Precalculus Examples

$f (x) = \sqrt{x + 7}$ ; $2 \leq x \leq 10$

Step 1

Write $f (x) = \sqrt{x + 7}$ as an equation.

$y = \sqrt{x + 7}$

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.

$\frac{f (10) - f (2)}{(10) - (2)}$

Step 2.2

Substitute the equation $y = \sqrt{x + 7}$ for $f (10)$ and $f (2)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(\sqrt{(10) + 7}) - (\sqrt{(2) + 7})}{(10) - (2)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Add $10$ and $7$ .

$\frac{\sqrt{17} - \sqrt{2 + 7}}{10 - (2)}$

Step 3.1.2

Add $2$ and $7$ .

$\frac{\sqrt{17} - \sqrt{9}}{10 - (2)}$

Step 3.1.3

Rewrite $9$ as $3^{2}$ .

$\frac{\sqrt{17} - \sqrt{3^{2}}}{10 - (2)}$

Step 3.1.4

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

$\frac{\sqrt{17} - 1 \cdot 3}{10 - (2)}$

Step 3.1.5

Multiply $- 1$ by $3$ .

$\frac{\sqrt{17} - 3}{10 - (2)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $2$ .

$\frac{\sqrt{17} - 3}{10 - 2}$

Step 3.2.2

Subtract $2$ from $10$ .

$\frac{\sqrt{17} - 3}{8}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{17} - 3}{8}$

Decimal Form:

$0.14038820 \dots$