Question

by determining $f(x)=lim_{h
ightarrow0}\frac{f(x + h)-f(x)}{h}$, find $f(8)$ for the given function.

$f(x)=3x^{2}$

$f(8)=square$ (simplify your 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: Calculate f(x + h)-f(x)

$f(x + h)-f(x)=(3x^{2}+6xh + 3h^{2})-3x^{2}=6xh+3h^{2}$

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

$\frac{f(x + h)-f(x)}{h}=\frac{6xh + 3h^{2}}{h}=6x+3h$

Step4: Find f'(x)

$f'(x)=\lim_{h
ightarrow0}\frac{f(x + h)-f(x)}{h}=\lim_{h
ightarrow0}(6x + 3h)=6x$

Step5: Find f'(8)

Substitute x = 8 into f'(x):
$f'(8)=6\times8 = 48$

Answer:

48