Slow driver, or maniac driver? That is the question...

Hello everybody, I'm MrSiwWy and I'll be scribing for today's (and yesterday's) class(es). I hope everyone out there is doing great and feeling well. I was going to title this post "the indrocution to the derivative", but I think that would have been a tad too bland for a class such as today's. Well, enough about the unnecessary, without further ado, here is my long awaited (Ha! ya right) scribe post.

Tuesday, September 25, 2007

During this class, we attempted to complete as much of the review as we could. And though much of the class would deem this class to be a "slack" class, I think everyone did their share of their own work which undoubtedly benefitted each of us in terms of perparation for the upcoming test. Now, during this class, in case anyone was absent (though I don't believe so) was essentially a review class consisting of a work period on questions from the textbook (pg.76-82). Though some of the time was consumed by the utterly gravitational math riddles Van put up on the whiteboard, most of us completed a considerable amount of work in class. There weren't really any problems that we encountered, except for the "bump" in question number 7, so the description of this class is actually quite compendious and sufficient as of this point. Thus, it's time to move onto the subsequent class on September 26.

Wednesday, September 26, 2007

Class began as usual, we all casually made way to our habitual abode in the calculus classroom, though some of us could vaguely hint upon an incoming lesson for the day, despite expecting another review period for this class. Many of us were proven wrong when Mr. K zealously presented to us the single, seemingly monumentous slide which our entire class would encircle upon. The slide of course consisted of only a graph, and after Mr. K quickly went over a question concerning one of the homework questions (question 7 on pg. 77)* we were fully enveloped by the day's lesson.

*Mr. K was explaining the question which I also referred to as before as the question with the "bump". Craig asked how one would come about sketching that graph without using a calculator, since the bump on the graph crossed a horiztonal asymptote and this shape would be terribly difficult to determine without a graphing instrument. Mr. K basically said that sketching this graph normally, as we have learned to do so in the past, would be sufficient to approximate specific qualities (such as asymptotes, domain, range, etc.) and thus be completely alright.

Onto this so-called monumentous slide. Though there was quite a lot of information implanted onto the graph after it's reveal, here is how the slide looked at the end of the class (I will of course outline the progression of the lecture and convey the implications and significance of the aforementioned information shortly):Mr. K began projecting to us the meaning of the distance vs. time graph displayed by assigning a story to definitively represent the data. He eventually decided upon Mark and I journeying along the road from Winnipeg to Winnipeg Beach. And he then asked the class to determine the average velocity of the graph, and as observed from the above graph, a total distance of 100 m was covered over a time interval of 2 hours, therefore the average velocity was quickly calculated to be 50 km/hr. Mr. K then elaborating upon how this average was attained, through the calculation of the slope over the interval at hand, or rather the amount of distance traveled during the given amount of time. He specifically explained (as it is a vital concept in the comprehension of a derivative) that these lines (that join the graph at exactly two points, as we shall also see later) are referred to as secant lines. It easy to think of secant lines in terms of tangent lines that meet at two points instead of one, as to determine the slope over an interval instead of at one specific point (through the direct utilization of a tangent line).

He then began asking questions pertaining to the overall volubility of the function, for one instance, he asked us what kind of driver did we conclude Mark to be with respect to the average velocity of the journey. We all essentially replied with a unified "slow driver", though Mr. K began to dissect this comment and attempted to prove otherwise. He then started to ask us for the average velocity of Mark's driving endeavor over specific intervals, such as depicted above by the green triangles. We determined the average velocities to be 40km/hr and 60 km/hr respectively.

Subsequent to the identification of the velocities over specific intervals, Mr. K then asked what happened at approximately 45 minutes into the journey, and why this occurred. The answer to this is of course due to the fact that the car stopped, since no distance was covered over the time interval from 45 minutes to about 75 minutes. We also know that Mark stopped the car because the graph representing out travel's progression began to flatten out, meaning that we didn't move for that amount of time. The class then decided that Mark and I could have stopped at a gas station, or even could have stopped to eat before we continue on our quest to Winnipeg beach.

We began to dissect the graph over specific intervals once again, but this time over smaller intervals at specific portions of the graph. As denoted by the many coloured triangles in the graph, as well as the corresponding colour-coded velocity values, these values change over different intervals on the graph. This caused the class to suggest that I took over the car after we stopped, and I of course was a maniac driver on the road. This was derived from the fact that as we calculated the average velocity for the intervals subsequent to the "pit stop"/plateau found roughly in the middle of the travel, we determined them to become increasingly grand. The velocity of the car rose very significantly as we shortened the intervals of which we analyzed. This then led to the introduction of the key concept of the day's class; the derivative.

The above image is of a zoomed-in portion of the graph (upper-right portion), which was used to then transition how we would calculate the velocity for time intervals in units of seconds (such as 1 second, or 0.5 seconds). As we may know, seconds are extremely short amounts of time, and calculating this velocity is approximately and yet increasingly similar to determing the instantaneous velocity immediate to that interval. This meaning that since we are calculating the ratio of the change in distance to the change in time, as the change in time decreases, the distance values are ever closer, narrowing into some specific value.

We attempted to absorb the true connection of the secant line and it's importance to the concept of the derivative to a certain extent. Mr. K then altered the distance and time variables of the graph help dignify this connection by explaining how we are determining the rate of change within the graph.

This graph illustrates this explanation. The alteration is visible as a function of the volume of a balloon in terms of it's radius as it is being filled. The graph portrays how we would expect to determine the rate of change of the overall filling process over intervals, on average, or as a cumulative generality. This rate of change was given by the triangles we found earlier, which were formed by specific secant lines. As shown below, and through the above explanation of determining increasingly miniscule time intervals of units in seconds, Mr. K once again elaborated on the secant line and revisited an aforementioned comparison.

From the long magenta line, which is a secant, we can see that as we found smaller and smaller intervals (leading to smaller secant lengths) we achieved larger average velocity values. But as we also outlined, examining these intervals using secant lines (determining the slope/rate of change) with an increasingly insignificant amount of change, we can see how the line then becomes closer and closer to acting as if it were a tangent line. This means that by using secant lines with an ascendingly small change in values, is like determining the instantaneous velocity approximate to that point. As this change approaches an infinitely small amount, the instantaneous rate of change becomes more apparent and more accurate. This, in essence, is the concept of the derivative.

To conclude our class, we shortly discussed what day our first calculus test of the year lands on, which is friday in case anyone missed that, and we also covered what questions will be due for homework tomorrow (chapter 1, pages 76-82) and the homework for the weekend on derivatives (exercise 2.1, all odd questions including 6 and 12). I will now conclude this scribe post by conveying the aptly subtle, yet beautiful analogy Mr. K used to contrast the significance and vastness of the concept of the derivative to another profoundly familiar concept. He stated that his son asked him what a decimal was, and Mr. K simply explained to him that a decimal is used to indicate values less than 1 instead of using fractions. He said that decimals have a more universal and prolific usage than just to indicate number less than 1, but to comprehend the subject fully, one must start somewhere and build up their knowledge. This is the same as the derivative, though we can analyze it's uses and it's essential meaning and implications, it's just the birth of our progressively intricate understanding of the derivative.

Enough of that, I think I covered the entire class now, I hope I helped anyone if they didn't understand something in class, or if anyone other than my classmates that require help, have come no further but to seek sufficiently helpful information here at our very own blog. Please, feel obligated to leave any comments, feedback, corrections or questions that you may have. But for now, I bid everyone goodday and have a great night!

The next scribe will be Mark!