Calculus Examples

$y = \frac{5}{2} \cdot \sin (\frac{1}{2} x)$

Step 1

Write $y = \frac{5}{2} \cdot \sin (\frac{1}{2} x)$ as a function.

$f (x) = \frac{5}{2} \cdot \sin (\frac{1}{2} x)$

Step 2

Find the first derivative of the function.

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Step 2.1

Since $\frac{5}{2}$ is constant with respect to $x$ , the derivative of $\frac{5}{2} \cdot \sin (\frac{1}{2} x)$ with respect to $x$ is $\frac{5}{2} \frac{d}{d x} [\sin (\frac{1}{2} x)]$ .

$\frac{5}{2} \frac{d}{d x} [\sin (\frac{1}{2} x)]$

Step 2.2

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 $g (x) = \frac{1}{2} x$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $\frac{1}{2} x$ .

$\frac{5}{2} (\frac{d}{d u} [\sin (u)] \frac{d}{d x} [\frac{1}{2} x])$

Step 2.2.2

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

$\frac{5}{2} (\cos (u) \frac{d}{d x} [\frac{1}{2} x])$

Step 2.2.3

Replace all occurrences of $u$ with $\frac{1}{2} x$ .

$\frac{5}{2} (\cos (\frac{1}{2} x) \frac{d}{d x} [\frac{1}{2} x])$

Step 2.3

Differentiate.

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Step 2.3.1

Combine $\frac{1}{2}$ and $x$ .

$\frac{5}{2} (\cos (\frac{x}{2}) \frac{d}{d x} [\frac{1}{2} x])$

Step 2.3.2

Combine fractions.

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Step 2.3.2.1

Combine $\frac{1}{2}$ and $x$ .

$\frac{5}{2} (\cos (\frac{x}{2}) \frac{d}{d x} [\frac{x}{2}])$

Step 2.3.2.2

Combine $\cos (\frac{x}{2})$ and $\frac{5}{2}$ .

$\frac{\cos (\frac{x}{2}) \cdot 5}{2} \frac{d}{d x} [\frac{x}{2}]$

Step 2.3.2.3

Move $5$ to the left of $\cos (\frac{x}{2})$ .

$\frac{5 \cdot \cos (\frac{x}{2})}{2} \frac{d}{d x} [\frac{x}{2}]$

Step 2.3.3

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

$\frac{5 \cos (\frac{x}{2})}{2} (\frac{1}{2} \frac{d}{d x} [x])$

Step 2.3.4

Combine fractions.

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Step 2.3.4.1

Multiply $\frac{1}{2}$ by $\frac{5 \cos (\frac{x}{2})}{2}$ .

$\frac{5 \cos (\frac{x}{2})}{2 \cdot 2} \frac{d}{d x} [x]$

Step 2.3.4.2

Multiply $2$ by $2$ .

$\frac{5 \cos (\frac{x}{2})}{4} \frac{d}{d x} [x]$

Step 2.3.5

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

$\frac{5 \cos (\frac{x}{2})}{4} \cdot 1$

Step 2.3.6

Multiply $\frac{5 \cos (\frac{x}{2})}{4}$ by $1$ .

$\frac{5 \cos (\frac{x}{2})}{4}$

Step 3

Find the second derivative of the function.

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Step 3.1

Since $\frac{5}{4}$ is constant with respect to $x$ , the derivative of $\frac{5 \cos (\frac{x}{2})}{4}$ with respect to $x$ is $\frac{5}{4} \frac{d}{d x} [\cos (\frac{x}{2})]$ .

$f'' (x) = \frac{5}{4} \cdot \frac{d}{d x} (\cos (\frac{x}{2}))$

Step 3.2

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

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Step 3.2.1

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

$f'' (x) = \frac{5}{4} \cdot (\frac{d}{d u} (\cos (u)) \frac{d}{d x} (\frac{x}{2}))$

Step 3.2.2

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

$f'' (x) = \frac{5}{4} \cdot (- \sin (u) \frac{d}{d x} \frac{x}{2})$

Step 3.2.3

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

$f'' (x) = \frac{5}{4} \cdot (- \sin (\frac{x}{2}) \frac{d}{d x} \frac{x}{2})$

Step 3.3

Differentiate.

