Calculus Examples

$y = - 1 + 3 \cos (\frac{3}{2} x)$

Step 1

Find the first derivative of the function.

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Step 1.1

Differentiate.

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Step 1.1.1

By the Sum Rule, the derivative of $- 1 + 3 \cos (\frac{3}{2} x)$ with respect to $x$ is $\frac{d}{d x} [- 1] + \frac{d}{d x} [3 \cos (\frac{3}{2} x)]$ .

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

Step 1.1.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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 $g (x) = \frac{3 x}{2}$ .

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Step 1.2.3.1

To apply the Chain Rule, set $u$ as $\frac{3 x}{2}$ .

$0 + 3 (\frac{d}{d u} [\cos (u)] \frac{d}{d x} [\frac{3 x}{2}])$

Step 1.2.3.2

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

$0 + 3 (- \sin (u) \frac{d}{d x} [\frac{3 x}{2}])$

Step 1.2.3.3

Replace all occurrences of $u$ with $\frac{3 x}{2}$ .

$0 + 3 (- \sin (\frac{3 x}{2}) \frac{d}{d x} [\frac{3 x}{2}])$

Step 1.2.4

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

$0 + 3 (- \sin (\frac{3 x}{2}) (\frac{3}{2} \frac{d}{d x} [x]))$

Step 1.2.5

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

$0 + 3 (- \sin (\frac{3 x}{2}) (\frac{3}{2} \cdot 1))$

Step 1.2.6

Multiply $\frac{3}{2}$ by $1$ .

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

Step 1.2.7

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

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

Step 1.2.8

Multiply $- 1$ by $3$ .

$0 - 3 \frac{3 \sin (\frac{3 x}{2})}{2}$

Step 1.2.9

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

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

Step 1.2.10

Multiply $3$ by $- 3$ .

$0 + \frac{- 9 \sin (\frac{3 x}{2})}{2}$

Step 1.2.11

Move the negative in front of the fraction.

$0 - \frac{9 \sin (\frac{3 x}{2})}{2}$

Step 1.3

Subtract $\frac{9 \sin (\frac{3 x}{2})}{2}$ from $0$ .

$- \frac{9 \sin (\frac{3 x}{2})}{2}$

Step 2

Find the second derivative of the function.

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Step 2.1

Since $- \frac{9}{2}$ is constant with respect to $x$ , the derivative of $- \frac{9 \sin (\frac{3 x}{2})}{2}$ with respect to $x$ is $- \frac{9}{2} \frac{d}{d x} [\sin (\frac{3 x}{2})]$ .

$f'' (x) = - \frac{9}{2} \cdot \frac{d}{d x} \sin (\frac{3 x}{2})$

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 $f (x) = \sin (x)$ and $g (x) = \frac{3 x}{2}$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $\frac{3 x}{2}$ .

$f'' (x) = - \frac{9}{2} \cdot (\frac{d}{d u} (\sin (u)) \frac{d}{d x} (\frac{3 x}{2}))$

Step 2.2.2

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

$f'' (x) = - \frac{9}{2} \cdot (\cos (u) \frac{d}{d x} (\frac{3 x}{2}))$

Step 2.2.3

Replace all occurrences of $u$ with $\frac{3 x}{2}$ .

$f'' (x) = - \frac{9}{2} \cdot (\cos (\frac{3 x}{2}) \frac{d}{d x} (\frac{3 x}{2}))$

Step 2.3

Differentiate.

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Step 2.3.1

Combine $\cos (\frac{3 x}{2})$ and $\frac{9}{2}$ .

$f'' (x) = - \frac{\cos (\frac{3 x}{2}) \cdot 9}{2} \cdot \frac{d}{d x} \frac{3 x}{2}$

Step 2.3.2

Move $9$ to the left of $\cos (\frac{3 x}{2})$ .

$f'' (x) = - \frac{9 \cdot \cos (\frac{3 x}{2})}{2} \cdot \frac{d}{d x} \frac{3 x}{2}$

Step 2.3.3

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

$f'' (x) = - \frac{9 \cos (\frac{3 x}{2})}{2} \cdot (\frac{3}{2} \cdot \frac{d}{d x} (x))$

Step 2.3.4

Combine fractions.

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Step 2.3.4.1

Multiply $\frac{3}{2}$ by $\frac{9 \cos (\frac{3 x}{2})}{2}$ .

$f'' (x) = - \frac{3 (9 \cos (\frac{3 x}{2}))}{2 \cdot 2} \cdot \frac{d}{d x} x$

Step 2.3.4.2

Multiply.

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Step 2.3.4.2.1

Multiply $9$ by $3$ .

$f'' (x) = - \frac{27 \cos (\frac{3 x}{2})}{2 \cdot 2} \cdot \frac{d}{d x} x$

Step 2.3.4.2.2

Multiply $2$ by $2$ .

$f'' (x) = - \frac{27 \cos (\frac{3 x}{2})}{4} \cdot \frac{d}{d x} x$

Step 2.3.5

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

$f'' (x) = - \frac{27 \cos (\frac{3 x}{2})}{4} \cdot 1$

Step 2.3.6

Multiply $- 1$ by $1$ .

