Calculus Examples

$f (x) = 2 \sin (x) - x + 2$ , $[0, \frac{3 π}{2}]$

Step 1

Find the critical points.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

By the Sum Rule, the derivative of $2 \sin (x) - x + 2$ with respect to $x$ is $\frac{d}{d x} [2 \sin (x)] + \frac{d}{d x} [- x] + \frac{d}{d x} [2]$ .

$\frac{d}{d x} [2 \sin (x)] + \frac{d}{d x} [- x] + \frac{d}{d x} [2]$

Step 1.1.1.2

Evaluate $\frac{d}{d x} [2 \sin (x)]$ .

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Step 1.1.1.2.1

Since $2$ is constant with respect to $x$ , the derivative of $2 \sin (x)$ with respect to $x$ is $2 \frac{d}{d x} [\sin (x)]$ .

$2 \frac{d}{d x} [\sin (x)] + \frac{d}{d x} [- x] + \frac{d}{d x} [2]$

Step 1.1.1.2.2

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$2 \cos (x) + \frac{d}{d x} [- x] + \frac{d}{d x} [2]$

Step 1.1.1.3

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

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Step 1.1.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$2 \cos (x) - \frac{d}{d x} [x] + \frac{d}{d x} [2]$

Step 1.1.1.3.2

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

$2 \cos (x) - 1 \cdot 1 + \frac{d}{d x} [2]$

Step 1.1.1.3.3

Multiply $- 1$ by $1$ .

$2 \cos (x) - 1 + \frac{d}{d x} [2]$

Step 1.1.1.4

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$2 \cos (x) - 1 + 0$

Step 1.1.1.5

Simplify.

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Step 1.1.1.5.1

Add $2 \cos (x) - 1$ and $0$ .

$2 \cos (x) - 1$

Step 1.1.1.5.2

Reorder terms.

$f' (x) = - 1 + 2 \cos (x)$

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $- 1 + 2 \cos (x)$ .

$- 1 + 2 \cos (x)$

Step 1.2

Set the first derivative equal to $0$ then solve the equation $- 1 + 2 \cos (x) = 0$ .

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Step 1.2.1

Set the first derivative equal to $0$ .

$- 1 + 2 \cos (x) = 0$

Step 1.2.2

Add $1$ to both sides of the equation.

$2 \cos (x) = 1$

Step 1.2.3

Divide each term in $2 \cos (x) = 1$ by $2$ and simplify.

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Step 1.2.3.1

Divide each term in $2 \cos (x) = 1$ by $2$ .

$\frac{2 \cos (x)}{2} = \frac{1}{2}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $2$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{2 \cos (x)}{2} = \frac{1}{2}$

Step 1.2.3.2.1.2

Divide $\cos (x)$ by $1$ .

$\cos (x) = \frac{1}{2}$

Step 1.2.4

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

$x = \arccos (\frac{1}{2})$

Step 1.2.5

Simplify the right side.

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Step 1.2.5.1

The exact value of $\arccos (\frac{1}{2})$ is $\frac{π}{3}$ .

$x = \frac{π}{3}$

Step 1.2.6

The cosine 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 1.2.7

Simplify $2 π - \frac{π}{3}$ .

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Step 1.2.7.1

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 1.2.7.2

Combine fractions.

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Step 1.2.7.2.1

Combine $2 π$ and $\frac{3}{3}$ .

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

Step 1.2.7.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 3 - π}{3}$

Step 1.2.7.3

Simplify the numerator.

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Step 1.2.7.3.1

Multiply $3$ by $2$ .

$x = \frac{6 π - π}{3}$

Step 1.2.7.3.2

Subtract $π$ from $6 π$ .

$x = \frac{5 π}{3}$

Step 1.2.8

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

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Step 1.2.8.1

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

$\frac{2 π}{| b |}$

Step 1.2.8.2

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

$\frac{2 π}{| 1 |}$

Step 1.2.8.3

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

$\frac{2 π}{1}$

Step 1.2.8.4

Divide $2 π$ by $1$ .

$2 π$

Step 1.2.9

The period of the $\cos (x)$ 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 1.3

Find the values where the derivative is undefined.

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Step 1.3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 1.4

Evaluate $2 \sin (x) - x + 2$ at each $x$ value where the derivative is $0$ or undefined.

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Step 1.4.1

Evaluate at $x = \frac{π}{3}$ .

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Step 1.4.1.1

Substitute $\frac{π}{3}$ for $x$ .

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

Step 1.4.1.2

Simplify each term.

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Step 1.4.1.2.1

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

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

Step 1.4.1.2.2

Cancel the common factor of $2$ .

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Step 1.4.1.2.2.1

Cancel the common factor.

