Calculus Examples

$f (x) = 3 x - 6 \cos (x)$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $3 x - 6 \cos (x)$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 6 \cos (x)]$ .

$\frac{d}{d x} [3 x] + \frac{d}{d x} [- 6 \cos (x)]$

Step 1.2

Evaluate $\frac{d}{d x} [3 x]$ .

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Step 1.2.1

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

$3 \frac{d}{d x} [x] + \frac{d}{d x} [- 6 \cos (x)]$

Step 1.2.2

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

$3 \cdot 1 + \frac{d}{d x} [- 6 \cos (x)]$

Step 1.2.3

Multiply $3$ by $1$ .

$3 + \frac{d}{d x} [- 6 \cos (x)]$

Step 1.3

Evaluate $\frac{d}{d x} [- 6 \cos (x)]$ .

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Step 1.3.1

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

$3 - 6 \frac{d}{d x} [\cos (x)]$

Step 1.3.2

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

$3 - 6 (- \sin (x))$

Step 1.3.3

Multiply $- 1$ by $- 6$ .

$3 + 6 \sin (x)$

Step 2

Find the second derivative of the function.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $3 + 6 \sin (x)$ with respect to $x$ is $\frac{d}{d x} [3] + \frac{d}{d x} [6 \sin (x)]$ .

$f'' (x) = \frac{d}{d x} (3) + \frac{d}{d x} (6 \sin (x))$

Step 2.1.2

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$f'' (x) = 0 + \frac{d}{d x} (6 \sin (x))$

Step 2.2

Evaluate $\frac{d}{d x} [6 \sin (x)]$ .

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Step 2.2.1

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

$f'' (x) = 0 + 6 \frac{d}{d x} (\sin (x))$

Step 2.2.2

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

$f'' (x) = 0 + 6 \cos (x)$

Step 2.3

Add $0$ and $6 \cos (x)$ .

$f'' (x) = 6 \cos (x)$

Step 3

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

$3 + 6 \sin (x) = 0$

Step 4

Subtract $3$ from both sides of the equation.

$6 \sin (x) = - 3$

Step 5

Divide each term in $6 \sin (x) = - 3$ by $6$ and simplify.

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Step 5.1

Divide each term in $6 \sin (x) = - 3$ by $6$ .

$\frac{6 \sin (x)}{6} = \frac{- 3}{6}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $6$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{6 \sin (x)}{6} = \frac{- 3}{6}$

Step 5.2.1.2

Divide $\sin (x)$ by $1$ .

$\sin (x) = \frac{- 3}{6}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Cancel the common factor of $- 3$ and $6$ .

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Step 5.3.1.1

Factor $3$ out of $- 3$ .

$\sin (x) = \frac{3 (- 1)}{6}$

Step 5.3.1.2

Cancel the common factors.

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Step 5.3.1.2.1

Factor $3$ out of $6$ .

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

Step 5.3.1.2.2

Cancel the common factor.

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

Step 5.3.1.2.3

Rewrite the expression.

$\sin (x) = \frac{- 1}{2}$

Step 5.3.2

Move the negative in front of the fraction.

$\sin (x) = - \frac{1}{2}$

Step 6

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

$x = \arcsin (- \frac{1}{2})$

Step 7

Simplify the right side.

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Step 7.1

The exact value of $\arcsin (- \frac{1}{2})$ is $- \frac{π}{6}$ .

$x = - \frac{π}{6}$

Step 8

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $2 π$ , to find a reference angle. Next, add this reference angle to $π$ to find the solution in the third quadrant.

$x = 2 π + \frac{π}{6} + π$

Step 9

Simplify the expression to find the second solution.

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Step 9.1

Subtract $2 π$ from $2 π + \frac{π}{6} + π$ .

$x = 2 π + \frac{π}{6} + π - 2 π$

Step 9.2

The resulting angle of $\frac{7 π}{6}$ is positive, less than $2 π$ , and coterminal with $2 π + \frac{π}{6} + π$ .

