Section 2.1

There are lots ways to say "derivative" - Instantaneous Slope, Velocity, etc. The first section in chapter two focuses on finding it by means of a tangent line to a curve. As we know, the slope of a line is equal to the "rise" divided by the "run". or the change in Y divided by the change in X. But while this system works for finding the slope of a straight line, we have to use it in a very specific way to find the slope of a given point on a curve. For instance, to find the slope at X=1 on the line Y=X^2...

• We must have two points to find the slope of any line. Since we are looking for the slope at X=1, one of our points will be (1,1)
• To find the second point, we should try some different points. First, let's use (2,4)
• The slope between (1,1) and (2,4) is 3/1. Let's see how it changes as our second point gets closer to (1,1)
• If our second point is (1.5, 2.25), the slope is 1.25/.5, or 2.5
• If we get ridiculously close to (1,1) then our point could be (1.0001, 1.00020001), in which case our slope would be .00020001/.0001, or almost exactly 2.
• As we can see, the closer our second point gets to 1, the closer the calculated slope is to 2
• This can be expressed in the equation: Slope f(x) x->a = (f(x)-f(a))/(x-a) , where a is the point at which you want to find the slope, and x is a point that is (ideally) infinitely close to a.

Two interesting websites that I found that illustrate the concept of tangent lines in relation to instantaneous slope (both require the Java plugin).
http://www.ies.co.jp/math/java/calc/doukan/doukan.html
, and a much more technical version,
http://www-math.mit.edu/18.013A/HTML/tools/tools04.html

And now for the first of my series of the greatest and most meaningful quotes of all time...
“Then after World War Two, it got kinda quiet, ‘til Superman challenged FDR to a race around the world! FDR beat him by a furlong, or so the comic books would have you believe. The truth lies somewhere in between…”
—Abe Simpson (Better known as Grandpa Simpson)
And on that note, Lauren has the next post...