Precalculus Examples

$f (x) = \cot (x)$ , $[\frac{π}{6}, \frac{π}{2}]$

Step 1

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

$y = \cot (x)$

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.

$\frac{f (\frac{π}{2}) - f (\frac{π}{6})}{(\frac{π}{2}) - (\frac{π}{6})}$

Step 2.2

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

$\frac{(\cot (\frac{π}{2})) - (\cot (\frac{π}{6}))}{(\frac{π}{2}) - (\frac{π}{6})}$

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 $\frac{\cot (\frac{π}{2}) - \cot (\frac{π}{6})}{\frac{π}{2} - \frac{π}{6}}$ by $\frac{6}{6}$ .

$\frac{6}{6} \cdot \frac{\cot (\frac{π}{2}) - \cot (\frac{π}{6})}{\frac{π}{2} - \frac{π}{6}}$

Step 3.1.2

Combine.

$\frac{6 (\cot (\frac{π}{2}) - \cot (\frac{π}{6}))}{6 (\frac{π}{2} - \frac{π}{6})}$

Step 3.2

Apply the distributive property.

$\frac{6 \cot (\frac{π}{2}) + 6 (- \cot (\frac{π}{6}))}{6 \frac{π}{2} + 6 (- \frac{π}{6})}$

Step 3.3

Simplify by cancelling.

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Step 3.3.1

Cancel the common factor of $2$ .

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Step 3.3.1.1

Factor $2$ out of $6$ .

$\frac{6 \cot (\frac{π}{2}) + 6 (- \cot (\frac{π}{6}))}{2 (3) \frac{π}{2} + 6 (- \frac{π}{6})}$

Step 3.3.1.2

Cancel the common factor.

$\frac{6 \cot (\frac{π}{2}) + 6 (- \cot (\frac{π}{6}))}{2 \cdot 3 \frac{π}{2} + 6 (- \frac{π}{6})}$

Step 3.3.1.3

Rewrite the expression.

$\frac{6 \cot (\frac{π}{2}) + 6 (- \cot (\frac{π}{6}))}{3 π + 6 (- \frac{π}{6})}$

Step 3.3.2

Cancel the common factor of $6$ .

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Step 3.3.2.1

Move the leading negative in $- \frac{π}{6}$ into the numerator.

$\frac{6 \cot (\frac{π}{2}) + 6 (- \cot (\frac{π}{6}))}{3 π + 6 \frac{- π}{6}}$

Step 3.3.2.2

Cancel the common factor.

$\frac{6 \cot (\frac{π}{2}) + 6 (- \cot (\frac{π}{6}))}{3 π + 6 \frac{- π}{6}}$

Step 3.3.2.3

Rewrite the expression.

$\frac{6 \cot (\frac{π}{2}) + 6 (- \cot (\frac{π}{6}))}{3 π - π}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Factor $6$ out of $6 \cot (\frac{π}{2}) + 6 (- \cot (\frac{π}{6}))$ .

$\frac{6 (\cot (\frac{π}{2}) - \cot (\frac{π}{6}))}{3 π - π}$

Step 3.4.2

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

$\frac{6 (0 - \cot (\frac{π}{6}))}{3 π - π}$

Step 3.4.3

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

$\frac{6 (0 - \sqrt{3})}{3 π - π}$

Step 3.4.4

Subtract $\sqrt{3}$ from $0$ .

$\frac{6 \cdot - 1 \sqrt{3}}{3 π - π}$

Step 3.4.5

Combine exponents.

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Step 3.4.5.1

Factor out negative.

$\frac{- (6 \sqrt{3})}{3 π - π}$

Step 3.4.5.2

Multiply $6$ by $- 1$ .

$\frac{- 6 \sqrt{3}}{3 π - π}$

Step 3.5

Simplify terms.

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Step 3.5.1

Subtract $π$ from $3 π$ .

$\frac{- 6 \sqrt{3}}{2 π}$

Step 3.5.2

Cancel the common factor of $- 6$ and $2$ .

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Step 3.5.2.1

Factor $2$ out of $- 6 \sqrt{3}$ .

$\frac{2 (- 3 \sqrt{3})}{2 π}$

Step 3.5.2.2

Cancel the common factors.

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Step 3.5.2.2.1

Factor $2$ out of $2 π$ .

$\frac{2 (- 3 \sqrt{3})}{2 (π)}$

Step 3.5.2.2.2

Cancel the common factor.

$\frac{2 (- 3 \sqrt{3})}{2 π}$

Step 3.5.2.2.3

Rewrite the expression.

$\frac{- 3 \sqrt{3}}{π}$

Step 3.5.3

Move the negative in front of the fraction.

$- \frac{3 \sqrt{3}}{π}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$- \frac{3 \sqrt{3}}{π}$

Decimal Form:

$- 1.65398668 \dots$