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Step 3.3.1

Combine $\frac{5}{4}$ and $\sin (\frac{x}{2})$ .

$f'' (x) = - \frac{5 \sin (\frac{x}{2})}{4} \cdot \frac{d}{d x} \frac{x}{2}$

Step 3.3.2

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

$f'' (x) = - \frac{5 \sin (\frac{x}{2})}{4} \cdot (\frac{1}{2} \cdot \frac{d}{d x} (x))$

Step 3.3.3

Combine fractions.

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Step 3.3.3.1

Multiply $\frac{1}{2}$ by $\frac{5 \sin (\frac{x}{2})}{4}$ .

$f'' (x) = - \frac{5 \sin (\frac{x}{2})}{2 \cdot 4} \cdot \frac{d}{d x} x$

Step 3.3.3.2

Multiply $2$ by $4$ .

$f'' (x) = - \frac{5 \sin (\frac{x}{2})}{8} \cdot \frac{d}{d x} x$

Step 3.3.4

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

$f'' (x) = - \frac{5 \sin (\frac{x}{2})}{8} \cdot 1$

Step 3.3.5

Multiply $- 1$ by $1$ .

$f'' (x) = - \frac{5 \sin (\frac{x}{2})}{8}$

Step 4

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$\frac{5 \cos (\frac{x}{2})}{4} = 0$

Step 5

Set the numerator equal to zero.

$5 \cos (\frac{x}{2}) = 0$

Step 6

Solve the equation for $x$ .

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Step 6.1

Divide each term in $5 \cos (\frac{x}{2}) = 0$ by $5$ and simplify.

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Step 6.1.1

Divide each term in $5 \cos (\frac{x}{2}) = 0$ by $5$ .

$\frac{5 \cos (\frac{x}{2})}{5} = \frac{0}{5}$

Step 6.1.2

Simplify the left side.

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Step 6.1.2.1

Cancel the common factor of $5$ .

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Step 6.1.2.1.1

Cancel the common factor.

$\frac{5 \cos (\frac{x}{2})}{5} = \frac{0}{5}$

Step 6.1.2.1.2

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

$\cos (\frac{x}{2}) = \frac{0}{5}$

Step 6.1.3

Simplify the right side.

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Step 6.1.3.1

Divide $0$ by $5$ .

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

Step 6.2

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

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

Step 6.3

Simplify the right side.

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Step 6.3.1

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

$\frac{x}{2} = \frac{π}{2}$

Step 6.4

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

$x = π$

Step 6.5

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 6.6

Solve for $x$ .

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Step 6.6.1

Multiply both sides of the equation by $2$ .

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

Step 6.6.2

Simplify both sides of the equation.

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Step 6.6.2.1

Simplify the left side.

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Step 6.6.2.1.1

Cancel the common factor of $2$ .

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Step 6.6.2.1.1.1

Cancel the common factor.

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

Step 6.6.2.1.1.2

Rewrite the expression.

$x = 2 (2 π - \frac{π}{2})$

Step 6.6.2.2

Simplify the right side.

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Step 6.6.2.2.1

Simplify $2 (2 π - \frac{π}{2})$ .

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Step 6.6.2.2.1.1

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 6.6.2.2.1.2

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 6.6.2.2.1.3

Combine the numerators over the common denominator.

$x = 2 \frac{2 π \cdot 2 - π}{2}$

Step 6.6.2.2.1.4

Cancel the common factor of $2$ .

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Step 6.6.2.2.1.4.1

Cancel the common factor.

$x = 2 \frac{2 π \cdot 2 - π}{2}$

Step 6.6.2.2.1.4.2

Rewrite the expression.

$x = 2 π \cdot 2 - π$

Step 6.6.2.2.1.5

Multiply $2$ by $2$ .

$x = 4 π - π$

Step 6.6.2.2.1.6

Subtract $π$ from $4 π$ .

$x = 3 π$

Step 6.7

The solution to the equation $\frac{x}{2} = \frac{π}{2}$ .

