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AP Calculus AB

An interactive log for students and parents in my AP Calculus class. This ongoing dialogue is as rich as YOU make it. Visit often and post your comments freely.

Tuesday, October 11, 2005

All About the Derivative Function

Today in class we learned how to calculate the derivative of a function several different ways. One way to find the derivative, is by using the symmetric difference quotient. All you need is a table of values and then you simply calculate the forward secant line and the backward secant line, then take the average of the two and that is usually a fairly accurate way to determine the derivative. An example we used in class was:

f(x)= x^2 -6x +10

x y
0 10
1 5
2 2

Delta y/ Delta x
(2-5)/(2-1) = -3
(5-10)/(1-0) = -5


((-3)+(-5))/2 = -4

Another way of finding the derivative is by using the definition of the derivative with algebra to solve for d.
First off, evaluate using the function f(x) = x^2 -6x +10 for f(1) and for f(1+h).

f(1) = (1)^2 -6(1) +10
f(1) = 5


f(1+h) = (1+h)^2 -6(1+h) +10
f(1+h) = 1+ 2h +h^2 -6 -6h +10
f(1+h) = h^2 -4h +5


lim (f(1+h)- f(1))/h
h->0


lim ((h^2 -4h +5) -(5))/h
h->0


lim (h^2 -4h)/h
h->0


lim (h(h-4))/h
h->0


lim h-4
h->0

Assume h = 0, therefore d = -4
.

You can determine a function that will give you every derivative value for the function you are using. You use the formal definition of a derivative to solve for f^1 (x).

f(x) = x^2 -6x +10
f(x+h) = (x+h)^2 -6(x+h)+10
f(x+h) = x^2 +2xh +h^2 -6x -6h +10

lim ((x^2 -6x +10 +h^2 + 2xh -6h)-(x^2 -6x +10))/h
h->0

lim (h(h +2x -6))/h
h->0


lim h +2x -6
h->0


f^1(x) = 2x -6

You can also draw a tangent line if you are given a graph that is draw to scale. Your ruler can be important if you have a graph.
Your calculator has a SLOPE program and you can use this program to calculate the derivative but it is still only an estimate. You can also use your calculator to draw a tangent line on the graph of the function.
In case you forget what the formal definition of a derivative is:

(f(x+h)-f(x))/h

The next scribe is ... Jayson.



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