Precalculus Examples

$f (x) = \frac{1}{4} x^{4} - x^{3}$ ; $[- 2, 2]$

Step 1

Write $f (x) = \frac{1}{4} x^{4} - x^{3}$ as an equation.

$y = \frac{1}{4} x^{4} - x^{3}$

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 $y = \frac{1}{4} x^{4} - x^{3}$ for $f (2)$ and $f (- 2)$ , replacing $x$ in the function with the corresponding $x$ value.

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

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 $4$ .

$\frac{\frac{1}{4} \cdot 16 - 2^{3} - (\frac{1}{4} \cdot {(- 2)}^{4} - {(- 2)}^{3})}{2 - (- 2)}$

Step 3.1.2

Cancel the common factor of $4$ .

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Step 3.1.2.1

Factor $4$ out of $16$ .

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

Step 3.1.2.2

Cancel the common factor.

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

Step 3.1.2.3

Rewrite the expression.

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

Step 3.1.3

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

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

Step 3.1.4

Multiply $- 1$ by $8$ .

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

Step 3.1.5

Simplify each term.

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Step 3.1.5.1

Raise $- 2$ to the power of $4$ .

$\frac{4 - 8 - (\frac{1}{4} \cdot 16 - {(- 2)}^{3})}{2 - (- 2)}$

Step 3.1.5.2

Cancel the common factor of $4$ .

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Step 3.1.5.2.1

Factor $4$ out of $16$ .

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

Step 3.1.5.2.2

Cancel the common factor.

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

Step 3.1.5.2.3

Rewrite the expression.

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

Step 3.1.5.3

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

$\frac{4 - 8 - (4 - - 8)}{2 - (- 2)}$

Step 3.1.5.4

Multiply $- 1$ by $- 8$ .

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

Step 3.1.6

Add $4$ and $8$ .

$\frac{4 - 8 - 1 \cdot 12}{2 - (- 2)}$

Step 3.1.7

Multiply $- 1$ by $12$ .

$\frac{4 - 8 - 12}{2 - (- 2)}$

Step 3.1.8

Subtract $8$ from $4$ .

$\frac{- 4 - 12}{2 - (- 2)}$

Step 3.1.9

Subtract $12$ from $- 4$ .

$\frac{- 16}{2 - (- 2)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $- 2$ .

$\frac{- 16}{2 + 2}$

Step 3.2.2

Add $2$ and $2$ .

$\frac{- 16}{4}$

Step 3.3

Divide $- 16$ by $4$ .

$- 4$