Precalculus Examples

$R (θ) = \sqrt{4 θ + 1}$ ;, $[0, 2]$

Step 1

Write $R (θ) = \sqrt{4 θ + 1}$ as an equation.

$y = \sqrt{4 θ + 1}$

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 $θ$ values of the two points.

$\frac{f (2) - f (0)}{(2) - (0)}$

Step 2.2

Substitute the equation $y = \sqrt{4 θ + 1}$ for $f (2)$ and $f (0)$ , replacing $θ$ in the function with the corresponding $θ$ value.

$\frac{(\sqrt{4 (2) + 1}) - (\sqrt{4 (0) + 1})}{(2) - (0)}$

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 $2$ .

$\frac{\sqrt{8 + 1} - \sqrt{4 (0) + 1}}{2 - (0)}$

Step 3.1.2

Add $8$ and $1$ .

$\frac{\sqrt{9} - \sqrt{4 (0) + 1}}{2 - (0)}$

Step 3.1.3

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

$\frac{\sqrt{3^{2}} - \sqrt{4 (0) + 1}}{2 - (0)}$

Step 3.1.4

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

$\frac{3 - \sqrt{4 (0) + 1}}{2 - (0)}$

Step 3.1.5

Multiply $4$ by $0$ .

$\frac{3 - \sqrt{0 + 1}}{2 - (0)}$

Step 3.1.6

Add $0$ and $1$ .

$\frac{3 - \sqrt{1}}{2 - (0)}$

Step 3.1.7

Any root of $1$ is $1$ .

$\frac{3 - 1 \cdot 1}{2 - (0)}$

Step 3.1.8

Multiply $- 1$ by $1$ .

$\frac{3 - 1}{2 - (0)}$

Step 3.1.9

Subtract $1$ from $3$ .

$\frac{2}{2 - (0)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $0$ .

$\frac{2}{2 + 0}$

Step 3.2.2

Add $2$ and $0$ .

$\frac{2}{2}$

Step 3.3

Divide $2$ by $2$ .

$1$