Precalculus Examples

$g (x) = 8 \cos (x)$ , $[- \frac{π}{2}, \frac{π}{2}]$

Step 1

Write $g (x) = 8 \cos (x)$ as an equation.

$y = 8 \cos (x)$

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.

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

Step 2.2

Substitute the equation $y = 8 \cos (x)$ for $f (\frac{π}{2})$ and $f (- \frac{π}{2})$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(8 \cos (\frac{π}{2})) - (8 \cos (- \frac{π}{2}))}{(\frac{π}{2}) - (- \frac{π}{2})}$

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 $\frac{8 \cos (\frac{π}{2}) - 1 \cdot 8 \cos (- \frac{π}{2})}{\frac{π}{2} - - \frac{π}{2}}$ by $\frac{2}{2}$ .

$\frac{2}{2} \cdot \frac{8 \cos (\frac{π}{2}) - 1 \cdot 8 \cos (- \frac{π}{2})}{\frac{π}{2} - - \frac{π}{2}}$

Step 3.1.2

Combine.

$\frac{2 (8 \cos (\frac{π}{2}) - 1 \cdot 8 \cos (- \frac{π}{2}))}{2 (\frac{π}{2} - - \frac{π}{2})}$

Step 3.2

Apply the distributive property.

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

Step 3.3

Cancel the common factor of $2$ .

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Step 3.3.1

Cancel the common factor.

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

Step 3.3.2

Rewrite the expression.

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

Step 3.4

Simplify the numerator.

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Step 3.4.1

Multiply $2$ by $8$ .

$\frac{16 \cos (\frac{π}{2}) + 2 (- 1 \cdot 8 \cos (- \frac{π}{2}))}{π + 2 (- - \frac{π}{2})}$

Step 3.4.2

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$\frac{16 \cdot 0 + 2 (- 1 \cdot 8 \cos (- \frac{π}{2}))}{π + 2 (- - \frac{π}{2})}$

Step 3.4.3

Multiply $16$ by $0$ .

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

Step 3.4.4

Multiply $- 1$ by $8$ .

$\frac{0 + 2 (- 8 \cos (- \frac{π}{2}))}{π + 2 (- - \frac{π}{2})}$

Step 3.4.5

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\frac{0 + 2 (- 8 \cos (\frac{3 π}{2}))}{π + 2 (- - \frac{π}{2})}$

Step 3.4.6

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

$\frac{0 + 2 (- 8 \cos (\frac{π}{2}))}{π + 2 (- - \frac{π}{2})}$

Step 3.4.7

The exact value of $\cos (\frac{π}{2})$ is $0$ .

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

Step 3.4.8

Multiply $2 (- 8 \cdot 0)$ .

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Step 3.4.8.1

Multiply $- 8$ by $0$ .

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

Step 3.4.8.2

Multiply $2$ by $0$ .

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

Step 3.4.9

Add $0$ and $0$ .

$\frac{0}{π + 2 (- - \frac{π}{2})}$

Step 3.5

Simplify the denominator.

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Step 3.5.1

Multiply $- - \frac{π}{2}$ .

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Step 3.5.1.1

Multiply $- 1$ by $- 1$ .

$\frac{0}{π + 2 (1 \frac{π}{2})}$

Step 3.5.1.2

Multiply $\frac{π}{2}$ by $1$ .

$\frac{0}{π + 2 \frac{π}{2}}$

Step 3.5.2

Cancel the common factor of $2$ .

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Step 3.5.2.1

Cancel the common factor.

$\frac{0}{π + 2 \frac{π}{2}}$

Step 3.5.2.2

Rewrite the expression.

$\frac{0}{π + π}$

Step 3.5.3

Add $π$ and $π$ .

$\frac{0}{2 π}$

Step 3.6

Reduce the expression by cancelling the common factors.

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Step 3.6.1

Cancel the common factor of $0$ and $2$ .

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Step 3.6.1.1

Factor $2$ out of $0$ .

$\frac{2 (0)}{2 π}$

Step 3.6.1.2

Cancel the common factors.

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Step 3.6.1.2.1

Factor $2$ out of $2 π$ .

$\frac{2 (0)}{2 (π)}$

Step 3.6.1.2.2

Cancel the common factor.

$\frac{2 \cdot 0}{2 π}$

Step 3.6.1.2.3

Rewrite the expression.

$\frac{0}{π}$

Step 3.6.2

Divide $0$ by $π$ .

$0$