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

Step 1.4.1.2.2.2

Rewrite the expression.

$\sqrt{3} - (\frac{π}{3}) + 2$

$\sqrt{3} - \frac{π}{3} + 2$

Step 1.4.2

Evaluate at $x = \frac{7 π}{3}$ .

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Step 1.4.2.1

Substitute $\frac{7 π}{3}$ for $x$ .

$2 \sin (\frac{7 π}{3}) - (\frac{7 π}{3}) + 2$

Step 1.4.2.2

Simplify each term.

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Step 1.4.2.2.1

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

$2 \sin (\frac{π}{3}) - (\frac{7 π}{3}) + 2$

Step 1.4.2.2.2

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

$2 \frac{\sqrt{3}}{2} - (\frac{7 π}{3}) + 2$

Step 1.4.2.2.3

Cancel the common factor of $2$ .

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Step 1.4.2.2.3.1

Cancel the common factor.

$2 \frac{\sqrt{3}}{2} - (\frac{7 π}{3}) + 2$

Step 1.4.2.2.3.2

Rewrite the expression.

$\sqrt{3} - (\frac{7 π}{3}) + 2$

$\sqrt{3} - \frac{7 π}{3} + 2$

Step 1.4.3

Evaluate at $x = \frac{13 π}{3}$ .

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Step 1.4.3.1

Substitute $\frac{13 π}{3}$ for $x$ .

$2 \sin (\frac{13 π}{3}) - (\frac{13 π}{3}) + 2$

Step 1.4.3.2

Simplify each term.

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Step 1.4.3.2.1

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

$2 \sin (\frac{π}{3}) - (\frac{13 π}{3}) + 2$

Step 1.4.3.2.2

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

$2 \frac{\sqrt{3}}{2} - (\frac{13 π}{3}) + 2$

Step 1.4.3.2.3

Cancel the common factor of $2$ .

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Step 1.4.3.2.3.1

Cancel the common factor.

$2 \frac{\sqrt{3}}{2} - (\frac{13 π}{3}) + 2$

Step 1.4.3.2.3.2

Rewrite the expression.

$\sqrt{3} - (\frac{13 π}{3}) + 2$

$\sqrt{3} - \frac{13 π}{3} + 2$

Step 1.4.4

Evaluate at $x = \frac{19 π}{3}$ .

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Step 1.4.4.1

Substitute $\frac{19 π}{3}$ for $x$ .

$2 \sin (\frac{19 π}{3}) - (\frac{19 π}{3}) + 2$

Step 1.4.4.2

Simplify each term.

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Step 1.4.4.2.1

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

$2 \sin (\frac{π}{3}) - (\frac{19 π}{3}) + 2$

Step 1.4.4.2.2

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

$2 \frac{\sqrt{3}}{2} - (\frac{19 π}{3}) + 2$

Step 1.4.4.2.3

Cancel the common factor of $2$ .

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Step 1.4.4.2.3.1

Cancel the common factor.

$2 \frac{\sqrt{3}}{2} - (\frac{19 π}{3}) + 2$

Step 1.4.4.2.3.2

Rewrite the expression.

$\sqrt{3} - (\frac{19 π}{3}) + 2$

$\sqrt{3} - \frac{19 π}{3} + 2$

Step 1.4.5

Evaluate at $x = \frac{25 π}{3}$ .

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Step 1.4.5.1

Substitute $\frac{25 π}{3}$ for $x$ .

$2 \sin (\frac{25 π}{3}) - (\frac{25 π}{3}) + 2$

Step 1.4.5.2

Simplify each term.

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Step 1.4.5.2.1

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

$2 \sin (\frac{π}{3}) - (\frac{25 π}{3}) + 2$

Step 1.4.5.2.2

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

$2 \frac{\sqrt{3}}{2} - (\frac{25 π}{3}) + 2$

Step 1.4.5.2.3

Cancel the common factor of $2$ .

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Step 1.4.5.2.3.1

Cancel the common factor.

$2 \frac{\sqrt{3}}{2} - (\frac{25 π}{3}) + 2$

Step 1.4.5.2.3.2

Rewrite the expression.

$\sqrt{3} - (\frac{25 π}{3}) + 2$

$\sqrt{3} - \frac{25 π}{3} + 2$

Step 1.4.6

Evaluate at $x = \frac{5 π}{3}$ .

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Step 1.4.6.1

Substitute $\frac{5 π}{3}$ for $x$ .

$2 \sin (\frac{5 π}{3}) - (\frac{5 π}{3}) + 2$

Step 1.4.6.2

Simplify each term.

