Calculus Examples

$f (x) = 4.1 \sin (x) - 1.6 \cos (x)$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $4.1 \sin (x) - 1.6 \cos (x)$ with respect to $x$ is $\frac{d}{d x} [4.1 \sin (x)] + \frac{d}{d x} [- 1.6 \cos (x)]$ .

$\frac{d}{d x} [4.1 \sin (x)] + \frac{d}{d x} [- 1.6 \cos (x)]$

Step 1.2

Evaluate $\frac{d}{d x} [4.1 \sin (x)]$ .

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Step 1.2.1

Since $4.1$ is constant with respect to $x$ , the derivative of $4.1 \sin (x)$ with respect to $x$ is $4.1 \frac{d}{d x} [\sin (x)]$ .

$4.1 \frac{d}{d x} [\sin (x)] + \frac{d}{d x} [- 1.6 \cos (x)]$

Step 1.2.2

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

$4.1 \cos (x) + \frac{d}{d x} [- 1.6 \cos (x)]$

Step 1.3

Evaluate $\frac{d}{d x} [- 1.6 \cos (x)]$ .

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Step 1.3.1

Since $- 1.6$ is constant with respect to $x$ , the derivative of $- 1.6 \cos (x)$ with respect to $x$ is $- 1.6 \frac{d}{d x} [\cos (x)]$ .

$4.1 \cos (x) - 1.6 \frac{d}{d x} [\cos (x)]$

Step 1.3.2

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

$4.1 \cos (x) - 1.6 (- \sin (x))$

Step 1.3.3

Multiply $- 1$ by $- 1.6$ .

$4.1 \cos (x) + 1.6 \sin (x)$

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $4.1 \cos (x) + 1.6 \sin (x)$ with respect to $x$ is $\frac{d}{d x} [4.1 \cos (x)] + \frac{d}{d x} [1.6 \sin (x)]$ .

$f'' (x) = \frac{d}{d x} (4.1 \cos (x)) + \frac{d}{d x} (1.6 \sin (x))$

Step 2.2

Evaluate $\frac{d}{d x} [4.1 \cos (x)]$ .

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Step 2.2.1

Since $4.1$ is constant with respect to $x$ , the derivative of $4.1 \cos (x)$ with respect to $x$ is $4.1 \frac{d}{d x} [\cos (x)]$ .

$f'' (x) = 4.1 \frac{d}{d x} (\cos (x)) + \frac{d}{d x} (1.6 \sin (x))$

Step 2.2.2

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

$f'' (x) = 4.1 (- \sin (x)) + \frac{d}{d x} (1.6 \sin (x))$

Step 2.2.3

Multiply $- 1$ by $4.1$ .

$f'' (x) = - 4.1 \sin (x) + \frac{d}{d x} (1.6 \sin (x))$

Step 2.3

Evaluate $\frac{d}{d x} [1.6 \sin (x)]$ .

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Step 2.3.1

Since $1.6$ is constant with respect to $x$ , the derivative of $1.6 \sin (x)$ with respect to $x$ is $1.6 \frac{d}{d x} [\sin (x)]$ .

$f'' (x) = - 4.1 \sin (x) + 1.6 \frac{d}{d x} (\sin (x))$

Step 2.3.2

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

$f'' (x) = - 4.1 \sin (x) + 1.6 \cos (x)$

Step 3

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

$4.1 \cos (x) + 1.6 \sin (x) = 0$

Step 4

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

$\frac{4.1 \cos (x)}{\cos (x)} + \frac{1.6 \sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 5

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

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Step 5.1

Cancel the common factor.

$\frac{4.1 \cos (x)}{\cos (x)} + \frac{1.6 \sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 5.2

Divide $4.1$ by $1$ .

$4.1 + \frac{1.6 \sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 6

Separate fractions.

$4.1 + \frac{1.6}{1} \cdot \frac{\sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 7

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

$4.1 + \frac{1.6}{1} \cdot \tan (x) = \frac{0}{\cos (x)}$

Step 8

Divide $1.6$ by $1$ .

$4.1 + 1.6 \tan (x) = \frac{0}{\cos (x)}$

Step 9

Separate fractions.

$4.1 + 1.6 \tan (x) = \frac{0}{1} \cdot \frac{1}{\cos (x)}$

Step 10

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

$4.1 + 1.6 \tan (x) = \frac{0}{1} \cdot \sec (x)$

Step 11

Divide $0$ by $1$ .

$4.1 + 1.6 \tan (x) = 0 \sec (x)$

Step 12

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

$4.1 + 1.6 \tan (x) = 0$

Step 13

Subtract $4.1$ from both sides of the equation.

$1.6 \tan (x) = - 4.1$

Step 14

Divide each term in $1.6 \tan (x) = - 4.1$ by $1.6$ and simplify.

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Step 14.1

Divide each term in $1.6 \tan (x) = - 4.1$ by $1.6$ .

