Precalculus Examples

$f (x) = \sqrt[3]{x}$ ; $3 \leq x \leq 12$

Step 1

Write $f (x) = \sqrt[3]{x}$ as an equation.

$y = \sqrt[3]{x}$

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.

$\frac{f (12) - f (3)}{(12) - (3)}$

Step 2.2

Substitute the equation $y = \sqrt[3]{x}$ for $f (12)$ and $f (3)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(\sqrt[3]{12}) - (\sqrt[3]{3})}{(12) - (3)}$

Step 3

Simplify the denominator.

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Step 3.1

Multiply $- 1$ by $3$ .

$\frac{\sqrt[3]{12} - \sqrt[3]{3}}{12 - 3}$

Step 3.2

Subtract $3$ from $12$ .

$\frac{\sqrt[3]{12} - \sqrt[3]{3}}{9}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt[3]{12} - \sqrt[3]{3}}{9}$

Decimal Form:

$0.09413099 \dots$