$x = \frac{7 π}{6}$

Step 10

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

$x = - \frac{π}{6}, \frac{7 π}{6}$

Step 11

Evaluate the second derivative at $x = - \frac{π}{6}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$6 \cos (- \frac{π}{6})$

Step 12

Evaluate the second derivative.

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Step 12.1

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

$6 \cos (\frac{11 π}{6})$

Step 12.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$6 \cos (\frac{π}{6})$

Step 12.3

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$6 \frac{\sqrt{3}}{2}$

Step 12.4

Cancel the common factor of $2$ .

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Step 12.4.1

Factor $2$ out of $6$ .

$2 (3) \frac{\sqrt{3}}{2}$

Step 12.4.2

Cancel the common factor.

$2 \cdot 3 \frac{\sqrt{3}}{2}$

Step 12.4.3

Rewrite the expression.

$3 \sqrt{3}$

Step 13

$x = - \frac{π}{6}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - \frac{π}{6}$ is a local minimum

Step 14

Find the y-value when $x = - \frac{π}{6}$ .

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Step 14.1

Replace the variable $x$ with $- \frac{π}{6}$ in the expression.

$f (- \frac{π}{6}) = 3 (- \frac{π}{6}) - 6 \cos (- \frac{π}{6})$

Step 14.2

Simplify the result.

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Step 14.2.1

Simplify each term.

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Step 14.2.1.1

Cancel the common factor of $3$ .

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Step 14.2.1.1.1

Move the leading negative in $- \frac{π}{6}$ into the numerator.

$f (- \frac{π}{6}) = 3 (\frac{- π}{6}) - 6 \cos (- \frac{π}{6})$

Step 14.2.1.1.2

Factor $3$ out of $6$ .

$f (- \frac{π}{6}) = 3 (\frac{- π}{3 (2)}) - 6 \cos (- \frac{π}{6})$

Step 14.2.1.1.3

Cancel the common factor.

$f (- \frac{π}{6}) = 3 (\frac{- π}{3 \cdot 2}) - 6 \cos (- \frac{π}{6})$

Step 14.2.1.1.4

Rewrite the expression.

$f (- \frac{π}{6}) = \frac{- π}{2} - 6 \cos (- \frac{π}{6})$

Step 14.2.1.2

Move the negative in front of the fraction.

$f (- \frac{π}{6}) = - \frac{π}{2} - 6 \cos (- \frac{π}{6})$

Step 14.2.1.3

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

$f (- \frac{π}{6}) = - \frac{π}{2} - 6 \cos (\frac{11 π}{6})$

Step 14.2.1.4

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$f (- \frac{π}{6}) = - \frac{π}{2} - 6 \cos (\frac{π}{6})$

Step 14.2.1.5

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$f (- \frac{π}{6}) = - \frac{π}{2} - 6 \frac{\sqrt{3}}{2}$

Step 14.2.1.6

Cancel the common factor of $2$ .

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Step 14.2.1.6.1

Factor $2$ out of $- 6$ .

$f (- \frac{π}{6}) = - \frac{π}{2} + 2 (- 3) (\frac{\sqrt{3}}{2})$

Step 14.2.1.6.2

Cancel the common factor.

$f (- \frac{π}{6}) = - \frac{π}{2} + 2 \cdot (- 3 \frac{\sqrt{3}}{2})$

Step 14.2.1.6.3

Rewrite the expression.

$f (- \frac{π}{6}) = - \frac{π}{2} - 3 \sqrt{3}$

Step 14.2.2

The final answer is $- \frac{π}{2} - 3 \sqrt{3}$ .

$y = - \frac{π}{2} - 3 \sqrt{3}$

Step 15

Evaluate the second derivative at $x = \frac{7 π}{6}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$6 \cos (\frac{7 π}{6})$

Step 16

Evaluate the second derivative.

