Precalculus Examples

$g (t) = 2 \cos^{2} (t)$ ; $[0, \frac{π}{6}]$

Step 1

Write $g (t) = 2 \cos^{2} (t)$ as an equation.

$y = 2 \cos^{2} (t)$

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.

$\frac{f (\frac{π}{6}) - f (0)}{(\frac{π}{6}) - (0)}$

Step 2.2

Substitute the equation $y = 2 \cos^{2} (t)$ for $f (\frac{π}{6})$ and $f (0)$ , replacing $t$ in the function with the corresponding $t$ value.

$\frac{(2 \cos^{2} (\frac{π}{6})) - (2 \cos^{2} (0))}{(\frac{π}{6}) - (0)}$

Step 3

Simplify the expression.

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Step 3.1

Multiply the numerator and denominator of the fraction by $6$ .

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Step 3.1.1

Multiply $\frac{2 \cos^{2} (\frac{π}{6}) - 1 \cdot 2 \cos^{2} (0)}{\frac{π}{6} - (0)}$ by $\frac{6}{6}$ .

$\frac{6}{6} \cdot \frac{2 \cos^{2} (\frac{π}{6}) - 1 \cdot 2 \cos^{2} (0)}{\frac{π}{6} - (0)}$

Step 3.1.2

Combine.

$\frac{6 (2 \cos^{2} (\frac{π}{6}) - 1 \cdot 2 \cos^{2} (0))}{6 (\frac{π}{6} - (0))}$

Step 3.2

Apply the distributive property.

$\frac{6 (2 \cos^{2} (\frac{π}{6})) + 6 (- 1 \cdot 2 \cos^{2} (0))}{6 \frac{π}{6} + 6 (- (0))}$

Step 3.3

Cancel the common factor of $6$ .

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Step 3.3.1

Cancel the common factor.

$\frac{6 (2 \cos^{2} (\frac{π}{6})) + 6 (- 1 \cdot 2 \cos^{2} (0))}{6 \frac{π}{6} + 6 (- (0))}$

Step 3.3.2

Rewrite the expression.

$\frac{6 (2 \cos^{2} (\frac{π}{6})) + 6 (- 1 \cdot 2 \cos^{2} (0))}{π + 6 (- (0))}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Multiply $6$ by $2$ .

$\frac{12 \cos^{2} (\frac{π}{6}) + 6 (- 1 \cdot 2 \cos^{2} (0))}{π + 6 (- (0))}$

Step 3.4.2

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$\frac{12 {(\frac{\sqrt{3}}{2})}^{2} + 6 (- 1 \cdot 2 \cos^{2} (0))}{π + 6 (- (0))}$

Step 3.4.3

Apply the product rule to $\frac{\sqrt{3}}{2}$ .

$\frac{12 \frac{{\sqrt{3}}^{2}}{2^{2}} + 6 (- 1 \cdot 2 \cos^{2} (0))}{π + 6 (- (0))}$

Step 3.4.4

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.4.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{12 \frac{{(3^{\frac{1}{2}})}^{2}}{2^{2}} + 6 (- 1 \cdot 2 \cos^{2} (0))}{π + 6 (- (0))}$

Step 3.4.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{12 \frac{3^{\frac{1}{2} \cdot 2}}{2^{2}} + 6 (- 1 \cdot 2 \cos^{2} (0))}{π + 6 (- (0))}$

Step 3.4.4.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{12 \frac{3^{\frac{2}{2}}}{2^{2}} + 6 (- 1 \cdot 2 \cos^{2} (0))}{π + 6 (- (0))}$

Step 3.4.4.4

Cancel the common factor of $2$ .

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Step 3.4.4.4.1

Cancel the common factor.

$\frac{12 \frac{3^{\frac{2}{2}}}{2^{2}} + 6 (- 1 \cdot 2 \cos^{2} (0))}{π + 6 (- (0))}$

Step 3.4.4.4.2

Rewrite the expression.

$\frac{12 \frac{3^{1}}{2^{2}} + 6 (- 1 \cdot 2 \cos^{2} (0))}{π + 6 (- (0))}$

Step 3.4.4.5

Evaluate the exponent.

$\frac{12 \frac{3}{2^{2}} + 6 (- 1 \cdot 2 \cos^{2} (0))}{π + 6 (- (0))}$

Step 3.4.5

Raise $2$ to the power of $2$ .

$\frac{12 (\frac{3}{4}) + 6 (- 1 \cdot 2 \cos^{2} (0))}{π + 6 (- (0))}$

Step 3.4.6

Cancel the common factor of $4$ .

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Step 3.4.6.1

Factor $4$ out of $12$ .

$\frac{4 (3) \frac{3}{4} + 6 (- 1 \cdot 2 \cos^{2} (0))}{π + 6 (- (0))}$

Step 3.4.6.2

Cancel the common factor.

$\frac{4 \cdot 3 \frac{3}{4} + 6 (- 1 \cdot 2 \cos^{2} (0))}{π + 6 (- (0))}$

Step 3.4.6.3

Rewrite the expression.

$\frac{3 \cdot 3 + 6 (- 1 \cdot 2 \cos^{2} (0))}{π + 6 (- (0))}$

Step 3.4.7

Multiply $3$ by $3$ .

$\frac{9 + 6 (- 1 \cdot 2 \cos^{2} (0))}{π + 6 (- (0))}$

Step 3.4.8

Multiply $- 1$ by $2$ .

$\frac{9 + 6 (- 2 \cos^{2} (0))}{π + 6 (- (0))}$

Step 3.4.9

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

$\frac{9 + 6 (- 2 \cdot 1^{2})}{π + 6 (- (0))}$

Step 3.4.10

One to any power is one.

$\frac{9 + 6 (- 2 \cdot 1)}{π + 6 (- (0))}$

Step 3.4.11

Multiply $6 (- 2 \cdot 1)$ .

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Step 3.4.11.1

Multiply $- 2$ by $1$ .

$\frac{9 + 6 \cdot - 2}{π + 6 (- (0))}$

Step 3.4.11.2

Multiply $6$ by $- 2$ .

$\frac{9 - 12}{π + 6 (- (0))}$

Step 3.4.12

Subtract $12$ from $9$ .

$\frac{- 3}{π + 6 (- (0))}$

Step 3.5

Simplify the denominator.

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Step 3.5.1

Multiply $6 (- (0))$ .

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Step 3.5.1.1

Multiply $- 1$ by $0$ .

$\frac{- 3}{π + 6 \cdot 0}$

Step 3.5.1.2

Multiply $6$ by $0$ .

$\frac{- 3}{π + 0}$

Step 3.5.2

Add $π$ and $0$ .

$\frac{- 3}{π}$

Step 3.6

Move the negative in front of the fraction.

$- \frac{3}{π}$