Thursday, June 26, 2008

So Long ...

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.

I'm so glad we've had this time together,

Just to have a laugh or learn some math,

Seems we've just got started and before you know it,

Comes the time we have to say, "So Long!"

So long everybody! Watch this space in the fall for pointers to new blogs for each of my classes.

Farewell, Auf Wiedersehen, Adieu, and all those good bye things. ;-)

Tuesday, June 17, 2008

Class Survey

The 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 email me or leave a comment below this post.

Without any further ado, here are the results of our class's survey. Please share your thoughts by commenting (anonymously if you wish) below .....

Classroom Environment
The questions in this section were ranked using this 5 point scale:

Strongly DisagreeDisagreeNeutralAgreeStrongly Agree

The bold numbers after each item are the average ratings given by the entire class.

1. The teacher was enthusiastic about teaching the course. 4.86

2. The teacher made students feel welcome in seeking help in/outside of class. 4.71

3. My interest in math has increased because of this course. 4.14

4. Students were encouraged to ask questions and were given meaningful answers. 4.57

5. The teacher enhanced the class through the use of humour. 4.43

6. Course materials were well understood and explained clearly by the teacher. 4.29

7. Graded materials fairly represented student understanding and effort. 4.29

8. The teacher showed a genuine interest in individual students. 3.71

9. I have learned something that I consider valuable. 4.86

10. The teacher normally came to class well prepared. 4.43

Overall Impression of the Course
The questions in this section were ranked using this 5 point scale:

Very PoorPoorAverageGoodVery Good

1. Compared with other high school courses I have taken, I would say this course was: 4.57

2. Compared with other high school teachers I have had, I would say this teacher is: 5.00

3. As an overall rating, I would say this teacher is: 4.86

Course Characteristics

1. Course difficulty, compared to other high school courses:

Very Easy
Very Difficult

2. Course workload, compared to other high school courses:

Very Easy
Very Difficult

3. Hours per week required outside of class:

0 to 2
2 to 3
3 to 5
5 to 7
over 7

4. Expected grade in the course:


AP Exam Preparation
The bold numbers after each item are the average ratings given by the entire class.

How prepared were you to write this exam? 75.6%

How much effort did you put into preparing for this exam? 73.1%

How good a job did your teacher do preparing you for this exam? 90.1%

Did you have enough preparation using your calculator?


Did you have enough preparation without using your calculator?


Was your teacher too hard or too easy on you?

Too Hard
Too Easy
Just Right

Specific Feedback
[Ed. Note: Numbers in parentheses indicate the number of students, over 1, that gave the same answer.]

What was your best learning experience in this course?

Enthusiasm, effort, and humour by the teacher (2)
Workshop classes
Pre-Tests (3)
Developing Expert Voices project
This blog
Everything except the Developing Expert Voices project
Teaching others and explaining concepts
Exam Prep
Scribe posts were awesome
Group learning

What was your worst learning experience in this course?

Developing Expert Voices project (2)
Falling behind due to the fast progress of other classmates
Not being able to do homework in class
The wiki solutions manual
Lectures were not visual enough
When substitute teachers wouldn't let us talk to each other

What changes would you suggest to improve the way this course is taught?

Return tests more quickly
Go over solutions to every test question always
Give regular "marks updates" so I know when I need to invest a lot or a little effort
Cancel the Developing Expert Voices project
More practice work for home posted to the blog
Cancel the wiki assignment
Slower explanations that hit all key points
Tell students how fast their Developing Expert Voices dues dates creep up on them
More group learning, I learned a lot in groups
Have a review class every couple of units

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 ....

Sunday, June 15, 2008

Student Voices Episode 4: Justice, Lawrence, and Richard

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 ...

Episode 1: Jessie 2440 downloads

Episode 2: Tim_MATH_y 1766 downloads

Episode 3: Chris, Craig, Graeme 1367 downloads

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.

In this episode of Student Voices Justice, Lawrence, and Richard talk about how they put together their Developing Expert Voices project 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.

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.

(Download File 12.2Mb, 25 min. 30 sec.)

Photo Credit: Shadow singer by flickr user EugeniusD80

Monday, May 12, 2008

Student Voices Episode 3: Chris, Craig, and Graeme

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 Developing Expert Voices project. It ends with a challenge, the result of which will be featured in a future podcast.

Let Chris, Craig, and Graeme know what you thought about the podcast by leaving a comment here on this post.

(Download File 31.8Mb, 26 min. 30 sec.)

