Calculus Examples

$36 x - 9 \tan (x)$

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

Find the first derivative.

Tap for more steps...

Step 1.1.1

By the Sum Rule, the derivative of $36 x - 9 \tan (x)$ with respect to $x$ is $\frac{d}{d x} [36 x] + \frac{d}{d x} [- 9 \tan (x)]$ .

$\frac{d}{d x} [36 x] + \frac{d}{d x} [- 9 \tan (x)]$

Step 1.1.2

Evaluate $\frac{d}{d x} [36 x]$ .

Tap for more steps...

Step 1.1.2.1

Since $36$ is constant with respect to $x$ , the derivative of $36 x$ with respect to $x$ is $36 \frac{d}{d x} [x]$ .

$36 \frac{d}{d x} [x] + \frac{d}{d x} [- 9 \tan (x)]$

Step 1.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$36 \cdot 1 + \frac{d}{d x} [- 9 \tan (x)]$

Step 1.1.2.3

Multiply $36$ by $1$ .

$36 + \frac{d}{d x} [- 9 \tan (x)]$

Step 1.1.3

Evaluate $\frac{d}{d x} [- 9 \tan (x)]$ .

Tap for more steps...

Step 1.1.3.1

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9 \tan (x)$ with respect to $x$ is $- 9 \frac{d}{d x} [\tan (x)]$ .

$36 - 9 \frac{d}{d x} [\tan (x)]$

Step 1.1.3.2

The derivative of $\tan (x)$ with respect to $x$ is $\sec^{2} (x)$ .

$f' (x) = 36 - 9 \sec^{2} (x)$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $36 - 9 \sec^{2} (x)$ .

$36 - 9 \sec^{2} (x)$

Step 2

Set the first derivative equal to $0$ then solve the equation $36 - 9 \sec^{2} (x) = 0$ .

Tap for more steps...

Step 2.1

Set the first derivative equal to $0$ .

$36 - 9 \sec^{2} (x) = 0$

Step 2.2

Subtract $36$ from both sides of the equation.

$- 9 \sec^{2} (x) = - 36$

Step 2.3

Divide each term in $- 9 \sec^{2} (x) = - 36$ by $- 9$ and simplify.

Tap for more steps...

Step 2.3.1

Divide each term in $- 9 \sec^{2} (x) = - 36$ by $- 9$ .

$\frac{- 9 \sec^{2} (x)}{- 9} = \frac{- 36}{- 9}$

Step 2.3.2

Simplify the left side.

Tap for more steps...

Step 2.3.2.1

Cancel the common factor of $- 9$ .

Tap for more steps...

Step 2.3.2.1.1

Cancel the common factor.

$\frac{- 9 \sec^{2} (x)}{- 9} = \frac{- 36}{- 9}$

Step 2.3.2.1.2

Divide $\sec^{2} (x)$ by $1$ .

$\sec^{2} (x) = \frac{- 36}{- 9}$

Step 2.3.3

Simplify the right side.

Tap for more steps...

Step 2.3.3.1

Divide $- 36$ by $- 9$ .

$\sec^{2} (x) = 4$

Step 2.4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$\sec (x) = \pm \sqrt{4}$

Step 2.5

Simplify $\pm \sqrt{4}$ .

Tap for more steps...

Step 2.5.1

Rewrite $4$ as $2^{2}$ .

$\sec (x) = \pm \sqrt{2^{2}}$

Step 2.5.2

Pull terms out from under the radical, assuming positive real numbers.

$\sec (x) = \pm 2$

Step 2.6

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 2.6.1

First, use the positive value of the $\pm$ to find the first solution.

$\sec (x) = 2$

Step 2.6.2

Next, use the negative value of the $\pm$ to find the second solution.

$\sec (x) = - 2$

Step 2.6.3

The complete solution is the result of both the positive and negative portions of the solution.

$\sec (x) = 2, - 2$

Step 2.7

Set up each of the solutions to solve for $x$ .

$\sec (x) = 2$

$\sec (x) = - 2$

Step 2.8

Solve for $x$ in $\sec (x) = 2$ .

Tap for more steps...

Step 2.8.1

Take the inverse secant of both sides of the equation to extract $x$ from inside the secant.

$x = arcsec (2)$

Step 2.8.2

Simplify the right side.

