Calculus Examples

$k (x) = 2 \cos (x) - \sin (x)$

Step 1

Find the first derivative of the function.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

Multiply $- 1$ by $2$ .

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

$- 2 \sin (x) - \cos (x)$

Step 2

Find the second derivative of the function.

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Step 2.1

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

$k'' (x) = \frac{d}{d x} (- 2 \sin (x)) + \frac{d}{d x} (- \cos (x))$

Step 2.2

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

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Step 2.2.1

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

$k'' (x) = - 2 \frac{d}{d x} \sin (x) + \frac{d}{d x} (- \cos (x))$

Step 2.2.2

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

$k'' (x) = - 2 \cos (x) + \frac{d}{d x} (- \cos (x))$

Step 2.3

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

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Step 2.3.1

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

$k'' (x) = - 2 \cos (x) - \frac{d}{d x} \cos (x)$

Step 2.3.2

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

$k'' (x) = - 2 \cos (x) + \sin (x)$

Step 2.3.3

Multiply $- 1$ by $- 1$ .

$k'' (x) = - 2 \cos (x) + 1 \sin (x)$

Step 2.3.4

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

$k'' (x) = - 2 \cos (x) + \sin (x)$

Step 3

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

$- 2 \sin (x) - \cos (x) = 0$

Step 4

Divide each term in the equation by $\cos (x)$ .

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

Step 5

Separate fractions.

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

Step 6

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\frac{- 2}{1} \cdot \tan (x) + \frac{- \cos (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 7

Divide $- 2$ by $1$ .

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

Step 8

Cancel the common factor of $\cos (x)$ .

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Step 8.1

Cancel the common factor.

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

Step 8.2

Divide $- 1$ by $1$ .

$- 2 \tan (x) - 1 = \frac{0}{\cos (x)}$

Step 9

Separate fractions.

$- 2 \tan (x) - 1 = \frac{0}{1} \cdot \frac{1}{\cos (x)}$

Step 10

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$- 2 \tan (x) - 1 = \frac{0}{1} \cdot \sec (x)$

Step 11

Divide $0$ by $1$ .

$- 2 \tan (x) - 1 = 0 \sec (x)$

Step 12

Multiply $0$ by $\sec (x)$ .

$- 2 \tan (x) - 1 = 0$

Step 13

Add $1$ to both sides of the equation.

$- 2 \tan (x) = 1$

Step 14

Divide each term in $- 2 \tan (x) = 1$ by $- 2$ and simplify.

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Step 14.1

Divide each term in $- 2 \tan (x) = 1$ by $- 2$ .

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

Step 14.2

Simplify the left side.

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Step 14.2.1

Cancel the common factor of $- 2$ .

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Step 14.2.1.1

Cancel the common factor.

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

Step 14.2.1.2

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

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

Step 14.3

Simplify the right side.

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Step 14.3.1

Move the negative in front of the fraction.

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

Step 15

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

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

Step 16

Simplify the right side.

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Step 16.1

Evaluate $\arctan (- \frac{1}{2})$ .

$x = - 0.4636476$

Step 17

The tangent function is negative in the second and fourth quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the third quadrant.

$x = - 0.4636476 - (3.14159265)$

Step 18

Simplify the expression to find the second solution.

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Step 18.1

Add $2 π$ to $- 0.4636476 - (3.14159265)$ .

$x = - 0.4636476 - (3.14159265) + 2 π$

Step 18.2

The resulting angle of $2.67794504$ is positive and coterminal with $- 0.4636476 - (3.14159265)$ .

$x = 2.67794504$

Step 19

The solution to the equation $x = - 0.4636476$ .

$x = - 0.4636476, 2.67794504$

Step 20

Evaluate the second derivative at $x = - 0.4636476$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 2 \cos (- 0.4636476) + \sin (- 0.4636476)$

Step 21

$x = - 0.4636476$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = - 0.4636476$ is a local maximum

Step 22

Find the y-value when $x = - 0.4636476$ .

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Step 22.1

Replace the variable $x$ with $- 0.4636476$ in the expression.

$k (- 0.4636476) = 2 \cos (- 0.4636476) - \sin (- 0.4636476)$

Step 22.2

The final answer is $2 \cos (- 0.4636476) - \sin (- 0.4636476)$ .

$y = 2 \cos (- 0.4636476) - \sin (- 0.4636476)$

Step 23

Evaluate the second derivative at $x = 2.67794504$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 2 \cos (2.67794504) + \sin (2.67794504)$

Step 24

$x = 2.67794504$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 2.67794504$ is a local minimum

Step 25

Find the y-value when $x = 2.67794504$ .

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Step 25.1

Replace the variable $x$ with $2.67794504$ in the expression.

$k (2.67794504) = 2 \cos (2.67794504) - \sin (2.67794504)$

Step 25.2

The final answer is $2 \cos (2.67794504) - \sin (2.67794504)$ .

$y = 2 \cos (2.67794504) - \sin (2.67794504)$

Step 26

These are the local extrema for $k (x) = 2 \cos (x) - \sin (x)$ .

$(- 0.4636476, 2 \cos (- 0.4636476) - \sin (- 0.4636476))$ is a local maxima

$(2.67794504, 2 \cos (2.67794504) - \sin (2.67794504))$ is a local minima

Step 27