Calculus Examples

$y = (5.4) \cos (\frac{π x}{6})$

Step 1

Write $y = (5.4) \cos (\frac{π x}{6})$ as a function.

$f (x) = (5.4) \cos (\frac{π x}{6})$

Step 2

Find the first derivative of the function.

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Step 2.1

Since $5.4$ is constant with respect to $x$ , the derivative of $5.4 \cos (\frac{π x}{6})$ with respect to $x$ is $5.4 \frac{d}{d x} [\cos (\frac{π x}{6})]$ .

$5.4 \frac{d}{d x} [\cos (\frac{π x}{6})]$

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 $g (x) = \frac{π x}{6}$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $\frac{π x}{6}$ .

$5.4 (\frac{d}{d u} [\cos (u)] \frac{d}{d x} [\frac{π x}{6}])$

Step 2.2.2

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

$5.4 (- \sin (u) \frac{d}{d x} [\frac{π x}{6}])$

Step 2.2.3

Replace all occurrences of $u$ with $\frac{π x}{6}$ .

$5.4 (- \sin (\frac{π x}{6}) \frac{d}{d x} [\frac{π x}{6}])$

Step 2.3

Differentiate.

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Step 2.3.1

Multiply $- 1$ by $5.4$ .

$- 5.4 (\sin (\frac{π x}{6}) \frac{d}{d x} [\frac{π x}{6}])$

Step 2.3.2

Since $\frac{π}{6}$ is constant with respect to $x$ , the derivative of $\frac{π x}{6}$ with respect to $x$ is $\frac{π}{6} \frac{d}{d x} [x]$ .

$- 5.4 \sin (\frac{π x}{6}) (\frac{π}{6} \frac{d}{d x} [x])$

Step 2.3.3

Simplify terms.

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Step 2.3.3.1

Combine $\frac{π}{6}$ and $- 5.4$ .

$\frac{π \cdot - 5.4}{6} \sin (\frac{π x}{6}) \frac{d}{d x} [x]$

Step 2.3.3.2

Multiply $π$ by $- 5.4$ .

$\frac{- 16.96460032}{6} \sin (\frac{π x}{6}) \frac{d}{d x} [x]$

Step 2.3.3.3

Combine $\frac{- 16.96460032}{6}$ and $\sin (\frac{π x}{6})$ .

$\frac{- 16.96460032 \sin (\frac{π x}{6})}{6} \frac{d}{d x} [x]$

Step 2.3.3.4

Move the negative in front of the fraction.

$- \frac{16.96460032 \sin (\frac{π x}{6})}{6} \frac{d}{d x} [x]$

Step 2.3.4

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

$- \frac{16.96460032 \sin (\frac{π x}{6})}{6} \cdot 1$

Step 2.3.5

Multiply $- 1$ by $1$ .

$- \frac{16.96460032 \sin (\frac{π x}{6})}{6}$

Step 2.4

Simplify.

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Step 2.4.1

Factor $16.96460032$ out of $16.96460032 \sin (\frac{π x}{6})$ .

$- \frac{16.96460032 (\sin (\frac{π x}{6}))}{6}$

Step 2.4.2

Factor $6$ out of $6$ .

$- \frac{16.96460032 (\sin (\frac{π x}{6}))}{6 (1)}$

Step 2.4.3

Separate fractions.

$- (\frac{16.96460032}{6} \cdot \frac{\sin (\frac{π x}{6})}{1})$

Step 2.4.4

Divide $16.96460032$ by $6$ .

$- (2.82743338 \frac{\sin (\frac{π x}{6})}{1})$

Step 2.4.5

Divide $\sin (\frac{π x}{6})$ by $1$ .

$- (2.82743338 \sin (\frac{π x}{6}))$

Step 2.4.6

Multiply $2.82743338$ by $- 1$ .

$- 2.82743338 \sin (\frac{π x}{6})$

Step 3

Find the second derivative of the function.

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Step 3.1

Since $- 2.82743338$ is constant with respect to $x$ , the derivative of $- 2.82743338 \sin (\frac{π x}{6})$ with respect to $x$ is $- 2.82743338 \frac{d}{d x} [\sin (\frac{π x}{6})]$ .

