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.

Jaya Legge 2021-02-01 Answered
Suppose f(x)=2x3. 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.
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Expert Answer

Obiajulu
Answered 2021-02-02 Author has 98 answers
Step 1
Consider the function:
f(x)=2x3
The average rate of change of function over any intervals of length “h” is given by the formula,
Average rate =f(x + h)  f(x)h
Step 2
The average rate of changes of the given function is,
Average rate =2(x + h)3  2x3h
=2(x3 + h3 + 3x2h + 3xh2)  2x3h
=2x3 + 2h3 + 6x2h + 6xh2  2x3h
=h(2h2 + 6x2 + 6xh)h
=2h2 + 6x2 + 6xh
=6x2 + 6xh + 2h2
Hence the average rate of changes of the given function is 6x2 + 6xh + 2h2
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