# Suppose f(x) = 2x^{3}. Write an expression in terms of x and h that represents the average rate of change of f over any interval of length h. [That is, over any interval (x, x + h)] Simplify your answer as much as possible.

Question
Confidence intervals
Suppose $$f(x) = 2x^{3}.$$ Write an expression in terms of x and h that represents the average rate of change of f over any interval of length h. [That is, over any interval (x, x + h)] Simplify your answer as much as possible.

2021-02-02
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 $$6x^{2}\ +\ 6xh\ +\ 2h^{2}$$

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