Precalculus Examples

$f (x) = \sqrt[3]{4 x - 1}$ ; $[0, 7]$

Step 1

Write $f (x) = \sqrt[3]{4 x - 1}$ as an equation.

$y = \sqrt[3]{4 x - 1}$

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.

$\frac{f (7) - f (0)}{(7) - (0)}$

Step 2.2

Substitute the equation $y = \sqrt[3]{4 x - 1}$ for $f (7)$ and $f (0)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(\sqrt[3]{4 (7) - 1}) - (\sqrt[3]{4 (0) - 1})}{(7) - (0)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Multiply $4$ by $7$ .

$\frac{\sqrt[3]{28 - 1} - \sqrt[3]{4 (0) - 1}}{7 - (0)}$

Step 3.1.2

Subtract $1$ from $28$ .

$\frac{\sqrt[3]{27} - \sqrt[3]{4 (0) - 1}}{7 - (0)}$

Step 3.1.3

Rewrite $27$ as $3^{3}$ .

$\frac{\sqrt[3]{3^{3}} - \sqrt[3]{4 (0) - 1}}{7 - (0)}$

Step 3.1.4

Pull terms out from under the radical, assuming real numbers.

$\frac{3 - \sqrt[3]{4 (0) - 1}}{7 - (0)}$

Step 3.1.5

Multiply $4$ by $0$ .

$\frac{3 - \sqrt[3]{0 - 1}}{7 - (0)}$

Step 3.1.6

Subtract $1$ from $0$ .

$\frac{3 - \sqrt[3]{- 1}}{7 - (0)}$

Step 3.1.7

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$\frac{3 - \sqrt[3]{{(- 1)}^{3}}}{7 - (0)}$

Step 3.1.8

Pull terms out from under the radical, assuming real numbers.

$\frac{3 - - 1}{7 - (0)}$

Step 3.1.9

Multiply $- 1$ by $- 1$ .

$\frac{3 + 1}{7 - (0)}$

Step 3.1.10

Add $3$ and $1$ .

$\frac{4}{7 - (0)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $0$ .

$\frac{4}{7 + 0}$

Step 3.2.2

Add $7$ and $0$ .

$\frac{4}{7}$