$f'' (x) = - \frac{27 \cos (\frac{3 x}{2})}{4}$

Step 3

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

$- \frac{9 \sin (\frac{3 x}{2})}{2} = 0$

Step 4

Set the numerator equal to zero.

$9 \sin (\frac{3 x}{2}) = 0$

Step 5

Solve the equation for $x$ .

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Step 5.1

Divide each term in $9 \sin (\frac{3 x}{2}) = 0$ by $9$ and simplify.

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Step 5.1.1

Divide each term in $9 \sin (\frac{3 x}{2}) = 0$ by $9$ .

$\frac{9 \sin (\frac{3 x}{2})}{9} = \frac{0}{9}$

Step 5.1.2

Simplify the left side.

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Step 5.1.2.1

Cancel the common factor of $9$ .

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Step 5.1.2.1.1

Cancel the common factor.

$\frac{9 \sin (\frac{3 x}{2})}{9} = \frac{0}{9}$

Step 5.1.2.1.2

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

$\sin (\frac{3 x}{2}) = \frac{0}{9}$

Step 5.1.3

Simplify the right side.

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Step 5.1.3.1

Divide $0$ by $9$ .

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

Step 5.2

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

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

Step 5.3

Simplify the right side.

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Step 5.3.1

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

$\frac{3 x}{2} = 0$

Step 5.4

Set the numerator equal to zero.

$3 x = 0$

Step 5.5

Divide each term in $3 x = 0$ by $3$ and simplify.

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Step 5.5.1

Divide each term in $3 x = 0$ by $3$ .

$\frac{3 x}{3} = \frac{0}{3}$

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $3$ .

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Step 5.5.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{0}{3}$

Step 5.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{3}$

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Divide $0$ by $3$ .

$x = 0$

Step 5.6

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.

$\frac{3 x}{2} = π - 0$

Step 5.7

Solve for $x$ .

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Step 5.7.1

Multiply both sides of the equation by $\frac{2}{3}$ .

$\frac{2}{3} \cdot \frac{3 x}{2} = \frac{2}{3} (π - 0)$

Step 5.7.2

Simplify both sides of the equation.

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Step 5.7.2.1

Simplify the left side.

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Step 5.7.2.1.1

Simplify $\frac{2}{3} \cdot \frac{3 x}{2}$ .

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Step 5.7.2.1.1.1

Cancel the common factor of $2$ .

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Step 5.7.2.1.1.1.1

Cancel the common factor.

$\frac{2}{3} \cdot \frac{3 x}{2} = \frac{2}{3} (π - 0)$

Step 5.7.2.1.1.1.2

Rewrite the expression.

$\frac{1}{3} (3 x) = \frac{2}{3} (π - 0)$

Step 5.7.2.1.1.2

Cancel the common factor of $3$ .

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Step 5.7.2.1.1.2.1

Factor $3$ out of $3 x$ .

$\frac{1}{3} (3 (x)) = \frac{2}{3} (π - 0)$

Step 5.7.2.1.1.2.2

Cancel the common factor.

$\frac{1}{3} (3 x) = \frac{2}{3} (π - 0)$

Step 5.7.2.1.1.2.3

Rewrite the expression.

$x = \frac{2}{3} (π - 0)$

Step 5.7.2.2

Simplify the right side.

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Step 5.7.2.2.1

Simplify $\frac{2}{3} (π - 0)$ .

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Step 5.7.2.2.1.1

Subtract $0$ from $π$ .

$x = \frac{2}{3} π$

Step 5.7.2.2.1.2

Combine $\frac{2}{3}$ and $π$ .

$x = \frac{2 π}{3}$

Step 5.8

The solution to the equation $\frac{3 x}{2} = 0$ .

$x = 0, \frac{2 π}{3}$

Step 6

Evaluate the second derivative at $x = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{27 \cos (\frac{3 (0)}{2})}{4}$

Step 7

Evaluate the second derivative.

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Step 7.1

Cancel the common factor of $0$ and $2$ .

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Step 7.1.1

Factor $2$ out of $3 (0)$ .

$- \frac{27 \cos (\frac{2 (3 \cdot (0))}{2})}{4}$

Step 7.1.2

Cancel the common factors.

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Step 7.1.2.1

Factor $2$ out of $2$ .

$- \frac{27 \cos (\frac{2 (3 \cdot (0))}{2 (1)})}{4}$

Step 7.1.2.2

Cancel the common factor.

$- \frac{27 \cos (\frac{2 (3 \cdot (0))}{2 \cdot 1})}{4}$

Step 7.1.2.3

Rewrite the expression.

$- \frac{27 \cos (\frac{3 \cdot (0)}{1})}{4}$

Step 7.1.2.4

Divide $3 \cdot (0)$ by $1$ .

$- \frac{27 \cos (3 \cdot (0))}{4}$

Step 7.2

Simplify the numerator.

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Step 7.2.1

Multiply $3$ by $0$ .

