CPM Homework Help : PC Problem 9-81

Mathematics is full of all kinds of different functions, but it is comforting to know that some things remain constant for all of them. For example, to find the slope of the tangent line of a function at $x=2$ , you can follow the same procedure, regardless of the function. For each of the following functions, first find the formula for a secant line for the function from $2$ to $2+h$ , then let $h→0$ to find the slope of the tangent line at $x = 2$ . Use your calculator to estimate the slopes to three decimal places. Enter a small value for h (such as h = 0.001) in the slope formula to approximate your answer.

Hint:

$m=\frac{f(2+h)-f(2)}{h}$

$f(x)=2x^2$
Step (a):
$m=\frac{2(2+h)^{2}-f(2)}{h}$

$f(x)=3^x$
Step (b):
$m=\frac{3^{(2+h)}-3^{2}}{h}$

$f(x)=\log(x)$
Step (c):
$m=\frac{\log (2+h)-\log(2)}{h}$

$f(x)=\cos(x)$ ( $x$ in radians)
Step (d):
$m=\frac{\cos(2+h)-\cos(2)}{h}$ Use the eTool below to help you with this problem.

Use the eTool below to help you with this problem.
Click the link at right for the full version of the eTool: 9-81 HW eTool