Precalculus Examples

$f (x) = \cot (x)$ , $[\frac{2 π}{3}, \frac{3 π}{2}]$

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.

$\frac{f (\frac{3 π}{2}) - f (\frac{2 π}{3})}{(\frac{3 π}{2}) - (\frac{2 π}{3})}$

Step 2.2

Substitute the equation $y = \cot (x)$ for $f (\frac{3 π}{2})$ and $f (\frac{2 π}{3})$ , replacing $x$ in the function with the corresponding $x$ value.

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

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

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

Step 3.1.2

Combine.

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

Step 3.2

Apply the distributive property.

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

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 $6$ .

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

Step 3.3.1.2

Cancel the common factor.

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

Step 3.3.1.3

Rewrite the expression.

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

Step 3.3.2

Multiply $3$ by $3$ .

$\frac{6 \cot (\frac{3 π}{2}) + 6 (- \cot (\frac{2 π}{3}))}{9 π + 6 (- \frac{2 π}{3})}$

Step 3.3.3

Cancel the common factor of $3$ .

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Step 3.3.3.1

Move the leading negative in $- \frac{2 π}{3}$ into the numerator.

$\frac{6 \cot (\frac{3 π}{2}) + 6 (- \cot (\frac{2 π}{3}))}{9 π + 6 \frac{- 2 π}{3}}$

Step 3.3.3.2

Factor $3$ out of $6$ .

$\frac{6 \cot (\frac{3 π}{2}) + 6 (- \cot (\frac{2 π}{3}))}{9 π + 3 (2) \frac{- 2 π}{3}}$

Step 3.3.3.3

Cancel the common factor.

$\frac{6 \cot (\frac{3 π}{2}) + 6 (- \cot (\frac{2 π}{3}))}{9 π + 3 \cdot 2 \frac{- 2 π}{3}}$

Step 3.3.3.4

Rewrite the expression.

$\frac{6 \cot (\frac{3 π}{2}) + 6 (- \cot (\frac{2 π}{3}))}{9 π + 2 (- 2 π)}$

Step 3.3.4

Multiply $- 2$ by $2$ .

$\frac{6 \cot (\frac{3 π}{2}) + 6 (- \cot (\frac{2 π}{3}))}{9 π - 4 π}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Factor $6$ out of $6 \cot (\frac{3 π}{2}) + 6 (- \cot (\frac{2 π}{3}))$ .

$\frac{6 (\cot (\frac{3 π}{2}) - \cot (\frac{2 π}{3}))}{9 π - 4 π}$

Step 3.4.2

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

$\frac{6 (- \cot (\frac{π}{2}) - \cot (\frac{2 π}{3}))}{9 π - 4 π}$

Step 3.4.3

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

$\frac{6 (- 0 - \cot (\frac{2 π}{3}))}{9 π - 4 π}$

Step 3.4.4

Multiply $- 1$ by $0$ .

$\frac{6 (0 - \cot (\frac{2 π}{3}))}{9 π - 4 π}$

Step 3.4.5

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

$\frac{6 (0 - - \cot (\frac{π}{3}))}{9 π - 4 π}$

Step 3.4.6

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

$\frac{6 (0 - - \frac{1}{\sqrt{3}})}{9 π - 4 π}$

Step 3.4.7

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{6 (0 - - (\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}))}{9 π - 4 π}$

Step 3.4.8

Combine and simplify the denominator.

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Step 3.4.8.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{6 (0 - - \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}})}{9 π - 4 π}$

Step 3.4.8.2

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{6 (0 - - \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}})}{9 π - 4 π}$

Step 3.4.8.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{6 (0 - - \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}})}{9 π - 4 π}$

Step 3.4.8.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{6 (0 - - \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}})}{9 π - 4 π}$

Step 3.4.8.5

Add $1$ and $1$ .

$\frac{6 (0 - - \frac{\sqrt{3}}{{\sqrt{3}}^{2}})}{9 π - 4 π}$

Step 3.4.8.6

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

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Step 3.4.8.6.1

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

$\frac{6 (0 - - \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}})}{9 π - 4 π}$

Step 3.4.8.6.2

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

$\frac{6 (0 - - \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}})}{9 π - 4 π}$

Step 3.4.8.6.3

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

$\frac{6 (0 - - \frac{\sqrt{3}}{3^{\frac{2}{2}}})}{9 π - 4 π}$

Step 3.4.8.6.4

Cancel the common factor of $2$ .

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Step 3.4.8.6.4.1

Cancel the common factor.

$\frac{6 (0 - - \frac{\sqrt{3}}{3^{\frac{2}{2}}})}{9 π - 4 π}$

Step 3.4.8.6.4.2

Rewrite the expression.

$\frac{6 (0 - - \frac{\sqrt{3}}{3^{1}})}{9 π - 4 π}$

Step 3.4.8.6.5

Evaluate the exponent.

$\frac{6 (0 - - \frac{\sqrt{3}}{3})}{9 π - 4 π}$

Step 3.4.9

Multiply $- - \frac{\sqrt{3}}{3}$ .

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Step 3.4.9.1

Multiply $- 1$ by $- 1$ .

$\frac{6 (0 + 1 \frac{\sqrt{3}}{3})}{9 π - 4 π}$

Step 3.4.9.2

Multiply $\frac{\sqrt{3}}{3}$ by $1$ .

$\frac{6 (0 + \frac{\sqrt{3}}{3})}{9 π - 4 π}$

Step 3.4.10

Add $0$ and $\frac{\sqrt{3}}{3}$ .

$\frac{6 \frac{\sqrt{3}}{3}}{9 π - 4 π}$

Step 3.5

Simplify terms.

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Step 3.5.1

Subtract $4 π$ from $9 π$ .

$\frac{6 \frac{\sqrt{3}}{3}}{5 π}$

Step 3.5.2

Combine $6$ and $\frac{\sqrt{3}}{3}$ .

$\frac{\frac{6 \sqrt{3}}{3}}{5 π}$

Step 3.6

Reduce the expression by cancelling the common factors.

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Step 3.6.1

Reduce the expression $\frac{6 \sqrt{3}}{3}$ by cancelling the common factors.

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Step 3.6.1.1

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

$\frac{\frac{3 (2 \sqrt{3})}{3}}{5 π}$

Step 3.6.1.2

Factor $3$ out of $3$ .

$\frac{\frac{3 (2 \sqrt{3})}{3 (1)}}{5 π}$

Step 3.6.1.3

Cancel the common factor.

$\frac{\frac{3 (2 \sqrt{3})}{3 \cdot 1}}{5 π}$

Step 3.6.1.4

Rewrite the expression.

$\frac{\frac{2 \sqrt{3}}{1}}{5 π}$

Step 3.6.2

Divide $2 \sqrt{3}$ by $1$ .

$\frac{2 \sqrt{3}}{5 π}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{2 \sqrt{3}}{5 π}$

Decimal Form:

$0.22053155 \dots$