Precalculus Examples

$f (x) = \sqrt{2 x}$ from $x_{1} = 2$ to $x_{2} = 8$

Step 1

Substitute using the average rate of change formula.

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Step 1.1

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.

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

Step 1.2

Substitute the equation $y = \sqrt{2 x}$ for $f (8)$ and $f (2)$ , replacing $x$ in the function with the corresponding $x$ value.

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

Step 2

Simplify the expression.

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Step 2.1

Simplify the numerator.

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Step 2.1.1

Multiply $2$ by $8$ .

$\frac{\sqrt{16} - \sqrt{2 (2)}}{8 - (2)}$

Step 2.1.2

Rewrite $16$ as $4^{2}$ .

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

Step 2.1.3

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

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

Step 2.1.4

Multiply $2$ by $2$ .

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

Step 2.1.5

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

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

Step 2.1.6

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

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

Step 2.1.7

Multiply $- 1$ by $2$ .

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

Step 2.1.8

Subtract $2$ from $4$ .

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

Step 2.2

Simplify the denominator.

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Step 2.2.1

Multiply $- 1$ by $2$ .

$\frac{2}{8 - 2}$

Step 2.2.2

Subtract $2$ from $8$ .

$\frac{2}{6}$

Step 2.3

Cancel the common factor of $2$ and $6$ .

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Step 2.3.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{6}$

Step 2.3.2

Cancel the common factors.

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Step 2.3.2.1

Factor $2$ out of $6$ .

$\frac{2 \cdot 1}{2 \cdot 3}$

Step 2.3.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 3}$

Step 2.3.2.3

Rewrite the expression.

$\frac{1}{3}$