223969"><object style="margin:0px" width="425" height="355"><param name="movie" value="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-january-10-2008-1200001367363234-2"/><param name="allowFullScreen" value="true"/><param name="allowScriptAccess" value="always"/><embed src="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-january-10-2008-1200001367363234-2" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="355"></embed></object><div style="font-size:11px;font-family:tahoma,arial;height:26px;padding-top:2px;"><a href="http://www.slideshare.net/?src=embed"><img src="http://static.slideshare.net/swf/logo_embd.png" style="border:0px none;margin-bottom:-5px" alt="SlideShare"/></a> | <a href="http://www.slideshare.net/dkuropatwa/ap-calculus-slides-january-10-2008" title="View 'AP Calculus Slides January 10, 2008' on SlideShare">View</a> | <a href="http://www.slideshare.net/upload">Upload your own</a></div></div><br /></center><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-778875765166877237?l=apcalc07.blogspot.com' alt='' /></div>dkuropatwahttp://www.blogger.com/profile/08462283847470560887noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-63205572855255204462008-01-09T16:50:00.000-06:002008-11-18T18:01:23.534-06:00<span style="font-family:trebuchet ms;"><span style="color: rgb(102, 0, 204);">Hello today's class was a very short class due to <span style="font-weight: bold; font-style: italic;">Mr. K's </span>absence for most of the class. He had to attend a meeting. I am lucky that I do not have to do a very lengthy scribe.<br /><br /><span style="font-weight: bold; font-style: italic;">Mr. K </span>started off by asking us if this was true:<br />[see slide two for original slide.] </span></span><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://4.bp.blogspot.com/_0dApKxDd_VI/R4VYKgw673I/AAAAAAAAAGw/Rrfi3sfZfqk/s1600-h/slide+2.GIF"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://4.bp.blogspot.com/_0dApKxDd_VI/R4VYKgw673I/AAAAAAAAAGw/Rrfi3sfZfqk/s400/slide+2.GIF" alt="" id="BLOGGER_PHOTO_ID_5153622286212460402" border="0" /></a><span style="font-family:trebuchet ms;"><span style="color: rgb(102, 0, 204);">We then found out that it was not true. After that<span style="font-weight: bold; font-style: italic;"> Mr. K</span> asked us if we knew what derivative rule would give us a product in the for its derivative. We figured out that the chain rule would do just that. So instead of differentiating the function we would </span></span><span style="font-family:trebuchet ms;"><span style="color: rgb(102, 0, 204);">anti-differentiate it by running the chain rule backwards. <span style="font-weight: bold;">Remember: </span>the chain rule involves a composite of functions and do not forget to add the <span style="font-weight: bold;">constant.</span><br /><br />So to test this out <span style="font-weight: bold; font-style: italic;">Mr. K </span>gave us a number of problems to work on. The first three problems were pretty straight forward. All that had to be done was to run the chain rule backwards. The last three problems were a little bit different. There must be a product in order to run the chain rule backwards. If there isn't a product you can just simply multiply by the number one but you need to be clever about it. Here is an example[see slide three for original slide]: </span></span><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://4.bp.blogspot.com/_0dApKxDd_VI/R4Vgzgw674I/AAAAAAAAAG4/VPExzJ6PIP8/s1600-h/slide+3.GIF"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://4.bp.blogspot.com/_0dApKxDd_VI/R4Vgzgw674I/AAAAAAAAAG4/VPExzJ6PIP8/s400/slide+3.GIF" alt="" id="BLOGGER_PHOTO_ID_5153631786680119170" border="0" /></a><span style="font-family:trebuchet ms;"><span style="color: rgb(102, 0, 204);"> The last three problems were similar to the example shown above. Shortly after that, </span></span><span style="font-family:trebuchet ms;"><span style="color: rgb(102, 0, 204);"><span style="font-weight: bold; font-style: italic;">Mr. K</span></span></span><span style="font-family:trebuchet ms;"><span style="color: rgb(102, 0, 204);"> departed to his meeting and <span style="font-weight: bold; font-style: italic;">Ms. Pangan</span> came in to supervise us. We were assigned eleven question 1-21 odds only! If you did not finish those in class it is for homework. We also used the smartboard but it froze and somehow got <span style="font-weight: bold;">disconnalcted</span> [<span style="font-weight: bold; font-style: italic;">Chris</span>' word] [disconnected]! but have no fear <span style="font-weight: bold; font-style: italic;">Craig</span> saved the day and fixed it ! <span style="font-weight: bold;">ZING! </span>After that he successfully powered it off! I'm not quite too sure who has been scribe yet so I will ask in class and decide from there.