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Step 1.4.6.2.1

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

$2 (- \sin (\frac{π}{3})) - (\frac{5 π}{3}) + 2$

Step 1.4.6.2.2

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

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

Step 1.4.6.2.3

Cancel the common factor of $2$ .

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Step 1.4.6.2.3.1

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

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

Step 1.4.6.2.3.2

Cancel the common factor.

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

Step 1.4.6.2.3.3

Rewrite the expression.

$- \sqrt{3} - (\frac{5 π}{3}) + 2$

$- \sqrt{3} - \frac{5 π}{3} + 2$

Step 1.4.7

Evaluate at $x = \frac{11 π}{3}$ .

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Step 1.4.7.1

Substitute $\frac{11 π}{3}$ for $x$ .

$2 \sin (\frac{11 π}{3}) - (\frac{11 π}{3}) + 2$

Step 1.4.7.2

Simplify each term.

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Step 1.4.7.2.1

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

$2 \sin (\frac{5 π}{3}) - (\frac{11 π}{3}) + 2$

Step 1.4.7.2.2

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

$2 (- \sin (\frac{π}{3})) - (\frac{11 π}{3}) + 2$

Step 1.4.7.2.3

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

$2 (- \frac{\sqrt{3}}{2}) - (\frac{11 π}{3}) + 2$

Step 1.4.7.2.4

Cancel the common factor of $2$ .

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Step 1.4.7.2.4.1

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

$2 \frac{- \sqrt{3}}{2} - (\frac{11 π}{3}) + 2$

Step 1.4.7.2.4.2

Cancel the common factor.

$2 \frac{- \sqrt{3}}{2} - (\frac{11 π}{3}) + 2$

Step 1.4.7.2.4.3

Rewrite the expression.

$- \sqrt{3} - (\frac{11 π}{3}) + 2$

$- \sqrt{3} - \frac{11 π}{3} + 2$

Step 1.4.8

Evaluate at $x = \frac{17 π}{3}$ .

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Step 1.4.8.1

Substitute $\frac{17 π}{3}$ for $x$ .

$2 \sin (\frac{17 π}{3}) - (\frac{17 π}{3}) + 2$

Step 1.4.8.2

Simplify each term.

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Step 1.4.8.2.1

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

$2 \sin (\frac{5 π}{3}) - (\frac{17 π}{3}) + 2$

Step 1.4.8.2.2

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

$2 (- \sin (\frac{π}{3})) - (\frac{17 π}{3}) + 2$

Step 1.4.8.2.3

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

$2 (- \frac{\sqrt{3}}{2}) - (\frac{17 π}{3}) + 2$

Step 1.4.8.2.4

Cancel the common factor of $2$ .

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Step 1.4.8.2.4.1

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

$2 \frac{- \sqrt{3}}{2} - (\frac{17 π}{3}) + 2$

Step 1.4.8.2.4.2

Cancel the common factor.

$2 \frac{- \sqrt{3}}{2} - (\frac{17 π}{3}) + 2$

Step 1.4.8.2.4.3

Rewrite the expression.

$- \sqrt{3} - (\frac{17 π}{3}) + 2$

$- \sqrt{3} - \frac{17 π}{3} + 2$

Step 1.4.9

Evaluate at $x = \frac{23 π}{3}$ .

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Step 1.4.9.1

Substitute $\frac{23 π}{3}$ for $x$ .

$2 \sin (\frac{23 π}{3}) - (\frac{23 π}{3}) + 2$

Step 1.4.9.2

Simplify each term.

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Step 1.4.9.2.1

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

$2 \sin (\frac{5 π}{3}) - (\frac{23 π}{3}) + 2$

Step 1.4.9.2.2

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

$2 (- \sin (\frac{π}{3})) - (\frac{23 π}{3}) + 2$

Step 1.4.9.2.3

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

$2 (- \frac{\sqrt{3}}{2}) - (\frac{23 π}{3}) + 2$

Step 1.4.9.2.4

Cancel the common factor of $2$ .

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Step 1.4.9.2.4.1

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

$2 \frac{- \sqrt{3}}{2} - (\frac{23 π}{3}) + 2$

Step 1.4.9.2.4.2

Cancel the common factor.

$2 \frac{- \sqrt{3}}{2} - (\frac{23 π}{3}) + 2$

Step 1.4.9.2.4.3

Rewrite the expression.

$- \sqrt{3} - (\frac{23 π}{3}) + 2$

$- \sqrt{3} - \frac{23 π}{3} + 2$

Step 1.4.10

Evaluate at $x = \frac{29 π}{3}$ .

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Step 1.4.10.1

Substitute $\frac{29 π}{3}$ for $x$ .

