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.

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
asked 2021-02-01
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.

Answers (1)

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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