CPM Homework Help : APCALC Problem 4-122

Compare two different methods for determining a derivative in parts (a) and (b).

Use the definition of the derivative as a limit to write the slope function, $f^\prime$ , if $f\left(x\right)=$ $-x^2+3x+1$ .
Step 1(a):
Substitute $f\left(x\right)$ into the Definition of the Derivative (this is Hana's definition):
$f'(x)=\lim \limits_{h\rightarrow 0}\frac{(-(x+h)^{2}+3(x+h)+1)-(-x^{2}+3x+1)}{h}$
Step 2(a):
Find a way to 'cancel out the $h^\prime$ in the denominator:
1. Expand the numerator.
2. Combine like terms.
3. Factor out an $h$ .
4. Cancel out the $h$ .
Step 3(a):
Now that there is no longer an $h$ in the denominator, take the limit as $h→0$ .
This is the derivative of $y = −x^{2} + 3x + 1$ .
Use the Power Rule to write an equation for $f^\prime$ . Do your answers agree?
Hint (b):
If your answers are different, then you probably made an algebraic error in part (a). Find it and fix it.
Use your slope function to calculate $f^\prime\left(0\right)$ and $f^\prime(1)$ .
Answer (c):
$f^\prime\left(0\right) = 3, f^\prime\left(1\right) = 1$