Precalculus Examples

$f (x) = \sqrt{x^{3} - 4}$ , $2 \leq x \leq 5$

Step 1

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

$y = \sqrt{x^{3} - 4}$

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.

$\frac{f (5) - f (2)}{(5) - (2)}$

Step 2.2

Substitute the equation $y = \sqrt{x^{3} - 4}$ for $f (5)$ and $f (2)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(\sqrt{{(5)}^{3} - 4}) - (\sqrt{{(2)}^{3} - 4})}{(5) - (2)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $5$ to the power of $3$ .

$\frac{\sqrt{125 - 4} - \sqrt{2^{3} - 4}}{5 - (2)}$

Step 3.1.2

Subtract $4$ from $125$ .

$\frac{\sqrt{121} - \sqrt{2^{3} - 4}}{5 - (2)}$

Step 3.1.3

Rewrite $121$ as $11^{2}$ .

$\frac{\sqrt{11^{2}} - \sqrt{2^{3} - 4}}{5 - (2)}$

Step 3.1.4

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

$\frac{11 - \sqrt{2^{3} - 4}}{5 - (2)}$

Step 3.1.5

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

$\frac{11 - \sqrt{8 - 4}}{5 - (2)}$

Step 3.1.6

Subtract $4$ from $8$ .

$\frac{11 - \sqrt{4}}{5 - (2)}$

Step 3.1.7

Rewrite $4$ as $2^{2}$ .

$\frac{11 - \sqrt{2^{2}}}{5 - (2)}$

Step 3.1.8

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

$\frac{11 - 1 \cdot 2}{5 - (2)}$

Step 3.1.9

Multiply $- 1$ by $2$ .

$\frac{11 - 2}{5 - (2)}$

Step 3.1.10

Subtract $2$ from $11$ .

$\frac{9}{5 - (2)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $2$ .

$\frac{9}{5 - 2}$

Step 3.2.2

Subtract $2$ from $5$ .

$\frac{9}{3}$

Step 3.3

Divide $9$ by $3$ .

$3$