Calculus Examples

$f (x) = 3 \sin (\frac{x}{2}) + 4$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $3 \sin (\frac{x}{2}) + 4$ with respect to $x$ is $\frac{d}{d x} [3 \sin (\frac{x}{2})] + \frac{d}{d x} [4]$ .

$\frac{d}{d x} [3 \sin (\frac{x}{2})] + \frac{d}{d x} [4]$

Step 1.2

Evaluate $\frac{d}{d x} [3 \sin (\frac{x}{2})]$ .

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Step 1.2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 \sin (\frac{x}{2})$ with respect to $x$ is $3 \frac{d}{d x} [\sin (\frac{x}{2})]$ .

$3 \frac{d}{d x} [\sin (\frac{x}{2})] + \frac{d}{d x} [4]$

Step 1.2.2

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

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Step 1.2.2.1

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

$3 (\frac{d}{d u} [\sin (u)] \frac{d}{d x} [\frac{x}{2}]) + \frac{d}{d x} [4]$

Step 1.2.2.2

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

$3 (\cos (u) \frac{d}{d x} [\frac{x}{2}]) + \frac{d}{d x} [4]$

Step 1.2.2.3

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

$3 (\cos (\frac{x}{2}) \frac{d}{d x} [\frac{x}{2}]) + \frac{d}{d x} [4]$

Step 1.2.3

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

$3 (\cos (\frac{x}{2}) (\frac{1}{2} \frac{d}{d x} [x])) + \frac{d}{d x} [4]$

Step 1.2.4

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

$3 (\cos (\frac{x}{2}) (\frac{1}{2} \cdot 1)) + \frac{d}{d x} [4]$

Step 1.2.5

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

$3 (\cos (\frac{x}{2}) \frac{1}{2}) + \frac{d}{d x} [4]$

Step 1.2.6

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

$3 \frac{\cos (\frac{x}{2})}{2} + \frac{d}{d x} [4]$

Step 1.2.7

Combine $3$ and $\frac{\cos (\frac{x}{2})}{2}$ .

$\frac{3 \cos (\frac{x}{2})}{2} + \frac{d}{d x} [4]$

Step 1.3

Differentiate using the Constant Rule.

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Step 1.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$\frac{3 \cos (\frac{x}{2})}{2} + 0$

Step 1.3.2

Add $\frac{3 \cos (\frac{x}{2})}{2}$ and $0$ .

$\frac{3 \cos (\frac{x}{2})}{2}$

Step 2

Find the second derivative of the function.

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Step 2.1

Since $\frac{3}{2}$ is constant with respect to $x$ , the derivative of $\frac{3 \cos (\frac{x}{2})}{2}$ with respect to $x$ is $\frac{3}{2} \frac{d}{d x} [\cos (\frac{x}{2})]$ .

$f'' (x) = \frac{3}{2} \cdot \frac{d}{d x} (\cos (\frac{x}{2}))$

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 $f (x) = \cos (x)$ and $g (x) = \frac{x}{2}$ .

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Step 2.2.1

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

$f'' (x) = \frac{3}{2} \cdot (\frac{d}{d u} (\cos (u)) \frac{d}{d x} (\frac{x}{2}))$

Step 2.2.2

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

$f'' (x) = \frac{3}{2} \cdot (- \sin (u) \frac{d}{d x} \frac{x}{2})$

Step 2.2.3

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

$f'' (x) = \frac{3}{2} \cdot (- \sin (\frac{x}{2}) \frac{d}{d x} \frac{x}{2})$

Step 2.3

Differentiate.

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Step 2.3.1

Combine $\frac{3}{2}$ and $\sin (\frac{x}{2})$ .

$f'' (x) = - \frac{3 \sin (\frac{x}{2})}{2} \cdot \frac{d}{d x} \frac{x}{2}$

Step 2.3.2

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

$f'' (x) = - \frac{3 \sin (\frac{x}{2})}{2} \cdot (\frac{1}{2} \cdot \frac{d}{d x} (x))$

Step 2.3.3

Combine fractions.

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Step 2.3.3.1

Multiply $\frac{1}{2}$ by $\frac{3 \sin (\frac{x}{2})}{2}$ .

$f'' (x) = - \frac{3 \sin (\frac{x}{2})}{2 \cdot 2} \cdot \frac{d}{d x} x$

Step 2.3.3.2

Multiply $2$ by $2$ .

$f'' (x) = - \frac{3 \sin (\frac{x}{2})}{4} \cdot \frac{d}{d x} x$

Step 2.3.4

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

$f'' (x) = - \frac{3 \sin (\frac{x}{2})}{4} \cdot 1$

Step 2.3.5

Multiply $- 1$ by $1$ .

$f'' (x) = - \frac{3 \sin (\frac{x}{2})}{4}$

Step 3

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

$\frac{3 \cos (\frac{x}{2})}{2} = 0$

Step 4

Set the numerator equal to zero.

$3 \cos (\frac{x}{2}) = 0$

Step 5

Solve the equation for $x$ .

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Step 5.1

Divide each term in $3 \cos (\frac{x}{2}) = 0$ by $3$ and simplify.

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Step 5.1.1

Divide each term in $3 \cos (\frac{x}{2}) = 0$ by $3$ .