Tap for more steps...

Step 2.8.2.1

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

$x = \frac{π}{3}$

Step 2.8.3

The secant function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$x = 2 π - \frac{π}{3}$

Step 2.8.4

Simplify $2 π - \frac{π}{3}$ .

Tap for more steps...

Step 2.8.4.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = 2 π \cdot \frac{3}{3} - \frac{π}{3}$

Step 2.8.4.2

Combine fractions.

Tap for more steps...

Step 2.8.4.2.1

Combine $2 π$ and $\frac{3}{3}$ .

$x = \frac{2 π \cdot 3}{3} - \frac{π}{3}$

Step 2.8.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 3 - π}{3}$

Step 2.8.4.3

Simplify the numerator.

Tap for more steps...

Step 2.8.4.3.1

Multiply $3$ by $2$ .

$x = \frac{6 π - π}{3}$

Step 2.8.4.3.2

Subtract $π$ from $6 π$ .

$x = \frac{5 π}{3}$

Step 2.8.5

Find the period of $\sec (x)$ .

Tap for more steps...

Step 2.8.5.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 2.8.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 2.8.5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 2.8.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 2.8.6

The period of the $\sec (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = \frac{π}{3} + 2 π n, \frac{5 π}{3} + 2 π n$ , for any integer $n$

Step 2.9

Solve for $x$ in $\sec (x) = - 2$ .

Tap for more steps...

Step 2.9.1

Take the inverse secant of both sides of the equation to extract $x$ from inside the secant.

$x = arcsec (- 2)$

Step 2.9.2

Simplify the right side.

Tap for more steps...

Step 2.9.2.1

The exact value of $arcsec (- 2)$ is $\frac{2 π}{3}$ .

$x = \frac{2 π}{3}$

Step 2.9.3

The secant function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the third quadrant.

$x = 2 π - \frac{2 π}{3}$

Step 2.9.4

Simplify $2 π - \frac{2 π}{3}$ .

Tap for more steps...

Step 2.9.4.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = 2 π \cdot \frac{3}{3} - \frac{2 π}{3}$

Step 2.9.4.2

Combine fractions.

Tap for more steps...

Step 2.9.4.2.1

Combine $2 π$ and $\frac{3}{3}$ .

$x = \frac{2 π \cdot 3}{3} - \frac{2 π}{3}$

Step 2.9.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 3 - 2 π}{3}$

Step 2.9.4.3

Simplify the numerator.

Tap for more steps...

Step 2.9.4.3.1

Multiply $3$ by $2$ .

$x = \frac{6 π - 2 π}{3}$

Step 2.9.4.3.2

Subtract $2 π$ from $6 π$ .

$x = \frac{4 π}{3}$

Step 2.9.5

Find the period of $\sec (x)$ .

Tap for more steps...

Step 2.9.5.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 2.9.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 2.9.5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 2.9.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 2.9.6

The period of the $\sec (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = \frac{2 π}{3} + 2 π n, \frac{4 π}{3} + 2 π n$ , for any integer $n$

Step 2.10

List all of the solutions.

$x = \frac{π}{3} + 2 π n, \frac{5 π}{3} + 2 π n, \frac{2 π}{3} + 2 π n, \frac{4 π}{3} + 2 π n$ , for any integer $n$

Step 3

Find the values where the derivative is undefined.

Tap for more steps...

Step 3.1

Set the argument in $\sec (x)$ equal to $\frac{π}{2} + π n$ to find where the expression is undefined.

$x = \frac{π}{2} + π n$ , for any integer $n$

Step 3.2

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

${x | x = \frac{π}{2} + π n}$ $n$ , for any integer $n$

Step 4

Evaluate $36 x - 9 \tan (x)$ at each $x$ value where the derivative is $0$ or undefined.

Tap for more steps...

Step 4.1

Evaluate at $x = \frac{π}{3}$ .

Tap for more steps...

Step 4.1.1

Substitute $\frac{π}{3}$ for $x$ .

$36 (\frac{π}{3}) - 9 \tan (\frac{π}{3})$

Step 4.1.2

Simplify each term.

Tap for more steps...

Step 4.1.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.1.2.1.1

Factor $3$ out of $36$ .

