Calculus Examples

$f (x) = \sin (\frac{x}{2})$ , $[0, 4 π]$

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

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = \frac{x}{2}$ .

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Step 1.1.1.1.1

To apply the Chain Rule, set $u$ as $\frac{x}{2}$ .

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

Step 1.1.1.1.2

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

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

Step 1.1.1.1.3

Replace all occurrences of $u$ with $\frac{x}{2}$ .

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

Step 1.1.1.2

Differentiate.

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Step 1.1.1.2.1

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x]$ .

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

Step 1.1.1.2.2

Combine $\frac{1}{2}$ and $\cos (\frac{x}{2})$ .

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

Step 1.1.1.2.3

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

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

Step 1.1.1.2.4

Multiply $\frac{\cos (\frac{x}{2})}{2}$ by $1$ .

$f' (x) = \frac{\cos (\frac{x}{2})}{2}$

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{\cos (\frac{x}{2})}{2}$ .

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

Step 1.2

Set the first derivative equal to $0$ then solve the equation $\frac{\cos (\frac{x}{2})}{2} = 0$ .

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Step 1.2.1

Set the first derivative equal to $0$ .

$\frac{\cos (\frac{x}{2})}{2} = 0$

Step 1.2.2

Set the numerator equal to zero.

$\cos (\frac{x}{2}) = 0$

Step 1.2.3

Solve the equation for $x$ .

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Step 1.2.3.1

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

$\frac{x}{2} = \arccos (0)$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

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

$\frac{x}{2} = \frac{π}{2}$

Step 1.2.3.3

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$x = π$

Step 1.2.3.4

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.

$\frac{x}{2} = 2 π - \frac{π}{2}$

Step 1.2.3.5

Solve for $x$ .

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Step 1.2.3.5.1

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 (2 π - \frac{π}{2})$

Step 1.2.3.5.2

Simplify both sides of the equation.

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Step 1.2.3.5.2.1

Simplify the left side.

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Step 1.2.3.5.2.1.1

Cancel the common factor of $2$ .

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Step 1.2.3.5.2.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 (2 π - \frac{π}{2})$

Step 1.2.3.5.2.1.1.2

Rewrite the expression.

$x = 2 (2 π - \frac{π}{2})$

Step 1.2.3.5.2.2

Simplify the right side.

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Step 1.2.3.5.2.2.1

Simplify $2 (2 π - \frac{π}{2})$ .

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Step 1.2.3.5.2.2.1.1

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

$x = 2 (2 π \cdot \frac{2}{2} - \frac{π}{2})$

Step 1.2.3.5.2.2.1.2

Combine $2 π$ and $\frac{2}{2}$ .

$x = 2 (\frac{2 π \cdot 2}{2} - \frac{π}{2})$

Step 1.2.3.5.2.2.1.3

Combine the numerators over the common denominator.

$x = 2 \frac{2 π \cdot 2 - π}{2}$

Step 1.2.3.5.2.2.1.4

Cancel the common factor of $2$ .

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Step 1.2.3.5.2.2.1.4.1

Cancel the common factor.

$x = 2 \frac{2 π \cdot 2 - π}{2}$

Step 1.2.3.5.2.2.1.4.2

Rewrite the expression.

$x = 2 π \cdot 2 - π$

Step 1.2.3.5.2.2.1.5

Multiply $2$ by $2$ .

$x = 4 π - π$

Step 1.2.3.5.2.2.1.6

Subtract $π$ from $4 π$ .

$x = 3 π$

Step 1.2.3.6

Find the period of $\cos (\frac{x}{2})$ .

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Step 1.2.3.6.1

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

$\frac{2 π}{| b |}$

Step 1.2.3.6.2

Replace $b$ with $\frac{1}{2}$ in the formula for period.

$\frac{2 π}{| \frac{1}{2} |}$

Step 1.2.3.6.3

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{2}}$

Step 1.2.3.6.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 2$

Step 1.2.3.6.5

Multiply $2$ by $2$ .

$4 π$

Step 1.2.3.7

The period of the $\cos (\frac{x}{2})$ function is $4 π$ so values will repeat every $4 π$ radians in both directions.

$x = π + 4 π n, 3 π + 4 π n$ , for any integer $n$

Step 1.2.4

Consolidate the answers.

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

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Step 1.4.1

Evaluate at $x = π$ .

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Step 1.4.1.1

Substitute $π$ for $x$ .

$\sin (\frac{π}{2})$

Step 1.4.1.2

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

$1$

Step 1.4.2

Evaluate at $x = 3 π$ .

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Step 1.4.2.1

Substitute $3 π$ for $x$ .

$\sin (\frac{3 π}{2})$

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 sine is negative in the fourth quadrant.

$- \sin (\frac{π}{2})$

Step 1.4.2.2.2

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

$(π + 4 π n, 1), (3 π + 4 π n, - 1)$ , for any integer $n$

Step 2

Exclude the points that are not on the interval.

$(π, 1), (3 π, - 1)$

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 $x$ .

$\sin (\frac{0}{2})$

Step 3.1.2

Simplify.

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Step 3.1.2.1

Divide $0$ by $2$ .

$\sin (0)$

Step 3.1.2.2

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

$0$

Step 3.2

Evaluate at $x = 4 π$ .

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Step 3.2.1

Substitute $4 π$ for $x$ .

$\sin (\frac{4 π}{2})$

Step 3.2.2

Simplify.

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Step 3.2.2.1

Cancel the common factor of $4$ and $2$ .

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Step 3.2.2.1.1

Factor $2$ out of $4 π$ .

$\sin (\frac{2 (2 π)}{2})$

Step 3.2.2.1.2

Cancel the common factors.

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Step 3.2.2.1.2.1

Factor $2$ out of $2$ .

$\sin (\frac{2 (2 π)}{2 (1)})$

Step 3.2.2.1.2.2

Cancel the common factor.

$\sin (\frac{2 (2 π)}{2 \cdot 1})$

Step 3.2.2.1.2.3

Rewrite the expression.

$\sin (\frac{2 π}{1})$

Step 3.2.2.1.2.4

Divide $2 π$ by $1$ .

$\sin (2 π)$

Step 3.2.2.2

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

$\sin (0)$

Step 3.2.2.3

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

$0$

Step 3.3

List all of the points.

$(0, 0), (4 π, 0)$

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: $(π, 1)$

Absolute Minimum: $(3 π, - 1)$

Step 5