Precalculus Examples

$f (x) = \cos (x)$ , $[- \frac{π}{3}, \frac{π}{3}]$

Step 1

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

$y = \cos (x)$

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.

$\frac{f (\frac{π}{3}) - f (- \frac{π}{3})}{(\frac{π}{3}) - (- \frac{π}{3})}$

Step 2.2

Substitute the equation $y = \cos (x)$ for $f (\frac{π}{3})$ and $f (- \frac{π}{3})$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(\cos (\frac{π}{3})) - (\cos (- \frac{π}{3}))}{(\frac{π}{3}) - (- \frac{π}{3})}$

Step 3

Simplify the expression.

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Step 3.1

Multiply the numerator and denominator of the fraction by $3$ .

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Step 3.1.1

Multiply $\frac{\cos (\frac{π}{3}) - \cos (- \frac{π}{3})}{\frac{π}{3} - - \frac{π}{3}}$ by $\frac{3}{3}$ .

$\frac{3}{3} \cdot \frac{\cos (\frac{π}{3}) - \cos (- \frac{π}{3})}{\frac{π}{3} - - \frac{π}{3}}$

Step 3.1.2

Combine.

$\frac{3 (\cos (\frac{π}{3}) - \cos (- \frac{π}{3}))}{3 (\frac{π}{3} - - \frac{π}{3})}$

Step 3.2

Apply the distributive property.

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

Step 3.3

Cancel the common factor of $3$ .

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Step 3.3.1

Cancel the common factor.

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

Step 3.3.2

Rewrite the expression.

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

Step 3.4

Simplify the numerator.

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Step 3.4.1

Factor $3$ out of $3 \cos (\frac{π}{3}) + 3 (- \cos (- \frac{π}{3}))$ .

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

Step 3.4.2

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$\frac{3 (\frac{1}{2} - \cos (- \frac{π}{3}))}{π + 3 (- - \frac{π}{3})}$

Step 3.4.3

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

$\frac{3 (\frac{1}{2} - \cos (\frac{5 π}{3}))}{π + 3 (- - \frac{π}{3})}$

Step 3.4.4

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

$\frac{3 (\frac{1}{2} - \cos (\frac{π}{3}))}{π + 3 (- - \frac{π}{3})}$

Step 3.4.5

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

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

Step 3.4.6

Combine the numerators over the common denominator.

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

Step 3.4.7

Subtract $1$ from $1$ .

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

Step 3.4.8

Divide $0$ by $2$ .

$\frac{3 \cdot 0}{π + 3 (- - \frac{π}{3})}$

Step 3.5

Simplify the denominator.

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Step 3.5.1

Multiply $- - \frac{π}{3}$ .

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Step 3.5.1.1

Multiply $- 1$ by $- 1$ .

$\frac{3 \cdot 0}{π + 3 (1 \frac{π}{3})}$

Step 3.5.1.2

Multiply $\frac{π}{3}$ by $1$ .

$\frac{3 \cdot 0}{π + 3 \frac{π}{3}}$

Step 3.5.2

Cancel the common factor of $3$ .

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Step 3.5.2.1

Cancel the common factor.

$\frac{3 \cdot 0}{π + 3 \frac{π}{3}}$

Step 3.5.2.2

Rewrite the expression.

$\frac{3 \cdot 0}{π + π}$

Step 3.5.3

Add $π$ and $π$ .

$\frac{3 \cdot 0}{2 π}$

Step 3.6

Reduce the expression by cancelling the common factors.

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Step 3.6.1

Multiply $3$ by $0$ .

$\frac{0}{2 π}$

Step 3.6.2

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

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Step 3.6.2.1

Factor $2$ out of $0$ .

$\frac{2 (0)}{2 π}$

Step 3.6.2.2

Cancel the common factors.

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Step 3.6.2.2.1

Factor $2$ out of $2 π$ .

$\frac{2 (0)}{2 (π)}$

Step 3.6.2.2.2

Cancel the common factor.

$\frac{2 \cdot 0}{2 π}$

Step 3.6.2.2.3

Rewrite the expression.

$\frac{0}{π}$

Step 3.6.3

Divide $0$ by $π$ .

$0$