$f'' (x) = - 2.82743338 \frac{d}{d x} \sin (\frac{π x}{6})$

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 $f (x) = \sin (x)$ and $g (x) = \frac{π x}{6}$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $\frac{π x}{6}$ .

$f'' (x) = - 2.82743338 (\frac{d}{d u} (\sin (u)) \frac{d}{d x} (\frac{π x}{6}))$

Step 3.2.2

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

$f'' (x) = - 2.82743338 (\cos (u) \frac{d}{d x} (\frac{π x}{6}))$

Step 3.2.3

Replace all occurrences of $u$ with $\frac{π x}{6}$ .

$f'' (x) = - 2.82743338 (\cos (\frac{π x}{6}) \frac{d}{d x} (\frac{π x}{6}))$

Step 3.3

Differentiate.

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Step 3.3.1

Since $\frac{π}{6}$ is constant with respect to $x$ , the derivative of $\frac{π x}{6}$ with respect to $x$ is $\frac{π}{6} \frac{d}{d x} [x]$ .

$f'' (x) = - 2.82743338 \cos (\frac{π x}{6}) (\frac{π}{6} \cdot \frac{d}{d x} (x))$

Step 3.3.2

Simplify terms.

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Step 3.3.2.1

Combine $\frac{π}{6}$ and $- 2.82743338$ .

$f'' (x) = \frac{π \cdot - 2.82743338}{6} \cdot \cos (\frac{π x}{6}) \frac{d}{d x} (x)$

Step 3.3.2.2

Multiply $π$ by $- 2.82743338$ .

$f'' (x) = \frac{- 8.88264396}{6} \cdot \cos (\frac{π x}{6}) \frac{d}{d x} (x)$

Step 3.3.2.3

Combine $\frac{- 8.88264396}{6}$ and $\cos (\frac{π x}{6})$ .

$f'' (x) = \frac{- 8.88264396 \cos (\frac{π x}{6})}{6} \cdot \frac{d}{d x} (x)$

Step 3.3.2.4

Move the negative in front of the fraction.

$f'' (x) = - \frac{8.88264396 \cos (\frac{π x}{6})}{6} \cdot \frac{d}{d x} x$

Step 3.3.3

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

$f'' (x) = - \frac{8.88264396 \cos (\frac{π x}{6})}{6} \cdot 1$

Step 3.3.4

Multiply $- 1$ by $1$ .

$f'' (x) = - \frac{8.88264396 \cos (\frac{π x}{6})}{6}$

Step 3.4

Simplify.

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Step 3.4.1

Factor $8.88264396$ out of $8.88264396 \cos (\frac{π x}{6})$ .

$f'' (x) = - \frac{8.88264396 (\cos (\frac{π x}{6}))}{6}$

Step 3.4.2

Factor $6$ out of $6$ .

$f'' (x) = - \frac{8.88264396 (\cos (\frac{π x}{6}))}{6 (1)}$

Step 3.4.3

Separate fractions.

$f'' (x) = - (\frac{8.88264396}{6} \cdot \frac{\cos (\frac{π x}{6})}{1})$

Step 3.4.4

Divide $8.88264396$ by $6$ .

$f'' (x) = - (1.48044066 (\frac{\cos (\frac{π x}{6})}{1}))$

Step 3.4.5

Divide $\cos (\frac{π x}{6})$ by $1$ .

$f'' (x) = - (1.48044066 \cos (\frac{π x}{6}))$

Step 3.4.6

Multiply $1.48044066$ by $- 1$ .

$f'' (x) = - 1.48044066 \cos (\frac{π x}{6})$

Step 4

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

$- 2.82743338 \sin (\frac{π x}{6}) = 0$

Step 5

Divide each term in $- 2.82743338 \sin (\frac{π x}{6}) = 0$ by $- 2.82743338$ and simplify.

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Step 5.1

Divide each term in $- 2.82743338 \sin (\frac{π x}{6}) = 0$ by $- 2.82743338$ .

