Precalculus Examples

$s (t) = 4 \sin (t)$ , $[\frac{π}{2}, π]$

Step 1

Write $s (t) = 4 \sin (t)$ as an equation.

$y = 4 \sin (t)$

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.

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

Step 2.2

Substitute the equation $y = 4 \sin (t)$ for $f (π)$ and $f (\frac{π}{2})$ , replacing $t$ in the function with the corresponding $t$ value.

$\frac{(4 \sin (π)) - (4 \sin (\frac{π}{2}))}{(π) - (\frac{π}{2})}$

Step 3

Simplify the expression.

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Step 3.1

Multiply the numerator and denominator of the fraction by $2$ .

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Step 3.1.1

Multiply $\frac{4 \sin (π) - 1 \cdot 4 \sin (\frac{π}{2})}{π - \frac{π}{2}}$ by $\frac{2}{2}$ .

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

Step 3.1.2

Combine.

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

Step 3.2

Apply the distributive property.

$\frac{2 (4 \sin (π)) + 2 (- 1 \cdot 4 \sin (\frac{π}{2}))}{2 π + 2 (- \frac{π}{2})}$

Step 3.3

Cancel the common factor of $2$ .

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Step 3.3.1

Move the leading negative in $- \frac{π}{2}$ into the numerator.

$\frac{2 (4 \sin (π)) + 2 (- 1 \cdot 4 \sin (\frac{π}{2}))}{2 π + 2 \frac{- π}{2}}$

Step 3.3.2

Cancel the common factor.

$\frac{2 (4 \sin (π)) + 2 (- 1 \cdot 4 \sin (\frac{π}{2}))}{2 π + 2 \frac{- π}{2}}$

Step 3.3.3

Rewrite the expression.

$\frac{2 (4 \sin (π)) + 2 (- 1 \cdot 4 \sin (\frac{π}{2}))}{2 π - π}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Multiply $2$ by $4$ .

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

Step 3.4.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\frac{8 \sin (0) + 2 (- 1 \cdot 4 \sin (\frac{π}{2}))}{2 π - π}$

Step 3.4.3

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

$\frac{8 \cdot 0 + 2 (- 1 \cdot 4 \sin (\frac{π}{2}))}{2 π - π}$

Step 3.4.4

Multiply $8$ by $0$ .

$\frac{0 + 2 (- 1 \cdot 4 \sin (\frac{π}{2}))}{2 π - π}$

Step 3.4.5

Multiply $- 1$ by $4$ .

$\frac{0 + 2 (- 4 \sin (\frac{π}{2}))}{2 π - π}$

Step 3.4.6

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

$\frac{0 + 2 (- 4 \cdot 1)}{2 π - π}$

Step 3.4.7

Multiply $2 (- 4 \cdot 1)$ .

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Step 3.4.7.1

Multiply $- 4$ by $1$ .

$\frac{0 + 2 \cdot - 4}{2 π - π}$

Step 3.4.7.2

Multiply $2$ by $- 4$ .

$\frac{0 - 8}{2 π - π}$

Step 3.4.8

Subtract $8$ from $0$ .

$\frac{- 8}{2 π - π}$

Step 3.5

Simplify by adding terms.

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Step 3.5.1

Subtract $π$ from $2 π$ .

$\frac{- 8}{π}$

Step 3.5.2

Move the negative in front of the fraction.

$- \frac{8}{π}$