<br /></span></span><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-6320557285525520446?l=apcalc07.blogspot.com' alt='' /></div>aichelle s.http://www.blogger.com/profile/17302966674461415089noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-62208824257225773112008-01-09T14:20:00.000-06:002008-01-09T14:53:41.134-06:00Here they are ...<br /><br /><center><br /><div style="width:425px;text-align:left" id="__ss_222929"><object style="margin:0px" width="425" height="355"><param name="movie" value="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-january-9-2008-1199911960272352-2"/><param name="allowFullScreen" value="true"/><param name="allowScriptAccess" value="always"/><embed src="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-january-9-2008-1199911960272352-2" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="355"></embed></object><div style="font-size:11px;font-family:tahoma,arial;height:26px;padding-top:2px;"><a href="http://www.slideshare.net/?src=embed"><img src="http://static.slideshare.net/swf/logo_embd.png" style="border:0px none;margin-bottom:-5px" alt="SlideShare"/></a> | <a href="http://www.slideshare.net/dkuropatwa/ap-calculus-slides-january-9-2008" title="View 'AP Calculus Slides January 9, 2008' on SlideShare">View</a> | <a href="http://www.slideshare.net/upload">Upload your own</a></div></div><br /></center><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-6220882425722577311?l=apcalc07.blogspot.com' alt='' /></div>dkuropatwahttp://www.blogger.com/profile/08462283847470560887noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-76665170448679745342008-01-07T16:52:00.000-06:002008-11-18T18:01:23.993-06:00First things First, I want to say happy holidays to everyone who sees this blog :) hope you had a good vacation.<br /><br /><br />I still don't know how to do those fancy slide show stuff so my post is going to be old school writing<br /><br /><div> The first thing we did in class was to review the work we were getting into before the break began. that means reviewing finding the anti derivatives, and the rules that go with them. We also got into definite integrals and indefinite integrals.</div><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://2.bp.blogspot.com/_fvK_Af4KmnA/R4L_By1OPKI/AAAAAAAAADM/IZPct_3bOoM/s1600-h/blog+%235_1.jpeg"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 455px; height: 352px;" src="http://2.bp.blogspot.com/_fvK_Af4KmnA/R4L_By1OPKI/AAAAAAAAADM/IZPct_3bOoM/s400/blog+%235_1.jpeg" alt="" id="BLOGGER_PHOTO_ID_5152961329955224738" border="0" /></a><br />We were then given some questions.<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://4.bp.blogspot.com/_fvK_Af4KmnA/R4MC2C1OPNI/AAAAAAAAADk/UGKfH4niUq0/s1600-h/blog+%235_2.jpeg"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 406px; height: 313px;" src="http://4.bp.blogspot.com/_fvK_Af4KmnA/R4MC2C1OPNI/AAAAAAAAADk/UGKfH4niUq0/s400/blog+%235_2.jpeg" alt="" id="BLOGGER_PHOTO_ID_5152965526138272978" border="0" /></a>and some more questions, lets take a look at this next question.<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://4.bp.blogspot.com/_fvK_Af4KmnA/R4MDeC1OPOI/AAAAAAAAADs/sJJM3r1_o-c/s1600-h/blog+%235_3.jpeg"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 481px; height: 373px;" src="http://4.bp.blogspot.com/_fvK_Af4KmnA/R4MDeC1OPOI/AAAAAAAAADs/sJJM3r1_o-c/s400/blog+%235_3.jpeg" alt="" id="BLOGGER_PHOTO_ID_5152966213333040354" border="0" /></a> For this one I imagine the question (6+9)/3 this we can break down into (6/3) + (9/3) then reduce that down to 2+3= 5 If you look at the red, he does the exact same thing as the question I imagined. Remember that this is a integral as it is in a closed interval.<br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://3.bp.blogspot.com/_fvK_Af4KmnA/R4MEyy1OPPI/AAAAAAAAAD0/7hCu0EcLrsM/s1600-h/blog+%235_4.jpeg"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 425px; height: 328px;" src="http://3.bp.blogspot.com/_fvK_Af4KmnA/R4MEyy1OPPI/AAAAAAAAAD0/7hCu0EcLrsM/s400/blog+%235_4.jpeg" alt="" id="BLOGGER_PHOTO_ID_5152967669326953714" border="0" /></a><br />The rest of these questions use the same method, remember the chart above with the rules of finding anti derivatives.<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://1.bp.blogspot.com/_fvK_Af4KmnA/R4MGYS1OPRI/AAAAAAAAAEE/kGvj_dajXgg/s1600-h/blog+%235_5.png"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 453px; height: 350px;" src="http://1.bp.blogspot.com/_fvK_Af4KmnA/R4MGYS1OPRI/AAAAAAAAAEE/kGvj_dajXgg/s400/blog+%235_5.png" alt="" id="BLOGGER_PHOTO_ID_5152969413083675922" border="0" /></a>So one last time have a great New Year.