$\frac{- 2.82743338 \sin (\frac{π x}{6})}{- 2.82743338} = \frac{0}{- 2.82743338}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $- 2.82743338$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{- 2.82743338 \sin (\frac{π x}{6})}{- 2.82743338} = \frac{0}{- 2.82743338}$

Step 5.2.1.2

Divide $\sin (\frac{π x}{6})$ by $1$ .

$\sin (\frac{π x}{6}) = \frac{0}{- 2.82743338}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $0$ by $- 2.82743338$ .

$\sin (\frac{π x}{6}) = 0$

Step 6

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

$\frac{π x}{6} = \arcsin (0)$

Step 7

Simplify the right side.

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Step 7.1

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

$\frac{π x}{6} = 0$

Step 8

Set the numerator equal to zero.

$π x = 0$

Step 9

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

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Step 9.1

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

$\frac{π x}{π} = \frac{0}{π}$

Step 9.2

Simplify the left side.

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Step 9.2.1

Cancel the common factor of $π$ .

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Step 9.2.1.1

Cancel the common factor.

$\frac{π x}{π} = \frac{0}{π}$

Step 9.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{π}$

Step 9.3

Simplify the right side.

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Step 9.3.1

Divide $0$ by $π$ .

$x = 0$

Step 10

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.

$\frac{π x}{6} = π - 0$

Step 11

Solve for $x$ .

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Step 11.1

Multiply both sides of the equation by $\frac{6}{π}$ .

$\frac{6}{π} \cdot \frac{π x}{6} = \frac{6}{π} (π - 0)$

Step 11.2

Simplify both sides of the equation.

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Step 11.2.1

Simplify the left side.

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Step 11.2.1.1

Simplify $\frac{6}{π} \cdot \frac{π x}{6}$ .

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Step 11.2.1.1.1

Cancel the common factor of $6$ .

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Step 11.2.1.1.1.1

Cancel the common factor.

$\frac{6}{π} \cdot \frac{π x}{6} = \frac{6}{π} (π - 0)$

Step 11.2.1.1.1.2

Rewrite the expression.

$\frac{1}{π} (π x) = \frac{6}{π} (π - 0)$

Step 11.2.1.1.2

Cancel the common factor of $π$ .

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Step 11.2.1.1.2.1

Factor $π$ out of $π x$ .

$\frac{1}{π} (π (x)) = \frac{6}{π} (π - 0)$

Step 11.2.1.1.2.2

Cancel the common factor.

$\frac{1}{π} (π x) = \frac{6}{π} (π - 0)$

Step 11.2.1.1.2.3

Rewrite the expression.

$x = \frac{6}{π} (π - 0)$

Step 11.2.2

Simplify the right side.

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Step 11.2.2.1

Simplify $\frac{6}{π} (π - 0)$ .

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Step 11.2.2.1.1

Subtract $0$ from $π$ .

$x = \frac{6}{π} π$

Step 11.2.2.1.2

Cancel the common factor of $π$ .

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Step 11.2.2.1.2.1

Cancel the common factor.

$x = \frac{6}{π} π$

Step 11.2.2.1.2.2

Rewrite the expression.

$x = 6$

Step 12

The solution to the equation $\frac{π x}{6} = 0$ .

$x = 0, 6$

Step 13

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.

$- 1.48044066 \cos (\frac{π (0)}{6})$

Step 14

Evaluate the second derivative.

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Step 14.1

Cancel the common factor of $0$ and $6$ .

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Step 14.1.1

Factor $6$ out of $π (0)$ .

$- 1.48044066 \cos (\frac{6 (π \cdot (0))}{6})$

Step 14.1.2

Cancel the common factors.

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Step 14.1.2.1

Factor $6$ out of $6$ .

$- 1.48044066 \cos (\frac{6 (π \cdot (0))}{6 (1)})$

Step 14.1.2.2

Cancel the common factor.

$- 1.48044066 \cos (\frac{6 (π \cdot (0))}{6 \cdot 1})$

Step 14.1.2.3

Rewrite the expression.

$- 1.48044066 \cos (\frac{π \cdot (0)}{1})$

Step 14.1.2.4

Divide $π \cdot (0)$ by $1$ .