$- \frac{27 \cos (0)}{4}$

Step 7.2.2

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

$- \frac{27 \cdot 1}{4}$

Step 7.3

Multiply $27$ by $1$ .

$- \frac{27}{4}$

Step 8

$x = 0$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 0$ is a local maximum

Step 9

Find the y-value when $x = 0$ .

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Step 9.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = - 1 + 3 \cos (\frac{3}{2} \cdot (0))$

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

Multiply $\frac{3}{2}$ by $0$ .

$f (0) = - 1 + 3 \cos (0)$

Step 9.2.1.2

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

$f (0) = - 1 + 3 \cdot 1$

Step 9.2.1.3

Multiply $3$ by $1$ .

$f (0) = - 1 + 3$

Step 9.2.2

Add $- 1$ and $3$ .

$f (0) = 2$

Step 9.2.3

The final answer is $2$ .

$y = 2$

Step 10

Evaluate the second derivative at $x = \frac{2 π}{3}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{27 \cos (\frac{3 (\frac{2 π}{3})}{2})}{4}$

Step 11

Evaluate the second derivative.

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Step 11.1

Combine $3$ and $\frac{2 π}{3}$ .

$- \frac{27 \cos (\frac{\frac{3 (2 π)}{3}}{2})}{4}$

Step 11.2

Multiply $3$ by $2$ .

$- \frac{27 \cos (\frac{\frac{6 π}{3}}{2})}{4}$

Step 11.3

Reduce the expression by cancelling the common factors.

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Step 11.3.1

Reduce the expression $\frac{6 π}{3}$ by cancelling the common factors.

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Step 11.3.1.1

Factor $3$ out of $6 π$ .

$- \frac{27 \cos (\frac{\frac{3 (2 π)}{3}}{2})}{4}$

Step 11.3.1.2

Factor $3$ out of $3$ .

$- \frac{27 \cos (\frac{\frac{3 (2 π)}{3 (1)}}{2})}{4}$

Step 11.3.1.3

Cancel the common factor.

$- \frac{27 \cos (\frac{\frac{3 (2 π)}{3 \cdot 1}}{2})}{4}$

Step 11.3.1.4

Rewrite the expression.

$- \frac{27 \cos (\frac{\frac{2 π}{1}}{2})}{4}$

Step 11.3.2

Divide $2 π$ by $1$ .

$- \frac{27 \cos (\frac{2 π}{2})}{4}$

Step 11.4

Simplify the numerator.

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Step 11.4.1

Cancel the common factor of $2$ .

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Step 11.4.1.1

Cancel the common factor.

$- \frac{27 \cos (\frac{2 π}{2})}{4}$

Step 11.4.1.2

Divide $π$ by $1$ .

$- \frac{27 \cos (π)}{4}$

Step 11.4.2

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.

$- \frac{27 (- \cos (0))}{4}$

Step 11.4.3

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

$- \frac{27 (- 1 \cdot 1)}{4}$

Step 11.4.4

Multiply $- 1$ by $1$ .

$- \frac{27 \cdot - 1}{4}$

Step 11.5

Simplify the expression.

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Step 11.5.1

Multiply $27$ by $- 1$ .

$- \frac{- 27}{4}$

Step 11.5.2

Move the negative in front of the fraction.

$- - \frac{27}{4}$

Step 11.6

Multiply $- - \frac{27}{4}$ .

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Step 11.6.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{27}{4})$

Step 11.6.2

Multiply $\frac{27}{4}$ by $1$ .

$\frac{27}{4}$

Step 12

$x = \frac{2 π}{3}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \frac{2 π}{3}$ is a local minimum

Step 13

Find the y-value when $x = \frac{2 π}{3}$ .

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Step 13.1

Replace the variable $x$ with $\frac{2 π}{3}$ in the expression.

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

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

Cancel the common factor of $3$ .

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Step 13.2.1.1.1

Cancel the common factor.

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

Step 13.2.1.1.2

Rewrite the expression.

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

Step 13.2.1.2

Cancel the common factor of $2$ .

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Step 13.2.1.2.1

Factor $2$ out of $2 π$ .

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

Step 13.2.1.2.2

Cancel the common factor.

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

Step 13.2.1.2.3

Rewrite the expression.

$f (\frac{2 π}{3}) = - 1 + 3 \cos (π)$

Step 13.2.1.3

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.

$f (\frac{2 π}{3}) = - 1 + 3 (- \cos (0))$

Step 13.2.1.4

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

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

Step 13.2.1.5

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

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Step 13.2.1.5.1

Multiply $- 1$ by $1$ .

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

Step 13.2.1.5.2

Multiply $3$ by $- 1$ .

$f (\frac{2 π}{3}) = - 1 - 3$

Step 13.2.2

Subtract $3$ from $- 1$ .

$f (\frac{2 π}{3}) = - 4$

Step 13.2.3

The final answer is $- 4$ .

$y = - 4$

Step 14

These are the local extrema for $f (x) = - 1 + 3 \cos (\frac{3}{2} x)$ .

$(0, 2)$ is a local maxima

$(\frac{2 π}{3}, - 4)$ is a local minima

Step 15