Precalculus Examples

$f (x) = \sin (x)$ , $[0, \frac{π}{6}]$

Step 1

Write $f (x) = \sin (x)$ as an equation.

$y = \sin (x)$

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.

$\frac{f (\frac{π}{6}) - f (0)}{(\frac{π}{6}) - (0)}$

Step 2.2

Substitute the equation $y = \sin (x)$ for $f (\frac{π}{6})$ and $f (0)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(\sin (\frac{π}{6})) - (\sin (0))}{(\frac{π}{6}) - (0)}$

Step 3

Simplify the expression.

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Step 3.1

Multiply the numerator and denominator of the fraction by $6$ .

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Step 3.1.1

Multiply $\frac{\sin (\frac{π}{6}) - \sin (0)}{\frac{π}{6} - (0)}$ by $\frac{6}{6}$ .

$\frac{6}{6} \cdot \frac{\sin (\frac{π}{6}) - \sin (0)}{\frac{π}{6} - (0)}$

Step 3.1.2

Combine.

$\frac{6 (\sin (\frac{π}{6}) - \sin (0))}{6 (\frac{π}{6} - (0))}$

Step 3.2

Apply the distributive property.

$\frac{6 \sin (\frac{π}{6}) + 6 (- \sin (0))}{6 \frac{π}{6} + 6 (- (0))}$

Step 3.3

Cancel the common factor of $6$ .

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Step 3.3.1

Cancel the common factor.

$\frac{6 \sin (\frac{π}{6}) + 6 (- \sin (0))}{6 \frac{π}{6} + 6 (- (0))}$

Step 3.3.2

Rewrite the expression.

$\frac{6 \sin (\frac{π}{6}) + 6 (- \sin (0))}{π + 6 (- (0))}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Factor $6$ out of $6 \sin (\frac{π}{6}) + 6 (- \sin (0))$ .

$\frac{6 (\sin (\frac{π}{6}) - \sin (0))}{π + 6 (- (0))}$

Step 3.4.2

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

$\frac{6 (\frac{1}{2} - \sin (0))}{π + 6 (- (0))}$

Step 3.4.3

The exact value of $\sin (0)$ is $0$ .

$\frac{6 (\frac{1}{2} - 0)}{π + 6 (- (0))}$

Step 3.4.4

Multiply $- 1$ by $0$ .

$\frac{6 (\frac{1}{2} + 0)}{π + 6 (- (0))}$

Step 3.4.5

Add $\frac{1}{2}$ and $0$ .

$\frac{6 (\frac{1}{2})}{π + 6 (- (0))}$

Step 3.5

Simplify the denominator.

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Step 3.5.1

Multiply $6 (- (0))$ .

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Step 3.5.1.1

Multiply $- 1$ by $0$ .

$\frac{6 (\frac{1}{2})}{π + 6 \cdot 0}$

Step 3.5.1.2

Multiply $6$ by $0$ .

$\frac{6 (\frac{1}{2})}{π + 0}$

Step 3.5.2

Add $π$ and $0$ .

$\frac{6 (\frac{1}{2})}{π}$

Step 3.6

Combine fractions.

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Step 3.6.1

Combine $6$ and $\frac{1}{2}$ .

$\frac{\frac{6}{2}}{π}$

Step 3.6.2

Divide $6$ by $2$ .

$\frac{3}{π}$