Question

by determining $f(x) = \lim\limits_{h\to 0} \frac{f(x + h) - f(x)}{h}$, find $f(8)$ for the given function. $f(x) = 6x^2$ $f(8) = \square$ (simplify your answer.)

Explanation:

Step1: Compute f(x+h)

$f(x+h) = 6(x+h)^2 = 6(x^2 + 2xh + h^2) = 6x^2 + 12xh + 6h^2$

Step2: Calculate f(x+h)-f(x)

$f(x+h)-f(x) = (6x^2 +12xh +6h^2) -6x^2 =12xh +6h^2$

Step3: Divide by h

$\frac{f(x+h)-f(x)}{h} = \frac{12xh +6h^2}{h} =12x +6h$

Step4: Take limit as h→0

$f'(x) = \lim_{h→0}(12x +6h)=12x$

Step5: Substitute x=8

$f'(8)=12×8=96$

Answer:

96