$x = π, 3 π$

Step 7

Evaluate the second derivative at $x = π$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{5 \sin (\frac{π}{2})}{8}$

Step 8

Evaluate the second derivative.

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Step 8.1

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

$- \frac{5 \cdot 1}{8}$

Step 8.2

Multiply $5$ by $1$ .

$- \frac{5}{8}$

Step 9

$x = π$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = π$ is a local maximum

Step 10

Find the y-value when $x = π$ .

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Step 10.1

Replace the variable $x$ with $π$ in the expression.

$f (π) = \frac{5}{2} \cdot \sin (\frac{1}{2} \cdot (π))$

Step 10.2

Simplify the result.

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Step 10.2.1

Combine $\frac{1}{2}$ and $π$ .

$f (π) = \frac{5}{2} \cdot \sin (\frac{π}{2})$

Step 10.2.2

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

$f (π) = \frac{5}{2} \cdot 1$

Step 10.2.3

Multiply $\frac{5}{2}$ by $1$ .

$f (π) = \frac{5}{2}$

Step 10.2.4

The final answer is $\frac{5}{2}$ .

$y = \frac{5}{2}$

Step 11

Evaluate the second derivative at $x = 3 π$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{5 \sin (\frac{3 π}{2})}{8}$

Step 12

Evaluate the second derivative.

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Step 12.1

Simplify the numerator.

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Step 12.1.1

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.

$- \frac{5 (- \sin (\frac{π}{2}))}{8}$

Step 12.1.2

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

$- \frac{5 (- 1 \cdot 1)}{8}$

Step 12.1.3

Multiply $- 1$ by $1$ .

$- \frac{5 \cdot - 1}{8}$

Step 12.2

Simplify the expression.

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Step 12.2.1

Multiply $5$ by $- 1$ .

$- \frac{- 5}{8}$

Step 12.2.2

Move the negative in front of the fraction.

$- - \frac{5}{8}$

Step 12.3

Multiply $- - \frac{5}{8}$ .

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Step 12.3.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{5}{8})$

Step 12.3.2

Multiply $\frac{5}{8}$ by $1$ .

$\frac{5}{8}$

Step 13

$x = 3 π$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 3 π$ is a local minimum

Step 14

Find the y-value when $x = 3 π$ .

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Step 14.1

Replace the variable $x$ with $3 π$ in the expression.

$f (3 π) = \frac{5}{2} \cdot \sin (\frac{1}{2} \cdot (3 π))$

Step 14.2

Simplify the result.

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Step 14.2.1

Multiply $\frac{1}{2} (3 π)$ .

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Step 14.2.1.1

Combine $3$ and $\frac{1}{2}$ .

$f (3 π) = \frac{5}{2} \cdot \sin (\frac{3}{2} \cdot π)$

Step 14.2.1.2

Combine $\frac{3}{2}$ and $π$ .

$f (3 π) = \frac{5}{2} \cdot \sin (\frac{3 π}{2})$

Step 14.2.2

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.

$f (3 π) = \frac{5}{2} \cdot (- \sin (\frac{π}{2}))$

Step 14.2.3

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

$f (3 π) = \frac{5}{2} \cdot (- 1 \cdot 1)$

Step 14.2.4

Multiply $- 1$ by $1$ .

$f (3 π) = \frac{5}{2} \cdot - 1$

Step 14.2.5

Multiply $\frac{5}{2} \cdot - 1$ .

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Step 14.2.5.1

Combine $\frac{5}{2}$ and $- 1$ .

$f (3 π) = \frac{5 \cdot - 1}{2}$

Step 14.2.5.2

Multiply $5$ by $- 1$ .

$f (3 π) = \frac{- 5}{2}$

Step 14.2.6

Move the negative in front of the fraction.

$f (3 π) = - \frac{5}{2}$

Step 14.2.7

The final answer is $- \frac{5}{2}$ .

$y = - \frac{5}{2}$

Step 15

These are the local extrema for $f (x) = \frac{5}{2} \cdot \sin (\frac{1}{2} x)$ .

$(π, \frac{5}{2})$ is a local maxima

$(3 π, - \frac{5}{2})$ is a local minima

Step 16