Question

Find the rate of change for x3. You need to work out the change in f(x)=x3 when x is increased by a small number h to x+h. So you will work out f(x+h)-f(x). Then do some algebra to simplify this. Then divide this by h to get the average rate of change of f(x) between x and x+h.

Answer 1

Answer:

Explanation:

Given

$F(x)=x^3$

Rate of change of F(x) is given by

$F'(x)=\lim_{h\rightarrow 0}\\\frac{F(x+h)-F(x)}{x+h-x}$

$F'(x)=\lim_{h\rightarrow 0}\\\frac{(x+h)^3-x^3}{h}$

$F'(x)=\lim_{h\rightarrow 0}\\\frac{x^3+h^3+3x^2h+3xh^2-x^3}{h}$

$F'(x)=\lim_{h\rightarrow 0}\\\frac{h^3+3x^2h+3xh^2}{h}$

$F'(x)=\lim_{h\rightarrow 0}\\h^2+3x^2+3xh$

Putting limits

$F'(x)=3x^2$