CPM Homework Help : PC Problem 9-143

Let $h\left(x\right) = x^{2}$ .

Hint:

Find the general slope function at any $x$ .

Steps:

$\lim\limits_{h\rightarrow 0}\frac{h(x+h)-h(x)}{x+h-x}=\lim\limits_{h\rightarrow 0}\frac{(x+h)^{2}-x^{2}}{h}=\lim\limits_{h\rightarrow 0}\frac{x^{2}+2xh+h^{2}-x^{2}}{h}=\lim\limits_{h\rightarrow 0}\frac{2xh+h^{2}}{h}=\lim\limits_{h\rightarrow 0}\frac{h(2x+h)}{h}=\lim\limits_{h\to 0}2x+h=2x$

Compute $h ′\left(3\right)$ .
Hint (a):
Substitute the input value in the general slope function.
Compute $h ′\left(−2\right)$ .
Answer (b):
$−4$
Compute $h ′\left(a\right)$ for any fixed number $a$ .
Answer (c):
$2a$