<br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://4.bp.blogspot.com/_fvK_Af4KmnA/R4MC2C1OPNI/AAAAAAAAADk/UGKfH4niUq0/s1600-h/blog+%235_2.jpeg"></a><div> </div><div> </div><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-7666517044867974534?l=apcalc07.blogspot.com' alt='' /></div>etimzhttp://www.blogger.com/profile/09626175119223517919noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-79415901079106514562008-01-07T16:01:00.000-06:002008-01-07T16:02:44.658-06:00Here they are ...<br /><br /><center><br /><div style="width:425px;text-align:left" id="__ss_220922"><object style="margin:0px" width="425" height="355"><param name="movie" value="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-january-7-2008-1199743080214742-4"/><param name="allowFullScreen" value="true"/><param name="allowScriptAccess" value="always"/><embed src="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-january-7-2008-1199743080214742-4" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="355"></embed></object><div style="font-size:11px;font-family:tahoma,arial;height:26px;padding-top:2px;"><a href="http://www.slideshare.net/?src=embed"><img src="http://static.slideshare.net/swf/logo_embd.png" style="border:0px none;margin-bottom:-5px" alt="SlideShare"/></a> | <a href="http://www.slideshare.net/dkuropatwa/ap-calculus-slides-january-7-2008" title="View 'AP Calculus Slides January 7, 2008' on SlideShare">View</a> | <a href="http://www.slideshare.net/upload">Upload your own</a></div></div><br /></center><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-7941590107910651456?l=apcalc07.blogspot.com' alt='' /></div>dkuropatwahttp://www.blogger.com/profile/08462283847470560887noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-13061034057495389062007-12-20T14:13:00.000-06:002007-12-20T14:21:08.046-06:00Well time to bob except this time I almost lost the mark. Thanks, to Ms. K for letting me use her computer to bob. Well this unit was very quick, a little bit too quick if you ask me. Well the part of the unit that had me a bit in a stir was assigned area questions, as i had become very confused over them <span style="color: rgb(204, 0, 0);"></span>and am still not sure if i am able to do them all. Most of the other units in these applications of the integrals unit were not very difficult. Well all in all it is time for me to get back to physics. <span style="color: rgb(204, 0, 0);">Good luck everyone, hope everyone had studied hard.</span><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-1306103405749538906?l=apcalc07.blogspot.com' alt='' /></div>Dinohttp://www.blogger.com/profile/16720903626604404687noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-4819182643337557292007-12-20T14:03:00.000-06:002007-12-20T14:10:20.147-06:00Oh my goodness! I was 20 minutes away from losing a mark on the test. Very sorry Mr. K., but I was focused on my World Issues project that was due today, so I forgot.<br /><br />Anyway, this unit was a pretty good one, simply because it is a combination of a couple units we have learned previously. These last two weeks were pretty stressful for me, so I was not fully awake for some of the classes for this unit. There were points that I was quite unclear of, but thanks to Mr. K. and the rest of classmates, as well as the practice problems, I am pretty confident with the material.<br /><br />If this is anything like the pre-test (I hope it is), this test should very fairly easy. However, Mr. K. always seems to throw in a question that throws me off, or maybe I just make simple mistakes.<br /><br />In any event, I'm looking forward to seeing how I do on this test, AS WELL AS THE RESULTS OF THE LAST ONE =D...<br /><br /><br />Good luck, Keep pushing, see ya<div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-481918264333755729?l=apcalc07.blogspot.com' alt='' /></div>«Craig»http://www.blogger.com/profile/08860268378924842699noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-5344916108337697772007-12-19T23:19:00.000-06:002007-12-19T23:23:18.686-06:00This unit was definitely very short! I understood a majority of the unit and some parts were difficult but that usually happens in every unit. This is our last bob and test before the winter break and I hope it goes better than the other tests we've done. Always do your homework and it will help you a lot! .. Good luck everyone.<div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-534491610833769777?l=apcalc07.blogspot.com' alt='' /></div>honey♥bunchnoreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-14989221682588413692007-12-19T23:17:00.000-06:002007-12-19T23:22:01.892-06:00I like this unit. Other than having difficulty understanding the notation, I'm getting the hang of it all. The workshop and pretest have really helped. So, a little bit of studying should be my cure. Good night everyone, and see you all tomorrow. Best of luck and great wishes.