$3 (12) \frac{π}{3} - 9 \tan (\frac{π}{3})$

Step 4.1.2.1.2

Cancel the common factor.

$3 \cdot 12 \frac{π}{3} - 9 \tan (\frac{π}{3})$

Step 4.1.2.1.3

Rewrite the expression.

$12 π - 9 \tan (\frac{π}{3})$

Step 4.1.2.2

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

$12 π - 9 \sqrt{3}$

Step 4.2

Evaluate at $x = \frac{7 π}{3}$ .

Tap for more steps...

Step 4.2.1

Substitute $\frac{7 π}{3}$ for $x$ .

$36 (\frac{7 π}{3}) - 9 \tan (\frac{7 π}{3})$

Step 4.2.2

Simplify each term.

Tap for more steps...

Step 4.2.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.2.2.1.1

Factor $3$ out of $36$ .

$3 (12) \frac{7 π}{3} - 9 \tan (\frac{7 π}{3})$

Step 4.2.2.1.2

Cancel the common factor.

$3 \cdot 12 \frac{7 π}{3} - 9 \tan (\frac{7 π}{3})$

Step 4.2.2.1.3

Rewrite the expression.

$12 (7 π) - 9 \tan (\frac{7 π}{3})$

Step 4.2.2.2

Multiply $7$ by $12$ .

$84 π - 9 \tan (\frac{7 π}{3})$

Step 4.2.2.3

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

$84 π - 9 \tan (\frac{π}{3})$

Step 4.2.2.4

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

$84 π - 9 \sqrt{3}$

Step 4.3

Evaluate at $x = \frac{13 π}{3}$ .

Tap for more steps...

Step 4.3.1

Substitute $\frac{13 π}{3}$ for $x$ .

$36 (\frac{13 π}{3}) - 9 \tan (\frac{13 π}{3})$

Step 4.3.2

Simplify each term.

Tap for more steps...

Step 4.3.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.3.2.1.1

Factor $3$ out of $36$ .

$3 (12) \frac{13 π}{3} - 9 \tan (\frac{13 π}{3})$

Step 4.3.2.1.2

Cancel the common factor.

$3 \cdot 12 \frac{13 π}{3} - 9 \tan (\frac{13 π}{3})$

Step 4.3.2.1.3

Rewrite the expression.

$12 (13 π) - 9 \tan (\frac{13 π}{3})$

Step 4.3.2.2

Multiply $13$ by $12$ .

$156 π - 9 \tan (\frac{13 π}{3})$

Step 4.3.2.3

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

$156 π - 9 \tan (\frac{π}{3})$

Step 4.3.2.4

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

$156 π - 9 \sqrt{3}$

Step 4.4

Evaluate at $x = \frac{19 π}{3}$ .

Tap for more steps...

Step 4.4.1

Substitute $\frac{19 π}{3}$ for $x$ .

$36 (\frac{19 π}{3}) - 9 \tan (\frac{19 π}{3})$

Step 4.4.2

Simplify each term.

Tap for more steps...

Step 4.4.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.4.2.1.1

Factor $3$ out of $36$ .

$3 (12) \frac{19 π}{3} - 9 \tan (\frac{19 π}{3})$

Step 4.4.2.1.2

Cancel the common factor.

$3 \cdot 12 \frac{19 π}{3} - 9 \tan (\frac{19 π}{3})$

Step 4.4.2.1.3

Rewrite the expression.

$12 (19 π) - 9 \tan (\frac{19 π}{3})$

Step 4.4.2.2

Multiply $19$ by $12$ .

$228 π - 9 \tan (\frac{19 π}{3})$

Step 4.4.2.3

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

$228 π - 9 \tan (\frac{π}{3})$

Step 4.4.2.4

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

$228 π - 9 \sqrt{3}$

Step 4.5

Evaluate at $x = \frac{25 π}{3}$ .

Tap for more steps...

Step 4.5.1

Substitute $\frac{25 π}{3}$ for $x$ .

$36 (\frac{25 π}{3}) - 9 \tan (\frac{25 π}{3})$

Step 4.5.2

Simplify each term.

Tap for more steps...

Step 4.5.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.5.2.1.1

Factor $3$ out of $36$ .

$3 (12) \frac{25 π}{3} - 9 \tan (\frac{25 π}{3})$

Step 4.5.2.1.2

Cancel the common factor.

