CPM Homework Help : PC3 Problem 7-126

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 calculate the slope of the line tangent to any function at $x=2$ , the same procedure can always be used, regardless of the function. For each of the following functions, write a formula for the slope of the secant line from $x=2$ to $x=2+h$ . Then, let $h→0$ to calculate the slope of the tangent line at $x=2$ .

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.