Step 1

Consider the function:

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

The average rate of change of function over any intervals of length “h” is given by the formula,

Average rate \(= \frac{f(x\ +\ h)\ -\ f(x)}{h}\)

Step 2

The average rate of changes of the given function is,

Average rate \(= \frac{2(x\ +\ h)^{3}\ -\ 2x^{3}}{h}\)

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

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

\(=\frac{h(2h^{2}\ +\ 6x^{2}\ +\ 6xh)}{h}\)

\(= 2h^{2}\ +\ 6x^{2}\ +\ 6xh\)

\(= 6x^{2}\ +\ 6xh\ +\ 2h^{2}\)

Hence the average rate of changes of the given function is \(6x^{2}\ +\ 6xh\ +\ 2h^{2}\)

Consider the function:

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

The average rate of change of function over any intervals of length “h” is given by the formula,

Average rate \(= \frac{f(x\ +\ h)\ -\ f(x)}{h}\)

Step 2

The average rate of changes of the given function is,

Average rate \(= \frac{2(x\ +\ h)^{3}\ -\ 2x^{3}}{h}\)

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

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

\(=\frac{h(2h^{2}\ +\ 6x^{2}\ +\ 6xh)}{h}\)

\(= 2h^{2}\ +\ 6x^{2}\ +\ 6xh\)

\(= 6x^{2}\ +\ 6xh\ +\ 2h^{2}\)

Hence the average rate of changes of the given function is \(6x^{2}\ +\ 6xh\ +\ 2h^{2}\)