$- 1.48044066 \cos (π \cdot (0))$

Step 14.2

Multiply $π$ by $0$ .

$- 1.48044066 \cos (0)$

Step 14.3

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

$- 1.48044066 \cdot 1$

Step 14.4

Multiply $- 1.48044066$ by $1$ .

$- 1.48044066$

Step 15

$x = 0$ 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 16

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

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Step 16.1

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

$f (0) = (5.4) \cos (\frac{π (0)}{6})$

Step 16.2

Simplify the result.

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Step 16.2.1

Cancel the common factor of $0$ and $6$ .

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Step 16.2.1.1

Factor $6$ out of $π (0)$ .

$f (0) = 5.4 \cos (\frac{6 (π \cdot (0))}{6})$

Step 16.2.1.2

Cancel the common factors.

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Step 16.2.1.2.1

Factor $6$ out of $6$ .

$f (0) = 5.4 \cos (\frac{6 (π \cdot (0))}{6 (1)})$

Step 16.2.1.2.2

Cancel the common factor.

$f (0) = 5.4 \cos (\frac{6 (π \cdot (0))}{6 \cdot 1})$

Step 16.2.1.2.3

Rewrite the expression.

$f (0) = 5.4 \cos (\frac{π \cdot (0)}{1})$

Step 16.2.1.2.4

Divide $π \cdot (0)$ by $1$ .

$f (0) = 5.4 \cos (π \cdot (0))$

Step 16.2.2

Multiply $π$ by $0$ .

$f (0) = 5.4 \cos (0)$

Step 16.2.3

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

$f (0) = 5.4 \cdot 1$

Step 16.2.4

Multiply $5.4$ by $1$ .

$f (0) = 5.4$

Step 16.2.5

The final answer is $5.4$ .

$y = 5.4$

Step 17

Evaluate the second derivative at $x = 6$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 1.48044066 \cos (\frac{π (6)}{6})$

Step 18

Evaluate the second derivative.

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Step 18.1

Cancel the common factor of $6$ .

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Step 18.1.1

Cancel the common factor.

$- 1.48044066 \cos (\frac{π \cdot 6}{6})$

Step 18.1.2

Divide $π$ by $1$ .

$- 1.48044066 \cos (π)$

Step 18.2

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.

$- 1.48044066 (- \cos (0))$

Step 18.3

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

$- 1.48044066 (- 1 \cdot 1)$

Step 18.4

Multiply $- 1.48044066 (- 1 \cdot 1)$ .

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Step 18.4.1

Multiply $- 1$ by $1$ .

$- 1.48044066 \cdot - 1$

Step 18.4.2

Multiply $- 1.48044066$ by $- 1$ .

$1.48044066$

Step 19

$x = 6$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 6$ is a local minimum

Step 20

Find the y-value when $x = 6$ .

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Step 20.1

Replace the variable $x$ with $6$ in the expression.

$f (6) = (5.4) \cos (\frac{π (6)}{6})$

Step 20.2

Simplify the result.

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Step 20.2.1

Cancel the common factor of $6$ .

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Step 20.2.1.1

Cancel the common factor.

$f (6) = 5.4 \cos (\frac{π \cdot 6}{6})$

Step 20.2.1.2

Divide $π$ by $1$ .

$f (6) = 5.4 \cos (π)$

Step 20.2.2

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.

$f (6) = 5.4 (- \cos (0))$

Step 20.2.3

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

$f (6) = 5.4 (- 1 \cdot 1)$

Step 20.2.4

Multiply $5.4 (- 1 \cdot 1)$ .

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Step 20.2.4.1

Multiply $- 1$ by $1$ .

$f (6) = 5.4 \cdot - 1$

Step 20.2.4.2

Multiply $5.4$ by $- 1$ .

$f (6) = - 5.4$

Step 20.2.5

The final answer is $- 5.4$ .

$y = - 5.4$

Step 21

These are the local extrema for $f (x) = (5.4) \cos (\frac{π x}{6})$ .

$(0, 5.4)$ is a local maxima

$(6, - 5.4)$ is a local minima

Step 22