The video mentioned near the end of the podcast is called Daft Hands. Here it is:

Photo Credit: Shadow singer by flickr user EugeniusD80

Sunday, April 27, 2008

Student Voices Episode 2: Tim_MATH_y

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.

Please feel free to leave Tim_MATH_y your comments here on this post.

(Download File 7.2Mb, 15 min. 3 sec.)

Photo Credit: Shadow singer by flickr user EugeniusD80

Wednesday, April 23, 2008

Today's Slides: April 23

Here they are ...

The scribe... kinda

Hello 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.

That's about it.

Next scribe... Dino...

Saturday, April 19, 2008

Student Voices Podcast Episode 1: Jessie

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.

This is the first in what I hope will be a series of podcasts called Student Voices. 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.

In this episode Jessie shares how she uses her class blog 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.

Please feel free to leave Jessie your comments here or on this post on her class blog.

(Download File 5.6Mb, 11 min. 40 sec.)

Photo Credit: Kids of conversation by flickr user Kris Hoet

The Return of the Lost Scribe... whoops

So now we are well into exam review and for good reason. (There are only 6 classes left until the exam).

On Thursday our class consisted of five accumulation problems from the AP exams of five different years.

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.

The more important part of the exercise was to notice how the same type of problem changed over the different years. Looking at them, 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.
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.
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.
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.

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.

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

The next scribe is Van I guess...

Enjoy the rest of your weekends!

Thursday, April 17, 2008

Today's Slides: April 17

20 Days Until The Exam And Counting ...

Here are the slides ...

Saturday, April 12, 2008

Carpe Diem!

Craig mentioned in his BOB:
Anyway, this is the last unit and from now on it's all exam review.
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!

How's your hold on calculus?

image source

Thursday, April 10, 2008


Woah! We had the test moved back two days and I still almost forgot.
But yes, this last unit has been okay mainly because it is just an elaboration on material we've already covered.
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.
But, it's nothing a little studying won't fix.
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
Anyway, this is the last unit and from now on it's all exam review.
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!

Good Luck to all on the test tomorrow, and good luck to all on the upcoming exam.

Keep On Pushing!

Wednesday, April 9, 2008

BOB ^ 9

Oh 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 (=.

It's been a crazy year crammed full of information, and this final unit was certainly no exception to working hard. This unit, differential equations, 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.

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 slope fields, solving differential equations, separating the variables (which greatly aids the solving process) and Newton's Law of Cooling. 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.

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 I wish everyone good luck on the test tomorrow!

Tuesday, April 8, 2008

Blogging on Blogging

Oh 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 =(

The last BOB

Well, 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 anti differentiating first-order and second-order equations although that also is a working progress which I should have figured out by tomorrow. Well good luck everybody for tomorrow's test and good luck.

Monday, April 7, 2008



***Craig-4th (belated)***
***Chris-13th (advanced)***

Much love ! =)
Good luck with your DEV =)

---Aichelle ROCKS!

Today's Slides: April 7

Here they are ...

Sunday, March 23, 2008

SCRiBE: Last Of Content

Hello everyone! I'm known as Tim-math-y and I'm the scribe for Thursday's lessons. On Thursday, our topic was: Solving Differential Equations Symbolically and Newton's Law of Cooling.


Solving Differential Equations Symbolically and Newton's Law of Cooling:

On Thursday, we learned more about differential equations, this time using the symbolic (algebraic) approach. This topic tied into Newton's Law of Cooling. According to this law, a hot object cools at a rate proportional to the difference between its own temperature and that of its environment.

By applying this concept, we solved questions that dealt with cooling.

Content and Lessons:

First we began by deconstructing the equation of "dy/dx = ky." 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.

By seperating the two variables, including both 'dy' and 'dx', we found a familiar term that could be antidifferentiated.

Through algebraic play, we found the equation that defined the parent function of cooling.

*Note: the variable "c" and "C" are different in value but generally both represent a constant.

Next we looked at an example question.
  • First we assigned variables, F = Temperature in Degrees of F and t = T time in Hours
  • Second we applied the variables to "dy/dx=ky" to produce "dF/dt=k(F-20)" where 2o is the lowest temperature the object can attain in an environment
  • By applying the same steps to seperate 'dF' and 'dt', we antidifferentiated and worked the out an 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.
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.

At this point, we considerred both questions solved.

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 paragraph 3).

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. We found the equation that still included the constant value of C.

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.

Finally, by inputting the temperature of the cola at an unknown point in time, we solved for the uknown value of 't'.


