Precalculus Examples

$f (x) = \cot (x)$ , $[\frac{3 π}{4}, \frac{5 π}{4}]$

Step 1

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

$y = \cot (x)$

Step 2

Substitute using the average rate of change formula.

Tap for more steps...

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.

$\frac{f (\frac{5 π}{4}) - f (\frac{3 π}{4})}{(\frac{5 π}{4}) - (\frac{3 π}{4})}$

Step 2.2

Substitute the equation $y = \cot (x)$ for $f (\frac{5 π}{4})$ and $f (\frac{3 π}{4})$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(\cot (\frac{5 π}{4})) - (\cot (\frac{3 π}{4}))}{(\frac{5 π}{4}) - (\frac{3 π}{4})}$

Step 3

Simplify the expression.

Tap for more steps...

Step 3.1

Multiply the numerator and denominator of the fraction by $4$ .

Tap for more steps...

Step 3.1.1

Multiply $\frac{\cot (\frac{5 π}{4}) - \cot (\frac{3 π}{4})}{\frac{5 π}{4} - \frac{3 π}{4}}$ by $\frac{4}{4}$ .

$\frac{4}{4} \cdot \frac{\cot (\frac{5 π}{4}) - \cot (\frac{3 π}{4})}{\frac{5 π}{4} - \frac{3 π}{4}}$

Step 3.1.2

Combine.

$\frac{4 (\cot (\frac{5 π}{4}) - \cot (\frac{3 π}{4}))}{4 (\frac{5 π}{4} - \frac{3 π}{4})}$

Step 3.2

Apply the distributive property.

$\frac{4 \cot (\frac{5 π}{4}) + 4 (- \cot (\frac{3 π}{4}))}{4 \frac{5 π}{4} + 4 (- \frac{3 π}{4})}$

Step 3.3

Simplify by cancelling.

Tap for more steps...

Step 3.3.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 3.3.1.1

Cancel the common factor.

$\frac{4 \cot (\frac{5 π}{4}) + 4 (- \cot (\frac{3 π}{4}))}{4 \frac{5 π}{4} + 4 (- \frac{3 π}{4})}$

Step 3.3.1.2

Rewrite the expression.

$\frac{4 \cot (\frac{5 π}{4}) + 4 (- \cot (\frac{3 π}{4}))}{5 π + 4 (- \frac{3 π}{4})}$

Step 3.3.2

Cancel the common factor of $4$ .

Tap for more steps...

Step 3.3.2.1

Move the leading negative in $- \frac{3 π}{4}$ into the numerator.

$\frac{4 \cot (\frac{5 π}{4}) + 4 (- \cot (\frac{3 π}{4}))}{5 π + 4 \frac{- 3 π}{4}}$

Step 3.3.2.2

Cancel the common factor.

$\frac{4 \cot (\frac{5 π}{4}) + 4 (- \cot (\frac{3 π}{4}))}{5 π + 4 \frac{- 3 π}{4}}$

Step 3.3.2.3

Rewrite the expression.

$\frac{4 \cot (\frac{5 π}{4}) + 4 (- \cot (\frac{3 π}{4}))}{5 π - 3 π}$

Step 3.4

Simplify the numerator.

Tap for more steps...

Step 3.4.1

Factor $4$ out of $4 \cot (\frac{5 π}{4}) + 4 (- \cot (\frac{3 π}{4}))$ .

$\frac{4 (\cot (\frac{5 π}{4}) - \cot (\frac{3 π}{4}))}{5 π - 3 π}$

Step 3.4.2

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

$\frac{4 (\cot (\frac{π}{4}) - \cot (\frac{3 π}{4}))}{5 π - 3 π}$

Step 3.4.3

The exact value of $\cot (\frac{π}{4})$ is $1$ .

$\frac{4 (1 - \cot (\frac{3 π}{4}))}{5 π - 3 π}$

Step 3.4.4

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.

$\frac{4 (1 - - \cot (\frac{π}{4}))}{5 π - 3 π}$

Step 3.4.5

The exact value of $\cot (\frac{π}{4})$ is $1$ .

$\frac{4 (1 - (- 1 \cdot 1))}{5 π - 3 π}$

Step 3.4.6

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

Tap for more steps...

Step 3.4.6.1

Multiply $- 1$ by $1$ .

$\frac{4 (1 - - 1)}{5 π - 3 π}$

Step 3.4.6.2

Multiply $- 1$ by $- 1$ .

$\frac{4 (1 + 1)}{5 π - 3 π}$

Step 3.4.7

Add $1$ and $1$ .

$\frac{4 \cdot 2}{5 π - 3 π}$

Step 3.5

Simplify terms.

Tap for more steps...

Step 3.5.1

Subtract $3 π$ from $5 π$ .

$\frac{4 \cdot 2}{2 π}$

Step 3.5.2

Multiply $4$ by $2$ .

$\frac{8}{2 π}$

Step 3.5.3

Cancel the common factor of $8$ and $2$ .

Tap for more steps...

Step 3.5.3.1

Factor $2$ out of $8$ .

$\frac{2 \cdot 4}{2 π}$

Step 3.5.3.2

Cancel the common factors.

Tap for more steps...

Step 3.5.3.2.1

Factor $2$ out of $2 π$ .

$\frac{2 \cdot 4}{2 (π)}$

Step 3.5.3.2.2

Cancel the common factor.

$\frac{2 \cdot 4}{2 π}$

Step 3.5.3.2.3

Rewrite the expression.

$\frac{4}{π}$