$3 \cdot 12 \frac{25 π}{3} - 9 \tan (\frac{25 π}{3})$

Step 4.5.2.1.3

Rewrite the expression.

$12 (25 π) - 9 \tan (\frac{25 π}{3})$

Step 4.5.2.2

Multiply $25$ by $12$ .

$300 π - 9 \tan (\frac{25 π}{3})$

Step 4.5.2.3

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

$300 π - 9 \tan (\frac{π}{3})$

Step 4.5.2.4

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

$300 π - 9 \sqrt{3}$

Step 4.6

Evaluate at $x = \frac{5 π}{3}$ .

Tap for more steps...

Step 4.6.1

Substitute $\frac{5 π}{3}$ for $x$ .

$36 (\frac{5 π}{3}) - 9 \tan (\frac{5 π}{3})$

Step 4.6.2

Simplify each term.

Tap for more steps...

Step 4.6.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.6.2.1.1

Factor $3$ out of $36$ .

$3 (12) \frac{5 π}{3} - 9 \tan (\frac{5 π}{3})$

Step 4.6.2.1.2

Cancel the common factor.

$3 \cdot 12 \frac{5 π}{3} - 9 \tan (\frac{5 π}{3})$

Step 4.6.2.1.3

Rewrite the expression.

$12 (5 π) - 9 \tan (\frac{5 π}{3})$

Step 4.6.2.2

Multiply $5$ by $12$ .

$60 π - 9 \tan (\frac{5 π}{3})$

Step 4.6.2.3

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

$60 π - 9 (- \tan (\frac{π}{3}))$

Step 4.6.2.4

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

$60 π - 9 (- \sqrt{3})$

Step 4.6.2.5

Multiply $- 1$ by $- 9$ .

$60 π + 9 \sqrt{3}$

Step 4.7

Evaluate at $x = \frac{11 π}{3}$ .

Tap for more steps...

Step 4.7.1

Substitute $\frac{11 π}{3}$ for $x$ .

$36 (\frac{11 π}{3}) - 9 \tan (\frac{11 π}{3})$

Step 4.7.2

Simplify each term.

Tap for more steps...

Step 4.7.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.7.2.1.1

Factor $3$ out of $36$ .

$3 (12) \frac{11 π}{3} - 9 \tan (\frac{11 π}{3})$

Step 4.7.2.1.2

Cancel the common factor.

$3 \cdot 12 \frac{11 π}{3} - 9 \tan (\frac{11 π}{3})$

Step 4.7.2.1.3

Rewrite the expression.

$12 (11 π) - 9 \tan (\frac{11 π}{3})$

Step 4.7.2.2

Multiply $11$ by $12$ .

$132 π - 9 \tan (\frac{11 π}{3})$

Step 4.7.2.3

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

$132 π - 9 \tan (\frac{5 π}{3})$

Step 4.7.2.4

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

$132 π - 9 (- \tan (\frac{π}{3}))$

Step 4.7.2.5

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

$132 π - 9 (- \sqrt{3})$

Step 4.7.2.6

Multiply $- 1$ by $- 9$ .

$132 π + 9 \sqrt{3}$

Step 4.8

Evaluate at $x = \frac{17 π}{3}$ .

Tap for more steps...

Step 4.8.1

Substitute $\frac{17 π}{3}$ for $x$ .

$36 (\frac{17 π}{3}) - 9 \tan (\frac{17 π}{3})$

Step 4.8.2

Simplify each term.

Tap for more steps...

Step 4.8.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.8.2.1.1

Factor $3$ out of $36$ .

$3 (12) \frac{17 π}{3} - 9 \tan (\frac{17 π}{3})$

Step 4.8.2.1.2

Cancel the common factor.

$3 \cdot 12 \frac{17 π}{3} - 9 \tan (\frac{17 π}{3})$

Step 4.8.2.1.3

Rewrite the expression.

$12 (17 π) - 9 \tan (\frac{17 π}{3})$

Step 4.8.2.2

Multiply $17$ by $12$ .

$204 π - 9 \tan (\frac{17 π}{3})$

Step 4.8.2.3

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

$204 π - 9 \tan (\frac{5 π}{3})$

Step 4.8.2.4

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

$204 π - 9 (- \tan (\frac{π}{3}))$

Step 4.8.2.5

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

$204 π - 9 (- \sqrt{3})$

Step 4.8.2.6

Multiply $- 1$ by $- 9$ .

