Calculus Examples

$f (x) = \cos (x) - x$ , $[\frac{π}{2}, 2 π]$

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 $\cos (x) - x$ with respect to $x$ is $\frac{d}{d x} [\cos (x)] + \frac{d}{d x} [- x]$ .

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

Step 1.1.1.2

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

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

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

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

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 $n = 1$ .

$- \sin (x) - 1 \cdot 1$

Step 1.1.1.3.3

Multiply $- 1$ by $1$ .

$- \sin (x) - 1$

Step 1.1.1.4

Reorder terms.

$f' (x) = - 1 - \sin (x)$

Step 1.1.2

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

$- 1 - \sin (x)$

Step 1.2

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

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Step 1.2.1

Set the first derivative equal to $0$ .

$- 1 - \sin (x) = 0$

Step 1.2.2

Add $1$ to both sides of the equation.

$- \sin (x) = 1$

Step 1.2.3

Divide each term in $- \sin (x) = 1$ by $- 1$ and simplify.

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Step 1.2.3.1

Divide each term in $- \sin (x) = 1$ by $- 1$ .

$\frac{- \sin (x)}{- 1} = \frac{1}{- 1}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Dividing two negative values results in a positive value.

$\frac{\sin (x)}{1} = \frac{1}{- 1}$

Step 1.2.3.2.2

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

$\sin (x) = \frac{1}{- 1}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Divide $1$ by $- 1$ .

$\sin (x) = - 1$

Step 1.2.4

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

$x = \arcsin (- 1)$

Step 1.2.5

Simplify the right side.

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Step 1.2.5.1

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

$x = - \frac{π}{2}$

Step 1.2.6

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $2 π$ , to find a reference angle. Next, add this reference angle to $π$ to find the solution in the third quadrant.

$x = 2 π + \frac{π}{2} + π$

Step 1.2.7

Simplify the expression to find the second solution.

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Step 1.2.7.1

Subtract $2 π$ from $2 π + \frac{π}{2} + π$ .

$x = 2 π + \frac{π}{2} + π - 2 π$

Step 1.2.7.2

The resulting angle of $\frac{3 π}{2}$ is positive, less than $2 π$ , and coterminal with $2 π + \frac{π}{2} + π$ .

$x = \frac{3 π}{2}$

Step 1.2.8

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

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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

Add $2 π$ to every negative angle to get positive angles.

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Step 1.2.9.1

Add $2 π$ to $- \frac{π}{2}$ to find the positive angle.

$- \frac{π}{2} + 2 π$

Step 1.2.9.2

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

$2 π \cdot \frac{2}{2} - \frac{π}{2}$

Step 1.2.9.3

Combine fractions.

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Step 1.2.9.3.1

Combine $2 π$ and $\frac{2}{2}$ .

$\frac{2 π \cdot 2}{2} - \frac{π}{2}$

Step 1.2.9.3.2

Combine the numerators over the common denominator.

$\frac{2 π \cdot 2 - π}{2}$

Step 1.2.9.4

Simplify the numerator.

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Step 1.2.9.4.1

Multiply $2$ by $2$ .

$\frac{4 π - π}{2}$

Step 1.2.9.4.2

Subtract $π$ from $4 π$ .

$\frac{3 π}{2}$

Step 1.2.9.5

List the new angles.

$x = \frac{3 π}{2}$

Step 1.2.10

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

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

Step 1.2.11

Consolidate the answers.

$x = \frac{3 π}{2} + 2 π n$ , 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 $\cos (x) - x$ at each $x$ value where the derivative is $0$ or undefined.

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Step 1.4.1

Evaluate at $x = \frac{3 π}{2}$ .

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Step 1.4.1.1

Substitute $\frac{3 π}{2}$ for $x$ .

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

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

Simplify each term.

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Step 1.4.1.2.1.1

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

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

Step 1.4.1.2.1.2

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

$0 - \frac{3 π}{2}$

Step 1.4.1.2.2

Subtract $\frac{3 π}{2}$ from $0$ .

$- \frac{3 π}{2}$

Step 1.4.2

Evaluate at $x = \frac{7 π}{2}$ .

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Step 1.4.2.1

Substitute $\frac{7 π}{2}$ for $x$ .

$\cos (\frac{7 π}{2}) - (\frac{7 π}{2})$

Step 1.4.2.2

Simplify.

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Step 1.4.2.2.1

Simplify each term.

