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[I your author am not a mathematician; I am merely an admirer of Edwards & Penney's wonderful calculus book. Please consult the text or other references to be certain about anything in the summary below. I mean this emphatically.]
Calculus
Chapter 2. Prelude to Calculus
Section 2.1. Tangent Lines and Slope Predictors
When we have any general graph expressing y = f(x), we do not necessarily have some radius that will allow us to easily find a tangent. However, "the line tangent to the graph at the point P should be the straight line through P that has -- in some sense -- the same direction at P as the curve itself." (p.54a)
A line's slope tells us its direction. So to determine the line tangent to a curve, we will find a "slope-prediction formula" that will tell us the tangent's slope.
Example
"Determine the slope of the line L tangent to the parabola y = x2 at the point P(a, a2)." (p.54b)
Below we see the graph for y = x2
The first b is on the x-axis. Because it is found by adding to a the change in x, it is defined as
The b2 is found on the y axis. It is found through the function y = x2. So we define it as
because (a + h) is how we get the x value for the function y = x2 at point Q.
And because the y coordinates are found through the function y = x2, the change in y between P and Q is
The slope of a line we call m. And the slope is the change in y over the change in x.
We know both the change in y and the change in x for secant line K, so we can write the formulation.
In the numerator, we multiply (a + h) by itself to get (a2 + 2ah + h2). We then subtract out the a's in the denominator to leave just h.
We then subtract out the two a2's in the numerator to get 2ah + h2.
Then finally we factor out the h from the numerator to get h(2a + h).
We then cancel the h's from the numerator and the denominator, and thus the slope of secant K is
Now consider if we move point Q toward point P along the curve, which by the way is the same as h (change in x) approaching zero.
We want to define tangent line L as "the limiting position of the secant line K." (p.55d) Then:
"As h approaches zero,What we want to know is, as h approaches zero, what value is slope mPQ = 2a + h approaching? So we are looking for the "limiting value" of 2a + h.
Q approaches P, and so
K approaches L; meanwhile,
the slope of K approaches the slope of L"
Here lim stands for "limit", and "h → 0" stands for "h approaches zero". What the above formulation asks is "What is the limit of 2a + h as h approaches zero?" (p.55b)
We first consider what if a were either 1 or -2.
We would see that if a were 1, then 2 + h tends toward 2 as h tends toward zero. And if a were -2, then 2 + h tends toward 2-4 as h tends toward zero.
So as we see for the values for slope 2a + h,
we may say, more generally, that
Thus the "slope m = m(a) of the line tangent to the parabola y = x2 at the point (a, a2) is given by
The above formula is the "slope predictor" for tangents to parabola y = x2. "Once we know the slope ofthe line tangent to the curve at a given point of the curve, we can then use the point-slope formula to write an equation of this tangent line." (p.56d)
Text summary and images from:
Edwards & Penney: Calculus. New Jersey: Prentice Hall, 2002, pp. 54-56.
Geometrical Derivative animation:
http://faculty.uncfsu.edu/msiddiqu/Maple_Animations.htm
http://faculty.uncfsu.edu/msiddiqu/images/images/Gif_Folder/Definition%20of%20Derivative18.gif
Thanks Dr. Siddique of Fayetteville State University
Thanks you very much ! This article help me very well .
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