$204 π + 9 \sqrt{3}$

Step 4.9

Evaluate at $x = \frac{23 π}{3}$ .

Tap for more steps...

Step 4.9.1

Substitute $\frac{23 π}{3}$ for $x$ .

$36 (\frac{23 π}{3}) - 9 \tan (\frac{23 π}{3})$

Step 4.9.2

Simplify each term.

Tap for more steps...

Step 4.9.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.9.2.1.1

Factor $3$ out of $36$ .

$3 (12) \frac{23 π}{3} - 9 \tan (\frac{23 π}{3})$

Step 4.9.2.1.2

Cancel the common factor.

$3 \cdot 12 \frac{23 π}{3} - 9 \tan (\frac{23 π}{3})$

Step 4.9.2.1.3

Rewrite the expression.

$12 (23 π) - 9 \tan (\frac{23 π}{3})$

Step 4.9.2.2

Multiply $23$ by $12$ .

$276 π - 9 \tan (\frac{23 π}{3})$

Step 4.9.2.3

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

$276 π - 9 \tan (\frac{5 π}{3})$

Step 4.9.2.4

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

$276 π - 9 (- \tan (\frac{π}{3}))$

Step 4.9.2.5

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

$276 π - 9 (- \sqrt{3})$

Step 4.9.2.6

Multiply $- 1$ by $- 9$ .

$276 π + 9 \sqrt{3}$

Step 4.10

Evaluate at $x = \frac{29 π}{3}$ .

Tap for more steps...

Step 4.10.1

Substitute $\frac{29 π}{3}$ for $x$ .

$36 (\frac{29 π}{3}) - 9 \tan (\frac{29 π}{3})$

Step 4.10.2

Simplify each term.

Tap for more steps...

Step 4.10.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.10.2.1.1

Factor $3$ out of $36$ .

$3 (12) \frac{29 π}{3} - 9 \tan (\frac{29 π}{3})$

Step 4.10.2.1.2

Cancel the common factor.

$3 \cdot 12 \frac{29 π}{3} - 9 \tan (\frac{29 π}{3})$

Step 4.10.2.1.3

Rewrite the expression.

$12 (29 π) - 9 \tan (\frac{29 π}{3})$

Step 4.10.2.2

Multiply $29$ by $12$ .

$348 π - 9 \tan (\frac{29 π}{3})$

Step 4.10.2.3

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

$348 π - 9 \tan (\frac{5 π}{3})$

Step 4.10.2.4

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

$348 π - 9 (- \tan (\frac{π}{3}))$

Step 4.10.2.5

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

$348 π - 9 (- \sqrt{3})$

Step 4.10.2.6

Multiply $- 1$ by $- 9$ .

$348 π + 9 \sqrt{3}$

Step 4.11

Evaluate at $x = \frac{2 π}{3}$ .

Tap for more steps...

Step 4.11.1

Substitute $\frac{2 π}{3}$ for $x$ .

$36 (\frac{2 π}{3}) - 9 \tan (\frac{2 π}{3})$

Step 4.11.2

Simplify each term.

Tap for more steps...

Step 4.11.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.11.2.1.1

Factor $3$ out of $36$ .

$3 (12) \frac{2 π}{3} - 9 \tan (\frac{2 π}{3})$

Step 4.11.2.1.2

Cancel the common factor.

$3 \cdot 12 \frac{2 π}{3} - 9 \tan (\frac{2 π}{3})$

Step 4.11.2.1.3

Rewrite the expression.

$12 (2 π) - 9 \tan (\frac{2 π}{3})$

Step 4.11.2.2

Multiply $2$ by $12$ .

$24 π - 9 \tan (\frac{2 π}{3})$

Step 4.11.2.3

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

$24 π - 9 (- \tan (\frac{π}{3}))$

Step 4.11.2.4

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

$24 π - 9 (- \sqrt{3})$

Step 4.11.2.5

Multiply $- 1$ by $- 9$ .

$24 π + 9 \sqrt{3}$

Step 4.12

Evaluate at $x = \frac{8 π}{3}$ .

Tap for more steps...

