Question

assignment 4: problem 13 (1 point) let $f(x)=-3x^{2}$. a) find $f(x + h)$: b) find $f(x + h)-f(x)$: c) find $\frac{f(x + h)-f(x)}{h}$: d) find $f(x)$:

Explanation:

Step1: Find \(f(x + h)\)

Substitute \(x+h\) into \(f(x)\):
\[f(x + h)=-3(x + h)^{2}=-3(x^{2}+2xh + h^{2})=-3x^{2}-6xh - 3h^{2}\]

Step2: Find \(f(x + h)-f(x)\)

\[

$$\begin{align*} f(x + h)-f(x)&=(-3x^{2}-6xh - 3h^{2})-(-3x^{2})\\ &=-3x^{2}-6xh - 3h^{2}+ 3x^{2}\\ &=-6xh-3h^{2} \end{align*}$$

\]

Step3: Find \(\frac{f(x + h)-f(x)}{h}\)

\[

$$\begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{-6xh - 3h^{2}}{h}\\ &=\frac{h(-6x - 3h)}{h}\\ &=-6x-3h \end{align*}$$

\]

Step4: Find \(f'(x)\)

Take the limit as \(h
ightarrow0\) of \(\frac{f(x + h)-f(x)}{h}\):
\[f'(x)=\lim_{h
ightarrow0}\frac{f(x + h)-f(x)}{h}=\lim_{h
ightarrow0}(-6x - 3h)=-6x\]

Answer:

a) \(-3x^{2}-6xh - 3h^{2}\)
b) \(-6xh-3h^{2}\)
c) \(-6x - 3h\)
d) \(-6x\)