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Step 1.4.2.2.1.1

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

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

Step 1.4.2.2.1.2

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

$\cos (\frac{π}{2}) - (\frac{7 π}{2})$

Step 1.4.2.2.1.3

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

$0 - \frac{7 π}{2}$

Step 1.4.2.2.2

Subtract $\frac{7 π}{2}$ from $0$ .

$- \frac{7 π}{2}$

Step 1.4.3

Evaluate at $x = \frac{11 π}{2}$ .

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Step 1.4.3.1

Substitute $\frac{11 π}{2}$ for $x$ .

$\cos (\frac{11 π}{2}) - (\frac{11 π}{2})$

Step 1.4.3.2

Simplify.

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Step 1.4.3.2.1

Simplify each term.

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Step 1.4.3.2.1.1

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

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

Step 1.4.3.2.1.2

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

$\cos (\frac{π}{2}) - (\frac{11 π}{2})$

Step 1.4.3.2.1.3

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

$0 - \frac{11 π}{2}$

Step 1.4.3.2.2

Subtract $\frac{11 π}{2}$ from $0$ .

$- \frac{11 π}{2}$

Step 1.4.4

Evaluate at $x = \frac{15 π}{2}$ .

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Step 1.4.4.1

Substitute $\frac{15 π}{2}$ for $x$ .

$\cos (\frac{15 π}{2}) - (\frac{15 π}{2})$

Step 1.4.4.2

Simplify.

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Step 1.4.4.2.1

Simplify each term.

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Step 1.4.4.2.1.1

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

$\cos (\frac{3 π}{2}) - (\frac{15 π}{2})$

Step 1.4.4.2.1.2

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

$\cos (\frac{π}{2}) - (\frac{15 π}{2})$

Step 1.4.4.2.1.3

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

$0 - \frac{15 π}{2}$

Step 1.4.4.2.2

Subtract $\frac{15 π}{2}$ from $0$ .

$- \frac{15 π}{2}$

Step 1.4.5

Evaluate at $x = \frac{19 π}{2}$ .

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Step 1.4.5.1

Substitute $\frac{19 π}{2}$ for $x$ .

$\cos (\frac{19 π}{2}) - (\frac{19 π}{2})$

Step 1.4.5.2

Simplify.

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Step 1.4.5.2.1

Simplify each term.

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Step 1.4.5.2.1.1

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

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

Step 1.4.5.2.1.2

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

$\cos (\frac{π}{2}) - (\frac{19 π}{2})$

Step 1.4.5.2.1.3

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

$0 - \frac{19 π}{2}$

Step 1.4.5.2.2

Subtract $\frac{19 π}{2}$ from $0$ .

$- \frac{19 π}{2}$

Step 1.4.6

List all of the points.

$(\frac{3 π}{2}, - \frac{3 π}{2}), (\frac{7 π}{2}, - \frac{7 π}{2}), (\frac{11 π}{2}, - \frac{11 π}{2}), (\frac{15 π}{2}, - \frac{15 π}{2}), (\frac{19 π}{2}, - \frac{19 π}{2})$

Step 2

Exclude the points that are not on the interval.

$(\frac{3 π}{2}, - \frac{3 π}{2})$

Step 3

Evaluate at the included endpoints.

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Step 3.1

Evaluate at $x = \frac{π}{2}$ .

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Step 3.1.1

Substitute $\frac{π}{2}$ for $x$ .

$\cos (\frac{π}{2}) - (\frac{π}{2})$

Step 3.1.2

Simplify.

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Step 3.1.2.1

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

$0 - \frac{π}{2}$

Step 3.1.2.2

Subtract $\frac{π}{2}$ from $0$ .

$- \frac{π}{2}$

Step 3.2

Evaluate at $x = 2 π$ .

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Step 3.2.1

Substitute $2 π$ for $x$ .

$\cos (2 π) - (2 π)$

Step 3.2.2

Simplify each term.

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Step 3.2.2.1

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

$\cos (0) - (2 π)$

Step 3.2.2.2

The exact value of $\cos (0)$ is $1$ .

$1 - (2 π)$

Step 3.2.2.3

Multiply $2$ by $- 1$ .

$1 - 2 π$

Step 3.3

List all of the points.

$(\frac{π}{2}, - \frac{π}{2}), (2 π, 1 - 2 π)$

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: $(\frac{π}{2}, - \frac{π}{2})$

Absolute Minimum: $(2 π, 1 - 2 π)$

Step 5