Precalculus Examples

$h (t) = \sin (5 t)$ , $[0, \frac{π}{10}]$

Step 1

Write $h (t) = \sin (5 t)$ as an equation.

$y = \sin (5 t)$

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 $t$ values of the two points.

$\frac{f (\frac{π}{10}) - f (0)}{(\frac{π}{10}) - (0)}$

Step 2.2

Substitute the equation $y = \sin (5 t)$ for $f (\frac{π}{10})$ and $f (0)$ , replacing $t$ in the function with the corresponding $t$ value.

$\frac{(\sin (5 (\frac{π}{10}))) - (\sin (5 (0)))}{(\frac{π}{10}) - (0)}$

Step 3

Simplify the expression.

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Step 3.1

Multiply the numerator and denominator of the fraction by $10$ .

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Step 3.1.1

Multiply $\frac{\sin (5 \frac{π}{10}) - \sin (5 (0))}{\frac{π}{10} - (0)}$ by $\frac{10}{10}$ .

$\frac{10}{10} \cdot \frac{\sin (5 \frac{π}{10}) - \sin (5 (0))}{\frac{π}{10} - (0)}$

Step 3.1.2

Combine.

$\frac{10 (\sin (5 \frac{π}{10}) - \sin (5 (0)))}{10 (\frac{π}{10} - (0))}$

Step 3.2

Apply the distributive property.

$\frac{10 \sin (5 \frac{π}{10}) + 10 (- \sin (5 (0)))}{10 \frac{π}{10} + 10 (- (0))}$

Step 3.3

Cancel the common factor of $10$ .

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Step 3.3.1

Cancel the common factor.

$\frac{10 \sin (5 \frac{π}{10}) + 10 (- \sin (5 (0)))}{10 \frac{π}{10} + 10 (- (0))}$

Step 3.3.2

Rewrite the expression.

$\frac{10 \sin (5 \frac{π}{10}) + 10 (- \sin (5 (0)))}{π + 10 (- (0))}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Factor $10$ out of $10 \sin (5 \frac{π}{10}) + 10 (- \sin (5 (0)))$ .

$\frac{10 (\sin (5 \frac{π}{10}) - \sin (5 (0)))}{π + 10 (- (0))}$

Step 3.4.2

Cancel the common factor of $5$ .

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Step 3.4.2.1

Factor $5$ out of $10$ .

$\frac{10 (\sin (5 \frac{π}{5 (2)}) - \sin (5 (0)))}{π + 10 (- (0))}$

Step 3.4.2.2

Cancel the common factor.

$\frac{10 (\sin (5 \frac{π}{5 \cdot 2}) - \sin (5 (0)))}{π + 10 (- (0))}$

Step 3.4.2.3

Rewrite the expression.

$\frac{10 (\sin (\frac{π}{2}) - \sin (5 (0)))}{π + 10 (- (0))}$

Step 3.4.3

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

$\frac{10 (1 - \sin (5 (0)))}{π + 10 (- (0))}$

Step 3.4.4

Multiply $5$ by $0$ .

$\frac{10 (1 - \sin (0))}{π + 10 (- (0))}$

Step 3.4.5

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

$\frac{10 (1 - 0)}{π + 10 (- (0))}$

Step 3.4.6

Multiply $- 1$ by $0$ .

$\frac{10 (1 + 0)}{π + 10 (- (0))}$

Step 3.4.7

Add $1$ and $0$ .

$\frac{10 \cdot 1}{π + 10 (- (0))}$

Step 3.5

Simplify the denominator.

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Step 3.5.1

Multiply $10 (- (0))$ .

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Step 3.5.1.1

Multiply $- 1$ by $0$ .

$\frac{10 \cdot 1}{π + 10 \cdot 0}$

Step 3.5.1.2

Multiply $10$ by $0$ .

$\frac{10 \cdot 1}{π + 0}$

Step 3.5.2

Add $π$ and $0$ .

$\frac{10 \cdot 1}{π}$

Step 3.6

Multiply $10$ by $1$ .

$\frac{10}{π}$