$2 \sin (\frac{29 π}{3}) - (\frac{29 π}{3}) + 2$

Step 1.4.10.2

Simplify each term.

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Step 1.4.10.2.1

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

$2 \sin (\frac{5 π}{3}) - (\frac{29 π}{3}) + 2$

Step 1.4.10.2.2

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

$2 (- \sin (\frac{π}{3})) - (\frac{29 π}{3}) + 2$

Step 1.4.10.2.3

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

$2 (- \frac{\sqrt{3}}{2}) - (\frac{29 π}{3}) + 2$

Step 1.4.10.2.4

Cancel the common factor of $2$ .

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Step 1.4.10.2.4.1

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

$2 \frac{- \sqrt{3}}{2} - (\frac{29 π}{3}) + 2$

Step 1.4.10.2.4.2

Cancel the common factor.

$2 \frac{- \sqrt{3}}{2} - (\frac{29 π}{3}) + 2$

Step 1.4.10.2.4.3

Rewrite the expression.

$- \sqrt{3} - (\frac{29 π}{3}) + 2$

$- \sqrt{3} - \frac{29 π}{3} + 2$

Step 1.4.11

List all of the points.

$(\frac{π}{3}, \sqrt{3} - \frac{π}{3} + 2), (\frac{7 π}{3}, \sqrt{3} - \frac{7 π}{3} + 2), (\frac{13 π}{3}, \sqrt{3} - \frac{13 π}{3} + 2), (\frac{19 π}{3}, \sqrt{3} - \frac{19 π}{3} + 2), (\frac{25 π}{3}, \sqrt{3} - \frac{25 π}{3} + 2), (\frac{5 π}{3}, - \sqrt{3} - \frac{5 π}{3} + 2), (\frac{11 π}{3}, - \sqrt{3} - \frac{11 π}{3} + 2), (\frac{17 π}{3}, - \sqrt{3} - \frac{17 π}{3} + 2), (\frac{23 π}{3}, - \sqrt{3} - \frac{23 π}{3} + 2), (\frac{29 π}{3}, - \sqrt{3} - \frac{29 π}{3} + 2)$

Step 2

Exclude the points that are not on the interval.

$(\frac{π}{3}, \sqrt{3} - \frac{π}{3} + 2)$

Step 3

Evaluate at the included endpoints.

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Step 3.1

Evaluate at $x = 0$ .

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Step 3.1.1

Substitute $0$ for $x$ .

$2 \sin (0) - (0) + 2$

Step 3.1.2

Simplify.

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

The exact value of $\sin (0)$ is $0$ .

$2 \cdot 0 - (0) + 2$

Step 3.1.2.1.2

Multiply $2$ by $0$ .

$0 - (0) + 2$

Step 3.1.2.1.3

Multiply $- 1$ by $0$ .

$0 + 0 + 2$

Step 3.1.2.2

Simplify by adding numbers.

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Step 3.1.2.2.1

Add $0$ and $0$ .

$0 + 2$

Step 3.1.2.2.2

Add $0$ and $2$ .

$2$

Step 3.2

Evaluate at $x = \frac{3 π}{2}$ .

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Step 3.2.1

Substitute $\frac{3 π}{2}$ for $x$ .

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

Step 3.2.2

Simplify.

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Step 3.2.2.1

Simplify each term.

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Step 3.2.2.1.1

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

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

Step 3.2.2.1.2

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$2 (- 1 \cdot 1) - (\frac{3 π}{2}) + 2$

Step 3.2.2.1.3

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

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Step 3.2.2.1.3.1

Multiply $- 1$ by $1$ .

$2 \cdot - 1 - (\frac{3 π}{2}) + 2$

Step 3.2.2.1.3.2

Multiply $2$ by $- 1$ .

$- 2 - (\frac{3 π}{2}) + 2$

$- 2 - \frac{3 π}{2} + 2$

Step 3.2.2.2

Simplify by adding numbers.

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Step 3.2.2.2.1

Add $- 2$ and $2$ .

$0 - \frac{3 π}{2}$

Step 3.2.2.2.2

Subtract $\frac{3 π}{2}$ from $0$ .

$- \frac{3 π}{2}$

Step 3.3

List all of the points.

$(0, 2), (\frac{3 π}{2}, - \frac{3 π}{2})$

Step 4

Compare the $f (x)$ values found for each value of $x$ in order to determine the absolute maximum and minimum over the given interval. The maximum will occur at the highest $f (x)$ value and the minimum will occur at the lowest $f (x)$ value.

Absolute Maximum: $(\frac{π}{3}, \sqrt{3} - \frac{π}{3} + 2)$

Absolute Minimum: $(\frac{3 π}{2}, - \frac{3 π}{2})$

Step 5