Precalculus Examples

$h (x) = \frac{1}{8} x^{3} - x^{2}$ , $- 2 \leq x \leq 2$

Step 1

Write $h (x) = \frac{1}{8} x^{3} - x^{2}$ as an equation.

$y = \frac{1}{8} x^{3} - x^{2}$

Step 2

Substitute using the average rate of change formula.

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Step 2.1

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.

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

Step 2.2

Substitute the equation $y = \frac{1}{8} x^{3} - x^{2}$ for $f (2)$ and $f (- 2)$ , replacing $x$ in the function with the corresponding $x$ value.

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

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

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

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

Step 3.1.2

Cancel the common factor of $8$ .

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Step 3.1.2.1

Cancel the common factor.

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

Step 3.1.2.2

Rewrite the expression.

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

Step 3.1.3

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

$\frac{1 - 1 \cdot 4 - (\frac{1}{8} \cdot {(- 2)}^{3} - {(- 2)}^{2})}{2 - (- 2)}$

Step 3.1.4

Multiply $- 1$ by $4$ .

$\frac{1 - 4 - (\frac{1}{8} \cdot {(- 2)}^{3} - {(- 2)}^{2})}{2 - (- 2)}$

Step 3.1.5

Simplify each term.

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Step 3.1.5.1

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

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

Step 3.1.5.2

Cancel the common factor of $8$ .

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Step 3.1.5.2.1

Factor $8$ out of $- 8$ .

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

Step 3.1.5.2.2

Cancel the common factor.

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

Step 3.1.5.2.3

Rewrite the expression.

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

Step 3.1.5.3

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

$\frac{1 - 4 - (- 1 - 1 \cdot 4)}{2 - (- 2)}$

Step 3.1.5.4

Multiply $- 1$ by $4$ .

$\frac{1 - 4 - (- 1 - 4)}{2 - (- 2)}$

Step 3.1.6

Subtract $4$ from $- 1$ .

$\frac{1 - 4 - - 5}{2 - (- 2)}$

Step 3.1.7

Multiply $- 1$ by $- 5$ .

$\frac{1 - 4 + 5}{2 - (- 2)}$

Step 3.1.8

Subtract $4$ from $1$ .

$\frac{- 3 + 5}{2 - (- 2)}$

Step 3.1.9

Add $- 3$ and $5$ .

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

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $- 2$ .

$\frac{2}{2 + 2}$

Step 3.2.2

Add $2$ and $2$ .

$\frac{2}{4}$

Step 3.3

Cancel the common factor of $2$ and $4$ .

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Step 3.3.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{4}$

Step 3.3.2

Cancel the common factors.

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Step 3.3.2.1

Factor $2$ out of $4$ .

$\frac{2 \cdot 1}{2 \cdot 2}$

Step 3.3.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 2}$

Step 3.3.2.3

Rewrite the expression.

$\frac{1}{2}$