We went through two problems that dealt with Newton's Law of Cooling, solving differential equations symbolically.


Homework is on the end of the slides posted in the previous post! Answer is on the following slide.

Tomorrow's Scribe is... Craig!

Happy Easter? Haha, good night.

Wednesday, March 19, 2008

The Edmonton Eulers Method

Hey 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, while text in green will pertain to explanations using an example for the first part of the post and text in blue will pertain to explanations using an example for the second part of the post. Hope it works out well. Enjoy!


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; Euler's identity.

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), i, and e. Euler's identity is as follows:

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 here. If you follow the link, it might ease the subsequent explanations tremendously.

- 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.

*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.

- 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.

*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 the slope of the line tangent to the function at that point. This is because the differential equation will solve for y'(t), which is the derivative of y(t) or rather the instantaneous rate of change of y(t) at a given t. This is particularly useful since 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:

y'(0) = 1 - (0) + 4(1)
y'(0) = 5

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 pay careful attention to the steps or the scale of each axis 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.

- 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 (or in this case t) 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 this website or this website 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 dx values (or dt) 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.

*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.

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:

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.

*Basically, when you were given the initial point, you could plug the point into the differential equation (which is y' = y - x this time) to find the derivative of the function at that point. This derivative will be m, or the slope of the tangent line at that point, which can be used by rearranging the m = Δy/Δx equation into Δy = m * Δx. Since m has just been calculated, and Δx (which is the step as I mentioned above) is given in the question as 0.25. Calculating Δy 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 Δy 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 each row will be one iteration of determining the solution to the differential 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 detailed section of this scribe post).

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:

- Store the initial y value into any variable in your calculator, say A.

- 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.

*Applying this step to the slide above, here's what it would look like in your calculator:
(((A - 0)0.25) + A) -> A

- 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.

*Again applying this step to the slide above, here's what the next line would look like:
(((A - 0.25)0.25) + A) -> A

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.

So then we tried using more segments to approximate a solution that is much closer than achieved above. We did this by using our new found 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).

Just before class ended, Mr. K distributed the "EULER" calculator program for us to use to quickly solve problems involving Euler's method.

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.

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:
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!

Monday, March 17, 2008

Article 13, Al Upton, and the minilegends

If 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.

You can also read more about Article 13 and the Convention on the Rights of the Child here. This movie illustrates what it's all about:

The Pi day scribe

Well, a certainly enjoyable pi day it was. Lots of treats and Friday!. Doesn't get much better than that.

With all the celebrating, we didn't do very much.

We learned, the slope fields were the graphs of the parent function. Just moved all over the place.

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)

Next scribe.


Friday, March 14, 2008

Today's Slides: March 14, π Day!

Here they are ...

Happy π Day!

Happy π Day Everyone!!

Have some π Pie!

Absolutely Late Scribe..... Sorry guys.

WOW! Very, very sorry about the tardiness of this scribe... I worked both last and this evening and today was very busy.

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.

Then we briefly discussed the Developing Expert Voices projects as Mr. K. handed back the proposals.

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.

Finally we started the lesson for the day. The first thing we did was a quick review of Differential Problems.
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.

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).
From there it is as simple as finding the positive root of that function and that will be the answer.

Next we moved onto the new content.

these are kind of tough to explain... I'll let some pictures speak some of my words.
This is the FIELD in which we plot the SLOPES. Easy enough right?
So we first tried it with the expression ( ∂y/∂x = y ). This means that the derivative of any point in the function is the value of the function at that point (y-coordinate).
•Let's say we want to plot the point (1,1). The derivative (SLOPE) of the function at that point is 1, so we draw a small line with slope 1 at that point.
•Next, take the point (3,2). The derivative at that point is 2, so there's another small line, but with a slope 2 at the point (3,2).
Now, you understand how they are constructed, this is how the SLOPE FIELD of the function(s) that have ( ∂y/∂x = y ).
From here, you can solve for the point (1,1).
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???
Correct! It is easy to see that the function that has a derivative equal to the value of the function and passes through the point (1,1) is y=e^x.
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.
This is because the SLOPE FIELD shows the entire family of functions [ ∫ƒ(x)∂x + C ].
So, by solving for a point, you can find the exact one function from that family.

We also did SLOPE FIELDS for ∂y/∂x = 2x and ∂y/∂x = -x/y... can you guess what functions they are???

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.

This concludes my scribe post, once again, sorry it was so late.

The scribe for PI (π) Day will be...


Tuesday, March 11, 2008