Question

Find the average rate of change of each function on the interval specified. Put your answer in simplified factored form. You may want to do your work on paper and submit an image of your written work rather than try to type it all out. To earn full credit please show all steps and calculations.j(x)= 3x^3 on the interval [1,1+h]

Answer 1

tion isThe rate of change of a function f(x) at the interval [a, b] is

$r=\frac{f(b)-f(a)}{b-a}$

Since the given function is

$f(x)=3x^3$

Since the given interval is [1, (1 + h)], then

a = 1

b = (1 + h)

Substitute x by a and b and use the rule above

$\begin{gathered} x=a=1 \\ f(1)=3(1)^3=3(1)=3 \end{gathered}$$\begin{gathered} x=b=(h+1) \\ f(h+1)=3(h+1)^3 \end{gathered}$

Substitute them in the rule above

$\begin{gathered} r=\frac{f(h+1)-f(1)}{(h+1)-1} \\ r=\frac{3(h+1)^3-3}{h+1-1} \\ r=\frac{3(h+1)^3-3}{h} \end{gathered}$

Take 3 as a common factor from the numerator

$r=\frac{3\lbrack(h+1)^3-1\rbrack}{h}$

The rate of change of the given function is

$undefined$