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 \(f(x + h)-f(x)\)

$$
LATEXBLOCK0
$$

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

$$
LATEXBLOCK1
$$

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) \(-6x\)