Precalculus Examples

$g (x) = 3 x^{3} - 1$ on $[- 3, 3]$

Step 1

Write $g (x) = 3 x^{3} - 1$ as an equation.

$y = 3 x^{3} - 1$

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.

$\frac{f (3) - f (- 3)}{(3) - (- 3)}$

Step 2.2

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

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

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Multiply $3$ by $3^{3}$ by adding the exponents.

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Step 3.1.1.1

Multiply $3$ by $3^{3}$ .

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Step 3.1.1.1.1

Raise $3$ to the power of $1$ .

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

Step 3.1.1.1.2

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

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

Step 3.1.1.2

Add $1$ and $3$ .

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

Step 3.1.2

Raise $3$ to the power of $4$ .

$\frac{81 - 1 - (3 {(- 3)}^{3} - 1)}{3 - (- 3)}$

Step 3.1.3

Simplify each term.

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Step 3.1.3.1

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

$\frac{81 - 1 - (3 \cdot - 27 - 1)}{3 - (- 3)}$

Step 3.1.3.2

Multiply $3$ by $- 27$ .

$\frac{81 - 1 - (- 81 - 1)}{3 - (- 3)}$

Step 3.1.4

Subtract $1$ from $- 81$ .

$\frac{81 - 1 - - 82}{3 - (- 3)}$

Step 3.1.5

Multiply $- 1$ by $- 82$ .

$\frac{81 - 1 + 82}{3 - (- 3)}$

Step 3.1.6

Subtract $1$ from $81$ .

$\frac{80 + 82}{3 - (- 3)}$

Step 3.1.7

Add $80$ and $82$ .

$\frac{162}{3 - (- 3)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $- 3$ .

$\frac{162}{3 + 3}$

Step 3.2.2

Add $3$ and $3$ .

$\frac{162}{6}$

Step 3.3

Divide $162$ by $6$ .

$27$