Step 4.12.1

Substitute $\frac{8 π}{3}$ for $x$ .

$36 (\frac{8 π}{3}) - 9 \tan (\frac{8 π}{3})$

Step 4.12.2

Simplify each term.

Tap for more steps...

Step 4.12.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.12.2.1.1

Factor $3$ out of $36$ .

$3 (12) \frac{8 π}{3} - 9 \tan (\frac{8 π}{3})$

Step 4.12.2.1.2

Cancel the common factor.

$3 \cdot 12 \frac{8 π}{3} - 9 \tan (\frac{8 π}{3})$

Step 4.12.2.1.3

Rewrite the expression.

$12 (8 π) - 9 \tan (\frac{8 π}{3})$

Step 4.12.2.2

Multiply $8$ by $12$ .

$96 π - 9 \tan (\frac{8 π}{3})$

Step 4.12.2.3

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

$96 π - 9 \tan (\frac{2 π}{3})$

Step 4.12.2.4

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

$96 π - 9 (- \tan (\frac{π}{3}))$

Step 4.12.2.5

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

$96 π - 9 (- \sqrt{3})$

Step 4.12.2.6

Multiply $- 1$ by $- 9$ .

$96 π + 9 \sqrt{3}$

Step 4.13

Evaluate at $x = \frac{14 π}{3}$ .

Tap for more steps...

Step 4.13.1

Substitute $\frac{14 π}{3}$ for $x$ .

$36 (\frac{14 π}{3}) - 9 \tan (\frac{14 π}{3})$

Step 4.13.2

Simplify each term.

Tap for more steps...

Step 4.13.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.13.2.1.1

Factor $3$ out of $36$ .

$3 (12) \frac{14 π}{3} - 9 \tan (\frac{14 π}{3})$

Step 4.13.2.1.2

Cancel the common factor.

$3 \cdot 12 \frac{14 π}{3} - 9 \tan (\frac{14 π}{3})$

Step 4.13.2.1.3

Rewrite the expression.

$12 (14 π) - 9 \tan (\frac{14 π}{3})$

Step 4.13.2.2

Multiply $14$ by $12$ .

$168 π - 9 \tan (\frac{14 π}{3})$

Step 4.13.2.3

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

$168 π - 9 \tan (\frac{2 π}{3})$

Step 4.13.2.4

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

$168 π - 9 (- \tan (\frac{π}{3}))$

Step 4.13.2.5

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

$168 π - 9 (- \sqrt{3})$

Step 4.13.2.6

Multiply $- 1$ by $- 9$ .

$168 π + 9 \sqrt{3}$

Step 4.14

Evaluate at $x = \frac{20 π}{3}$ .

Tap for more steps...

Step 4.14.1

Substitute $\frac{20 π}{3}$ for $x$ .

$36 (\frac{20 π}{3}) - 9 \tan (\frac{20 π}{3})$

Step 4.14.2

Simplify each term.

Tap for more steps...

Step 4.14.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.14.2.1.1

Factor $3$ out of $36$ .

$3 (12) \frac{20 π}{3} - 9 \tan (\frac{20 π}{3})$

Step 4.14.2.1.2

Cancel the common factor.

$3 \cdot 12 \frac{20 π}{3} - 9 \tan (\frac{20 π}{3})$

Step 4.14.2.1.3

Rewrite the expression.

$12 (20 π) - 9 \tan (\frac{20 π}{3})$

Step 4.14.2.2

Multiply $20$ by $12$ .

$240 π - 9 \tan (\frac{20 π}{3})$

Step 4.14.2.3

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

$240 π - 9 \tan (\frac{2 π}{3})$

Step 4.14.2.4

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

$240 π - 9 (- \tan (\frac{π}{3}))$

Step 4.14.2.5

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

$240 π - 9 (- \sqrt{3})$

Step 4.14.2.6

Multiply $- 1$ by $- 9$ .

$240 π + 9 \sqrt{3}$

Step 4.15

Evaluate at $x = \frac{26 π}{3}$ .

Tap for more steps...

Step 4.15.1

Substitute $\frac{26 π}{3}$ for $x$ .

$36 (\frac{26 π}{3}) - 9 \tan (\frac{26 π}{3})$

Step 4.15.2

Simplify each term.