<div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-1498922168258841369?l=apcalc07.blogspot.com' alt='' /></div>Vanhttp://www.blogger.com/profile/07679810325423377354noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-28868540299588358432007-12-19T20:47:00.000-06:002007-12-19T20:55:10.580-06:00I personally think that this sixth unit was very short but compact with tons of information. I think that i will have very little trouble with this test (hopefully) . There was really nothing too difficult in this unit other than the fact that i might mess up on using the proper notations, so i need to work on that tonight after i study for chemistry. Thats it for my last BOB for this year. Good night and study well.<div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-2886854029958835843?l=apcalc07.blogspot.com' alt='' /></div>m@rkhttp://www.blogger.com/profile/00509544958003356512noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-43819995175626959222007-12-19T18:45:00.000-06:002007-12-19T18:59:35.218-06:00<span style="color:#000000;"><span style="color:#000099;">MrSiwWy </span>here for my sixth and final bob before the winter break, though I almost forgot to bob for this test tonight, here it is. Now, given the difficulty of the yesterday's workshop as well as today's pre-test, I don't think that this test is going to be incredibly difficult. As long as you've stayed current with your homework, as I have really tried recently, the test shouldn't bring anything that we're entirely unfamiliar with or anything overly challenging. There weren't too many problems that I found a lot of trouble with while I was completing the exercises, so I'm not too worried about this calculus test, though I am worried about my self Physics C test later tonight (ahh scary). I can't wait until the winter break, though I'm sure I'll be busy all break with homework and definitely busy with Mechanics and Computer Science throughout, so thankfully I won't need to worry about calculus any time soon (at least I hope not). Don't forget to study everyone, and I wish <span style="color:#ff0000;">good luck</span><strong> </strong>to everyone and wish everyone to perform to the best of their ability on tomorrow's test. Have a great night all!</span><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-4381999517562695922?l=apcalc07.blogspot.com' alt='' /></div>MrSiwWyhttp://www.blogger.com/profile/18072494079396275311noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-53556764570371284802007-12-19T16:10:00.000-06:002007-12-19T16:14:27.192-06:00Hey everyone! Well once again another unit has gone by and it's that time again to bob and also study hard to do well on the test. This unit for me was nice and short and wasn't the hardest of the bunch. I grasp the concepts and I hope to do well on the test. Good luck to you all and have a fun time studying tonight. Bye!<div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-5355676457037128480?l=apcalc07.blogspot.com' alt='' /></div>Robert Phttp://www.blogger.com/profile/03046070808724541650noreply@blogger.com0tag:blogger.com,1999:blog-5788051255938485746.post-81168452707989772302007-12-19T15:33:00.001-06:002007-12-19T15:36:13.462-06:00Here they are ...<br /><br /><center><br /><div style="width:425px;text-align:left" id="__ss_208303"><object style="margin:0px" width="425" height="355"><param name="movie" value="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-december-19-2007-1198099943617832-4"/><param name="allowFullScreen" value="true"/><param name="allowScriptAccess" value="always"/><embed src="http://static.slideshare.net/swf/ssplayer2.swf?doc=ap-calculus-slides-december-19-2007-1198099943617832-4" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="355"></embed></object><div style="font-size:11px;font-family:tahoma,arial;height:26px;padding-top:2px;"><a href="http://www.slideshare.net/?src=embed"><img src="http://static.slideshare.net/swf/logo_embd.png" style="border:0px none;margin-bottom:-5px" alt="SlideShare"/></a> | <a href="http://www.slideshare.net/dkuropatwa/ap-calculus-slides-december-19-2007" title="View 'AP Calculus Slides December 19, 2007' on SlideShare">View</a> | <a href="http://www.slideshare.net/upload">Upload your own</a></div></div><br /></center><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/5788051255938485746-8116845270798977230?l=apcalc07.blogspot.com' alt='' /></div>dkuropatwahttp://www.blogger.com/profile/08462283847470560887noreply@blogger.com0