$\frac{3 \cos (\frac{x}{2})}{3} = \frac{0}{3}$

Step 5.1.2

Simplify the left side.

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Step 5.1.2.1

Cancel the common factor of $3$ .

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Step 5.1.2.1.1

Cancel the common factor.

$\frac{3 \cos (\frac{x}{2})}{3} = \frac{0}{3}$

Step 5.1.2.1.2

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

$\cos (\frac{x}{2}) = \frac{0}{3}$

Step 5.1.3

Simplify the right side.

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Step 5.1.3.1

Divide $0$ by $3$ .

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

Step 5.2

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

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

Step 5.3

Simplify the right side.

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Step 5.3.1

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

$\frac{x}{2} = \frac{π}{2}$

Step 5.4

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

$x = π$

Step 5.5

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 5.6

Solve for $x$ .

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Step 5.6.1

Multiply both sides of the equation by $2$ .

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

Step 5.6.2

Simplify both sides of the equation.

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Step 5.6.2.1

Simplify the left side.

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Step 5.6.2.1.1

Cancel the common factor of $2$ .

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Step 5.6.2.1.1.1

Cancel the common factor.

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

Step 5.6.2.1.1.2

Rewrite the expression.

$x = 2 (2 π - \frac{π}{2})$

Step 5.6.2.2

Simplify the right side.

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Step 5.6.2.2.1

Simplify $2 (2 π - \frac{π}{2})$ .

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Step 5.6.2.2.1.1

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 5.6.2.2.1.2

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 5.6.2.2.1.3

Combine the numerators over the common denominator.

$x = 2 \frac{2 π \cdot 2 - π}{2}$

Step 5.6.2.2.1.4

Cancel the common factor of $2$ .

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Step 5.6.2.2.1.4.1

Cancel the common factor.

$x = 2 \frac{2 π \cdot 2 - π}{2}$

Step 5.6.2.2.1.4.2

Rewrite the expression.

$x = 2 π \cdot 2 - π$

Step 5.6.2.2.1.5

Multiply $2$ by $2$ .

$x = 4 π - π$

Step 5.6.2.2.1.6

Subtract $π$ from $4 π$ .

$x = 3 π$

Step 5.7

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

$x = π, 3 π$

Step 6

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.

$- \frac{3 \sin (\frac{π}{2})}{4}$

Step 7

Evaluate the second derivative.

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Step 7.1

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

$- \frac{3 \cdot 1}{4}$

Step 7.2

Multiply $3$ by $1$ .

$- \frac{3}{4}$

Step 8

$x = π$ 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 9

Find the y-value when $x = π$ .

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Step 9.1

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

$f (π) = 3 \sin (\frac{π}{2}) + 4$

Step 9.2

Simplify the result.

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Step 9.2.1

Simplify each term.

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Step 9.2.1.1

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

$f (π) = 3 \cdot 1 + 4$

Step 9.2.1.2

Multiply $3$ by $1$ .

$f (π) = 3 + 4$

Step 9.2.2

Add $3$ and $4$ .

$f (π) = 7$

Step 9.2.3

The final answer is $7$ .

$y = 7$

Step 10

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.

$- \frac{3 \sin (\frac{3 π}{2})}{4}$

Step 11

Evaluate the second derivative.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

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.

$- \frac{3 (- \sin (\frac{π}{2}))}{4}$

Step 11.1.2

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

$- \frac{3 (- 1 \cdot 1)}{4}$

Step 11.1.3

Multiply $- 1$ by $1$ .

$- \frac{3 \cdot - 1}{4}$

Step 11.2

Simplify the expression.

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Step 11.2.1

Multiply $3$ by $- 1$ .

$- \frac{- 3}{4}$

Step 11.2.2

Move the negative in front of the fraction.

$- - \frac{3}{4}$

Step 11.3

Multiply $- - \frac{3}{4}$ .

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Step 11.3.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{3}{4})$

Step 11.3.2

Multiply $\frac{3}{4}$ by $1$ .

$\frac{3}{4}$

Step 12

$x = 3 π$ 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 13

Find the y-value when $x = 3 π$ .

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Step 13.1

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

$f (3 π) = 3 \sin (\frac{3 π}{2}) + 4$

Step 13.2

Simplify the result.

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Step 13.2.1

Simplify each term.

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Step 13.2.1.1

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.

$f (3 π) = 3 (- \sin (\frac{π}{2})) + 4$

Step 13.2.1.2

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

$f (3 π) = 3 (- 1 \cdot 1) + 4$

Step 13.2.1.3

Multiply $3 (- 1 \cdot 1)$ .

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Step 13.2.1.3.1

Multiply $- 1$ by $1$ .

$f (3 π) = 3 \cdot - 1 + 4$

Step 13.2.1.3.2

Multiply $3$ by $- 1$ .

$f (3 π) = - 3 + 4$

Step 13.2.2

Add $- 3$ and $4$ .

$f (3 π) = 1$

Step 13.2.3

The final answer is $1$ .

$y = 1$

Step 14

These are the local extrema for $f (x) = 3 \sin (\frac{x}{2}) + 4$ .

$(π, 7)$ is a local maxima

$(3 π, 1)$ is a local minima

Step 15