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Step 16.1

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 third quadrant.

$6 (- \cos (\frac{π}{6}))$

Step 16.2

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$6 (- \frac{\sqrt{3}}{2})$

Step 16.3

Cancel the common factor of $2$ .

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Step 16.3.1

Move the leading negative in $- \frac{\sqrt{3}}{2}$ into the numerator.

$6 \frac{- \sqrt{3}}{2}$

Step 16.3.2

Factor $2$ out of $6$ .

$2 (3) \frac{- \sqrt{3}}{2}$

Step 16.3.3

Cancel the common factor.

$2 \cdot 3 \frac{- \sqrt{3}}{2}$

Step 16.3.4

Rewrite the expression.

$3 (- \sqrt{3})$

Step 16.4

Multiply $- 1$ by $3$ .

$- 3 \sqrt{3}$

Step 17

$x = \frac{7 π}{6}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = \frac{7 π}{6}$ is a local maximum

Step 18

Find the y-value when $x = \frac{7 π}{6}$ .

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Step 18.1

Replace the variable $x$ with $\frac{7 π}{6}$ in the expression.

$f (\frac{7 π}{6}) = 3 (\frac{7 π}{6}) - 6 \cos (\frac{7 π}{6})$

Step 18.2

Simplify the result.

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Step 18.2.1

Simplify each term.

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Step 18.2.1.1

Cancel the common factor of $3$ .

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Step 18.2.1.1.1

Factor $3$ out of $6$ .

$f (\frac{7 π}{6}) = 3 (\frac{7 π}{3 (2)}) - 6 \cos (\frac{7 π}{6})$

Step 18.2.1.1.2

Cancel the common factor.

$f (\frac{7 π}{6}) = 3 (\frac{7 π}{3 \cdot 2}) - 6 \cos (\frac{7 π}{6})$

Step 18.2.1.1.3

Rewrite the expression.

$f (\frac{7 π}{6}) = \frac{7 π}{2} - 6 \cos (\frac{7 π}{6})$

Step 18.2.1.2

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 third quadrant.

$f (\frac{7 π}{6}) = \frac{7 π}{2} - 6 (- \cos (\frac{π}{6}))$

Step 18.2.1.3

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$f (\frac{7 π}{6}) = \frac{7 π}{2} - 6 (- \frac{\sqrt{3}}{2})$

Step 18.2.1.4

Cancel the common factor of $2$ .

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Step 18.2.1.4.1

Move the leading negative in $- \frac{\sqrt{3}}{2}$ into the numerator.

$f (\frac{7 π}{6}) = \frac{7 π}{2} - 6 \frac{- \sqrt{3}}{2}$

Step 18.2.1.4.2

Factor $2$ out of $- 6$ .

$f (\frac{7 π}{6}) = \frac{7 π}{2} + 2 (- 3) (\frac{- \sqrt{3}}{2})$

Step 18.2.1.4.3

Cancel the common factor.

$f (\frac{7 π}{6}) = \frac{7 π}{2} + 2 \cdot (- 3 \frac{- \sqrt{3}}{2})$

Step 18.2.1.4.4

Rewrite the expression.

$f (\frac{7 π}{6}) = \frac{7 π}{2} - 3 (- \sqrt{3})$

Step 18.2.1.5

Multiply $- 1$ by $- 3$ .

$f (\frac{7 π}{6}) = \frac{7 π}{2} + 3 \sqrt{3}$

Step 18.2.2

The final answer is $\frac{7 π}{2} + 3 \sqrt{3}$ .

$y = \frac{7 π}{2} + 3 \sqrt{3}$

Step 19

These are the local extrema for $f (x) = 3 x - 6 \cos (x)$ .

$(- \frac{π}{6}, - \frac{π}{2} - 3 \sqrt{3})$ is a local minima

$(\frac{7 π}{6}, \frac{7 π}{2} + 3 \sqrt{3})$ is a local maxima

Step 20