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…

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 $f(x)$:

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 $f(x)$:

Answer

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\rightarrow0) of (\frac{f(x + h)-f(x)}{h}): [f'(x)=\lim_{h\rightarrow0}\frac{f(x + h)-f(x)}{h}=\lim_{h\rightarrow0}(-6x - 3h)=-6x]

Answer:

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