Tap for more steps...

Step 4.15.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.15.2.1.1

Factor $3$ out of $36$ .

$3 (12) \frac{26 π}{3} - 9 \tan (\frac{26 π}{3})$

Step 4.15.2.1.2

Cancel the common factor.

$3 \cdot 12 \frac{26 π}{3} - 9 \tan (\frac{26 π}{3})$

Step 4.15.2.1.3

Rewrite the expression.

$12 (26 π) - 9 \tan (\frac{26 π}{3})$

Step 4.15.2.2

Multiply $26$ by $12$ .

$312 π - 9 \tan (\frac{26 π}{3})$

Step 4.15.2.3

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

$312 π - 9 \tan (\frac{2 π}{3})$

Step 4.15.2.4

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

$312 π - 9 (- \tan (\frac{π}{3}))$

Step 4.15.2.5

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

$312 π - 9 (- \sqrt{3})$

Step 4.15.2.6

Multiply $- 1$ by $- 9$ .

$312 π + 9 \sqrt{3}$

Step 4.16

Evaluate at $x = \frac{4 π}{3}$ .

Tap for more steps...

Step 4.16.1

Substitute $\frac{4 π}{3}$ for $x$ .

$36 (\frac{4 π}{3}) - 9 \tan (\frac{4 π}{3})$

Step 4.16.2

Simplify each term.

Tap for more steps...

Step 4.16.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.16.2.1.1

Factor $3$ out of $36$ .

$3 (12) \frac{4 π}{3} - 9 \tan (\frac{4 π}{3})$

Step 4.16.2.1.2

Cancel the common factor.

$3 \cdot 12 \frac{4 π}{3} - 9 \tan (\frac{4 π}{3})$

Step 4.16.2.1.3

Rewrite the expression.

$12 (4 π) - 9 \tan (\frac{4 π}{3})$

Step 4.16.2.2

Multiply $4$ by $12$ .

$48 π - 9 \tan (\frac{4 π}{3})$

Step 4.16.2.3

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

$48 π - 9 \tan (\frac{π}{3})$

Step 4.16.2.4

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

$48 π - 9 \sqrt{3}$

Step 4.17

Evaluate at $x = \frac{10 π}{3}$ .

Tap for more steps...

Step 4.17.1

Substitute $\frac{10 π}{3}$ for $x$ .

$36 (\frac{10 π}{3}) - 9 \tan (\frac{10 π}{3})$

Step 4.17.2

Simplify each term.

Tap for more steps...

Step 4.17.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.17.2.1.1

Factor $3$ out of $36$ .

$3 (12) \frac{10 π}{3} - 9 \tan (\frac{10 π}{3})$

Step 4.17.2.1.2

Cancel the common factor.

$3 \cdot 12 \frac{10 π}{3} - 9 \tan (\frac{10 π}{3})$

Step 4.17.2.1.3

Rewrite the expression.

$12 (10 π) - 9 \tan (\frac{10 π}{3})$

Step 4.17.2.2

Multiply $10$ by $12$ .

$120 π - 9 \tan (\frac{10 π}{3})$

Step 4.17.2.3

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

$120 π - 9 \tan (\frac{4 π}{3})$

Step 4.17.2.4

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

$120 π - 9 \tan (\frac{π}{3})$

Step 4.17.2.5

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

$120 π - 9 \sqrt{3}$

Step 4.18

Evaluate at $x = \frac{16 π}{3}$ .

Tap for more steps...

Step 4.18.1

Substitute $\frac{16 π}{3}$ for $x$ .

$36 (\frac{16 π}{3}) - 9 \tan (\frac{16 π}{3})$

Step 4.18.2

Simplify each term.

Tap for more steps...

Step 4.18.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.18.2.1.1

Factor $3$ out of $36$ .

$3 (12) \frac{16 π}{3} - 9 \tan (\frac{16 π}{3})$

Step 4.18.2.1.2

Cancel the common factor.

$3 \cdot 12 \frac{16 π}{3} - 9 \tan (\frac{16 π}{3})$

Step 4.18.2.1.3

Rewrite the expression.

$12 (16 π) - 9 \tan (\frac{16 π}{3})$

Step 4.18.2.2

Multiply $16$ by $12$ .

