Jaya Legge

2021-02-01

Suppose $f(x)=2{x}^{3}.$ 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.

Obiajulu

Skilled2021-02-02Added 98 answers

Step 1

Consider the function:

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

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

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

Step 2

The average rate of changes of the given function is,

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

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

$=\frac{2{x}^{3}\text{}+\text{}2{h}^{3}\text{}+\text{}6{x}^{2}h\text{}+\text{}6x{h}^{2}\text{}-\text{}2{x}^{3}}{h}$

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

$=2{h}^{2}\text{}+\text{}6{x}^{2}\text{}+\text{}6xh$

$=6{x}^{2}\text{}+\text{}6xh\text{}+\text{}2{h}^{2}$

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

Consider the function:

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

Average rate

Step 2

The average rate of changes of the given function is,

Average rate

Hence the average rate of changes of the given function is

Andre BalkonE

Skilled2023-05-26Added 110 answers

Jazz Frenia

Skilled2023-05-26Added 106 answers

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