CPM Homework Help : CALC Problem 3-175

Use the definition of a derivative as a limit to find the slope function, $f^\prime(x)$ , of $f(x) = 2x^2-3x + 4$ . Confirm your slope function with the Power Rule. Then use your slope function to find $f^\prime(3)$ and $f^\prime(−2)$ .

Hint:

Hana's definition of the derivative:
$f'(x) =\lim\limits_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}$

Step 1:

1. Substitute $f(x)$ into Hana's definition
2. Expand the numerator
3. Combine like terms
4. Factor out an $h$
5. Cancel out the $h$
6. Evaluate the limit as $h→0$
This is $f^\prime(x)$

Step 2:

Use the Power Rule to find $f^\prime(x)$ . Compare to the $f^\prime(x)$ you found algebraically. If they do not agree, you probably made an algebraic error.

Step 3:

Evaluate $f^\prime(3)$ and $f^\prime-(3)$ .