Find the Average Rate of Change f(x)=x^3 , [-1,2]

$f (x) = x^{3}$ , $[- 1, 2]$

Step 1

Write $f (x) = x^{3}$ as an equation.

$y = x^{3}$

Step 2

Substitute using the average rate of change formula.

The average rate of change of a function can be found by calculating the change in $y$ values of the two points divided by the change in $x$ values of the two points.

$\frac{f (2) - f (- 1)}{(2) - (- 1)}$

Substitute the equation $y = x^{3}$ for $f (2)$ and $f (- 1)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{({(2)}^{3}) - ({(- 1)}^{3})}{(2) - (- 1)}$

Step 3

Simplify the expression.

Simplify the numerator.

Raise $2$ to the power of $3$ .

$\frac{8 - {(- 1)}^{3}}{2 - (- 1)}$

Multiply $- 1$ by ${(- 1)}^{3}$ by adding the exponents.

Multiply $- 1$ by ${(- 1)}^{3}$ .

Raise $- 1$ to the power of $1$ .

$\frac{8 + {(- 1)}^{1} {(- 1)}^{3}}{2 - (- 1)}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{8 + {(- 1)}^{1 + 3}}{2 - (- 1)}$

Add $1$ and $3$ .

$\frac{8 + {(- 1)}^{4}}{2 - (- 1)}$

Raise $- 1$ to the power of $4$ .

$\frac{8 + 1}{2 - (- 1)}$

Add $8$ and $1$ .

$\frac{9}{2 - (- 1)}$

Simplify the denominator.

Multiply $- 1$ by $- 1$ .

$\frac{9}{2 + 1}$

Add $2$ and $1$ .

$\frac{9}{3}$

Divide $9$ by $3$ .

$3$

Find the Average Rate of Change f(x)=x^3 , [-1,2]

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