$\frac{1.6 \tan (x)}{1.6} = \frac{- 4.1}{1.6}$

Step 14.2

Simplify the left side.

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Step 14.2.1

Cancel the common factor of $1.6$ .

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Step 14.2.1.1

Cancel the common factor.

$\frac{1.6 \tan (x)}{1.6} = \frac{- 4.1}{1.6}$

Step 14.2.1.2

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

$\tan (x) = \frac{- 4.1}{1.6}$

Step 14.3

Simplify the right side.

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Step 14.3.1

Divide $- 4.1$ by $1.6$ .

$\tan (x) = - 2.5625$

Step 15

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

$x = \arctan (- 2.5625)$

Step 16

Simplify the right side.

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Step 16.1

Evaluate $\arctan (- 2.5625)$ .

$x = - 1.19872856$

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.

$x = - 1.19872856 - (3.14159265)$

Step 18

Simplify the expression to find the second solution.

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Step 18.1

Add $2 π$ to $- 1.19872856 - (3.14159265)$ .

$x = - 1.19872856 - (3.14159265) + 2 π$

Step 18.2

The resulting angle of $1.94286408$ is positive and coterminal with $- 1.19872856 - (3.14159265)$ .

$x = 1.94286408$

Step 19

The solution to the equation $x = - 1.19872856$ .

$x = - 1.19872856, 1.94286408$

Step 20

Evaluate the second derivative at $x = - 1.19872856$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 4.1 \sin (- 1.19872856) + 1.6 \cos (- 1.19872856)$

Step 21

Evaluate the second derivative.

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Step 21.1

Simplify each term.

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Step 21.1.1

Multiply $- 4.1$ by $\sin (- 1.19872856)$ .

$3.81946823 + 1.6 \cos (- 1.19872856)$

Step 21.1.2

Multiply $1.6$ by $\cos (- 1.19872856)$ .

$3.81946823 + 0.58166797$

Step 21.2

Add $3.81946823$ and $0.58166797$ .

$4.40113621$

Step 22

$x = - 1.19872856$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - 1.19872856$ is a local minimum

Step 23

Find the y-value when $x = - 1.19872856$ .

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Step 23.1

Replace the variable $x$ with $- 1.19872856$ in the expression.

$f (- 1.19872856) = 4.1 \sin (- 1.19872856) - 1.6 \cos (- 1.19872856)$

Step 23.2

Simplify the result.

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Step 23.2.1

Simplify each term.

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Step 23.2.1.1

Multiply $4.1$ by $\sin (- 1.19872856)$ .

$f (- 1.19872856) = - 3.81946823 - 1.6 \cos (- 1.19872856)$

Step 23.2.1.2

Multiply $- 1.6$ by $\cos (- 1.19872856)$ .

$f (- 1.19872856) = - 3.81946823 - 0.58166797$

Step 23.2.2

Subtract $0.58166797$ from $- 3.81946823$ .

$f (- 1.19872856) = - 4.40113621$

Step 23.2.3

The final answer is $- 4.40113621$ .

$y = - 4.40113621$

Step 24

Evaluate the second derivative at $x = 1.94286408$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 4.1 \sin (1.94286408) + 1.6 \cos (1.94286408)$

Step 25

Evaluate the second derivative.

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Step 25.1

Simplify each term.

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Step 25.1.1

Multiply $- 4.1$ by $\sin (1.94286408)$ .

$- 3.81946823 + 1.6 \cos (1.94286408)$

Step 25.1.2

Multiply $1.6$ by $\cos (1.94286408)$ .

$- 3.81946823 - 0.58166797$

Step 25.2

Subtract $0.58166797$ from $- 3.81946823$ .

$- 4.40113621$

Step 26

$x = 1.94286408$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 1.94286408$ is a local maximum

Step 27

Find the y-value when $x = 1.94286408$ .

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Step 27.1

Replace the variable $x$ with $1.94286408$ in the expression.

$f (1.94286408) = 4.1 \sin (1.94286408) - 1.6 \cos (1.94286408)$

Step 27.2

Simplify the result.

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Step 27.2.1

Simplify each term.

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Step 27.2.1.1

Multiply $4.1$ by $\sin (1.94286408)$ .

$f (1.94286408) = 3.81946823 - 1.6 \cos (1.94286408)$

Step 27.2.1.2

Multiply $- 1.6$ by $\cos (1.94286408)$ .

$f (1.94286408) = 3.81946823 + 0.58166797$

Step 27.2.2

Add $3.81946823$ and $0.58166797$ .

$f (1.94286408) = 4.40113621$

Step 27.2.3

The final answer is $4.40113621$ .

$y = 4.40113621$

Step 28

These are the local extrema for $f (x) = 4.1 \sin (x) - 1.6 \cos (x)$ .

$(- 1.19872856, - 4.40113621)$ is a local minima

$(1.94286408, 4.40113621)$ is a local maxima

Step 29