Calculus Examples

$f (x) = \cos (x)$ , $[π, 3 π]$

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

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

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

Step 1.1.2

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

$- \sin (x)$

Step 1.2

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

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Step 1.2.1

Set the first derivative equal to $0$ .

$- \sin (x) = 0$

Step 1.2.2

Divide each term in $- \sin (x) = 0$ by $- 1$ and simplify.

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Step 1.2.2.1

Divide each term in $- \sin (x) = 0$ by $- 1$ .

$\frac{- \sin (x)}{- 1} = \frac{0}{- 1}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{\sin (x)}{1} = \frac{0}{- 1}$

Step 1.2.2.2.2

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

$\sin (x) = \frac{0}{- 1}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $0$ by $- 1$ .

$\sin (x) = 0$

Step 1.2.3

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

$x = \arcsin (0)$

Step 1.2.4

Simplify the right side.

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Step 1.2.4.1

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

$x = 0$

Step 1.2.5

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

$x = π - 0$

Step 1.2.6

Subtract $0$ from $π$ .

$x = π$

Step 1.2.7

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

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Step 1.2.7.1

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

$\frac{2 π}{| b |}$

Step 1.2.7.2

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

$\frac{2 π}{| 1 |}$

Step 1.2.7.3

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

$\frac{2 π}{1}$

Step 1.2.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 1.2.8

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

$x = 2 π n, π + 2 π n$ , for any integer $n$

Step 1.2.9

Consolidate the answers.

$x = π n$ , 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 $\cos (x)$ at each $x$ value where the derivative is $0$ or undefined.

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Step 1.4.1

Evaluate at $x = 0$ .

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Step 1.4.1.1

Substitute $0$ for $x$ .

$\cos (0)$

Step 1.4.1.2

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

$1$

Step 1.4.2

Evaluate at $x = π$ .

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Step 1.4.2.1

Substitute $π$ for $x$ .

$\cos (π)$

Step 1.4.2.2

Simplify.

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Step 1.4.2.2.1

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

$- \cos (0)$

Step 1.4.2.2.2

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

$- 1 \cdot 1$

Step 1.4.2.2.3

Multiply $- 1$ by $1$ .

$- 1$

Step 1.4.3

List all of the points.

$(0 + 2 π n, 1), (π + 2 π n, - 1)$ , for any integer $n$

Step 2

Exclude the points that are not on the interval.

$(2 π, 1), (π, - 1), (3 π, - 1)$

Step 3

Evaluate at the included endpoints.

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Step 3.1

Evaluate at $x = π$ .

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Step 3.1.1

Substitute $π$ for $x$ .

$\cos (π)$

Step 3.1.2

Simplify.

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Step 3.1.2.1

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

$- \cos (0)$

Step 3.1.2.2

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

$- 1 \cdot 1$

Step 3.1.2.3

Multiply $- 1$ by $1$ .

$- 1$

Step 3.2

Evaluate at $x = 3 π$ .

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Step 3.2.1

Substitute $3 π$ for $x$ .

$\cos (3 π)$

Step 3.2.2

Simplify.

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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 $2 π$ .

$\cos (π)$

Step 3.2.2.2

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

$- \cos (0)$

Step 3.2.2.3

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

$- 1 \cdot 1$

Step 3.2.2.4

Multiply $- 1$ by $1$ .

$- 1$

Step 3.3

List all of the points.

$(π, - 1), (3 π, - 1)$

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: $(2 π, 1)$

Absolute Minimum: $(π, - 1), (3 π, - 1)$

Step 5