Precalculus Examples

$f (x) = \cos (x)$ , $[π, 3 π]$

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.

$\frac{f (3 π) - f (π)}{(3 π) - (π)}$

Step 2.2

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

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

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

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

$\frac{\cos (π) - \cos (π)}{3 π - π}$

Step 3.1.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

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

Step 3.1.3

The exact value of $\cos (0)$ is $1$ .

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

Step 3.1.4

Multiply $- 1$ by $1$ .

$\frac{- 1 - \cos (π)}{3 π - π}$

Step 3.1.5

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$\frac{- 1 - - \cos (0)}{3 π - π}$

Step 3.1.6

The exact value of $\cos (0)$ is $1$ .

$\frac{- 1 - (- 1 \cdot 1)}{3 π - π}$

Step 3.1.7

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

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Step 3.1.7.1

Multiply $- 1$ by $1$ .

$\frac{- 1 - - 1}{3 π - π}$

Step 3.1.7.2

Multiply $- 1$ by $- 1$ .

$\frac{- 1 + 1}{3 π - π}$

Step 3.1.8

Add $- 1$ and $1$ .

$\frac{0}{3 π - π}$

Step 3.2

Simplify terms.

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Step 3.2.1

Subtract $π$ from $3 π$ .

$\frac{0}{2 π}$

Step 3.2.2

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

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Step 3.2.2.1

Factor $2$ out of $0$ .

$\frac{2 (0)}{2 π}$

Step 3.2.2.2

Cancel the common factors.

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Step 3.2.2.2.1

Factor $2$ out of $2 π$ .

$\frac{2 (0)}{2 (π)}$

Step 3.2.2.2.2

Cancel the common factor.

$\frac{2 \cdot 0}{2 π}$

Step 3.2.2.2.3

Rewrite the expression.

$\frac{0}{π}$

Step 3.2.3

Divide $0$ by $π$ .

$0$