$192 π - 9 \tan (\frac{16 π}{3})$

Step 4.18.2.3

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

$192 π - 9 \tan (\frac{4 π}{3})$

Step 4.18.2.4

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

$192 π - 9 \tan (\frac{π}{3})$

Step 4.18.2.5

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

$192 π - 9 \sqrt{3}$

Step 4.19

Evaluate at $x = \frac{22 π}{3}$ .

Tap for more steps...

Step 4.19.1

Substitute $\frac{22 π}{3}$ for $x$ .

$36 (\frac{22 π}{3}) - 9 \tan (\frac{22 π}{3})$

Step 4.19.2

Simplify each term.

Tap for more steps...

Step 4.19.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.19.2.1.1

Factor $3$ out of $36$ .

$3 (12) \frac{22 π}{3} - 9 \tan (\frac{22 π}{3})$

Step 4.19.2.1.2

Cancel the common factor.

$3 \cdot 12 \frac{22 π}{3} - 9 \tan (\frac{22 π}{3})$

Step 4.19.2.1.3

Rewrite the expression.

$12 (22 π) - 9 \tan (\frac{22 π}{3})$

Step 4.19.2.2

Multiply $22$ by $12$ .

$264 π - 9 \tan (\frac{22 π}{3})$

Step 4.19.2.3

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

$264 π - 9 \tan (\frac{4 π}{3})$

Step 4.19.2.4

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

$264 π - 9 \tan (\frac{π}{3})$

Step 4.19.2.5

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

$264 π - 9 \sqrt{3}$

Step 4.20

Evaluate at $x = \frac{28 π}{3}$ .

Tap for more steps...

Step 4.20.1

Substitute $\frac{28 π}{3}$ for $x$ .

$36 (\frac{28 π}{3}) - 9 \tan (\frac{28 π}{3})$

Step 4.20.2

Simplify each term.

Tap for more steps...

Step 4.20.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.20.2.1.1

Factor $3$ out of $36$ .

$3 (12) \frac{28 π}{3} - 9 \tan (\frac{28 π}{3})$

Step 4.20.2.1.2

Cancel the common factor.

$3 \cdot 12 \frac{28 π}{3} - 9 \tan (\frac{28 π}{3})$

Step 4.20.2.1.3

Rewrite the expression.

$12 (28 π) - 9 \tan (\frac{28 π}{3})$

Step 4.20.2.2

Multiply $28$ by $12$ .

$336 π - 9 \tan (\frac{28 π}{3})$

Step 4.20.2.3

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

$336 π - 9 \tan (\frac{4 π}{3})$

Step 4.20.2.4

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

$336 π - 9 \tan (\frac{π}{3})$

Step 4.20.2.5

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

$336 π - 9 \sqrt{3}$

Step 4.21

List all of the points.

$(\frac{π}{3}, 12 π - 9 \sqrt{3}), (\frac{7 π}{3}, 84 π - 9 \sqrt{3}), (\frac{13 π}{3}, 156 π - 9 \sqrt{3}), (\frac{19 π}{3}, 228 π - 9 \sqrt{3}), (\frac{25 π}{3}, 300 π - 9 \sqrt{3}), (\frac{5 π}{3}, 60 π + 9 \sqrt{3}), (\frac{11 π}{3}, 132 π + 9 \sqrt{3}), (\frac{17 π}{3}, 204 π + 9 \sqrt{3}), (\frac{23 π}{3}, 276 π + 9 \sqrt{3}), (\frac{29 π}{3}, 348 π + 9 \sqrt{3}), (\frac{2 π}{3}, 24 π + 9 \sqrt{3}), (\frac{8 π}{3}, 96 π + 9 \sqrt{3}), (\frac{14 π}{3}, 168 π + 9 \sqrt{3}), (\frac{20 π}{3}, 240 π + 9 \sqrt{3}), (\frac{26 π}{3}, 312 π + 9 \sqrt{3}), (\frac{4 π}{3}, 48 π - 9 \sqrt{3}), (\frac{10 π}{3}, 120 π - 9 \sqrt{3}), (\frac{16 π}{3}, 192 π - 9 \sqrt{3}), (\frac{22 π}{3}, 264 π - 9 \sqrt{3}), (\frac{28 π}{3}, 336 π - 9 \sqrt{3})$

Step 5