AP Calculus 9/2/10

After being forced to be the first scribe of the year, I am now writing the scribe post that will set the bar for the rest of you. I ensure you, the bar will be high.

Today in class, we started out by walking through an investigation about the Instantaneous Rate of Change of a Function with a partner for the beginning of class.

This investigation begins by giving us an example of a door closing automatically, and asking us to examine the rate at which the door closing,and the relationship between the rate of the door closes and the time that it takes.

The class worked on this investigation for a while, until many people got caught on #6, because of the idea of instantaneous rate of change.

Mr. O'Brien stopped the class, and explained that instantaneous rate of change is another way of saying, "the d word", or Derivative. We learned that derivative is the instantaneous rate of change of a function.

To show us this, we started a quick class activity in Geogebra.
We had graphed the investigation, and had also made a table of values:






After going through a little of the investigation (testing values, using a slow method which gives a rough estimate of the derivative). This method was using a shorter and shorter time interval to get closer to the derivative. He then showed us that we could find the instantaneous rate of change by evaluating that limit, which is approximately this:

However, he explained that we wouldn't be able to understand this solution until later in the year, so we may want to come back to look at it then.

A possible solution to finding derivatives is using the nDeriv action on our calculator. Under the math tab, you can type: nDeriv (Y1, x, {the number you want the derivative for}, {the closeness you want it to search for, i.e. .00001})

for this problem, we used nDeriv (Y1,x,1, 0.0001)

Up until this point, we had only found a rough estimate of the derivative. Now the class looked at how to find a very close approximation of the derivative. Although we haven't found the actual derivative, we were being shown ways to find an accurate estimate, in a short amount of time. These methods will be very helpful in the following unit.

After learning about this, we tried to relate the derivative back to what we did last class, the tangent line. We did this because the slope of the tangent line is infact, the derivative. To do this, we used the point slope form to find the tangent line, which gave us a line tangent to the point (1, 100). Once we know the slope of the tangent line, we know what the derivative is for the point that it touches, and this will be very useful.

After finding this line, we moved on to Limits. When working with a limit, it will always have a function in it. There are three parts to a limit: the letters lim, a function, and and x with an arrow telling you what number the limit is approaching.
To see an example of some limits, you can look in your textbook at pg. 72

In the end, we defined that limits are a tool to find derivatives, and much more.

To show us what a limit really is, we followed a link through the blog, explaining to us what a limit is. http://archives.math.utk.edu/visual.calculus/1/limits.16/tut1-flash.html

After going through several examples, we realized that a limit has certain conditions.

As stated by the website:
An intuitive definition of Limit is:
If f is a function and a and L are numbers such that
1. if x is close to a but not equal to a, then f(x) is close to L
2. As x gets closer and closer to a but not equal to a then f(x) gets closer and closer to L
3. We can make f(x) as close to L by making x close to a but not equal to a.

This website was very helpful because it gave us a clearer idea of the relationships between Limits and problems we might be going over.

We then went over several example problems of what we will be going over on the homework for tonight. These problems were simple limit problems, designed to help us get a grasp of the concept before we dive in tonight.

These problems examined the idea that you can imput the function into your grapher, and in the table option on your TI-84,89 etc. you can imput values very close to the value the limit is going towards. Then, by imputing these numbers, you can see where the gap it, and make an accurate estimate of what the limit is.

Overall, today was a good day of reinforcement on what exactly derivatives are (and what they do), along with some introduction about limits, which when understood, are less complicated then they seem.

If you still don't understand limits and derivatives after this class, check out these websites, they can help:

http://www.coolmath.com/lesson-whats-a-limit-1.htm

http://mathforum.org/library/drmath/view/53398.html

Next scribe: Julia.

HW:
* p. 72/1, 3, 25, 41, 49, 53, 57, 71, 73, 75 using calculator's table feature to evaluate limits.
* Practice for Tuesday's trig values quiz by doing 10 Quizlet multiple choice questions until you can consistently get 100% in 3 minutes. Quizlet