Calculus Examples

$y = 5 x - 10 \cos (x)$

Step 1

Set $y$ as a function of $x$ .

$f (x) = 5 x - 10 \cos (x)$

Step 2

Find the derivative.

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Step 2.1

By the Sum Rule, the derivative of $5 x - 10 \cos (x)$ with respect to $x$ is $\frac{d}{d x} [5 x] + \frac{d}{d x} [- 10 \cos (x)]$ .

$\frac{d}{d x} [5 x] + \frac{d}{d x} [- 10 \cos (x)]$

Step 2.2

Evaluate $\frac{d}{d x} [5 x]$ .

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Step 2.2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$5 \frac{d}{d x} [x] + \frac{d}{d x} [- 10 \cos (x)]$

Step 2.2.2

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

$5 \cdot 1 + \frac{d}{d x} [- 10 \cos (x)]$

Step 2.2.3

Multiply $5$ by $1$ .

$5 + \frac{d}{d x} [- 10 \cos (x)]$

Step 2.3

Evaluate $\frac{d}{d x} [- 10 \cos (x)]$ .

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Step 2.3.1

Since $- 10$ is constant with respect to $x$ , the derivative of $- 10 \cos (x)$ with respect to $x$ is $- 10 \frac{d}{d x} [\cos (x)]$ .

$5 - 10 \frac{d}{d x} [\cos (x)]$

Step 2.3.2

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

$5 - 10 (- \sin (x))$

Step 2.3.3

Multiply $- 1$ by $- 10$ .

$5 + 10 \sin (x)$

Step 3

Set the derivative equal to $0$ then solve the equation $5 + 10 \sin (x) = 0$ .

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Step 3.1

Subtract $5$ from both sides of the equation.

$10 \sin (x) = - 5$

Step 3.2

Divide each term in $10 \sin (x) = - 5$ by $10$ and simplify.

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Step 3.2.1

Divide each term in $10 \sin (x) = - 5$ by $10$ .

$\frac{10 \sin (x)}{10} = \frac{- 5}{10}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $10$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{10 \sin (x)}{10} = \frac{- 5}{10}$

Step 3.2.2.1.2

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

$\sin (x) = \frac{- 5}{10}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Cancel the common factor of $- 5$ and $10$ .

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Step 3.2.3.1.1

Factor $5$ out of $- 5$ .

$\sin (x) = \frac{5 (- 1)}{10}$

Step 3.2.3.1.2

Cancel the common factors.

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Step 3.2.3.1.2.1

Factor $5$ out of $10$ .

$\sin (x) = \frac{5 \cdot - 1}{5 \cdot 2}$

Step 3.2.3.1.2.2

Cancel the common factor.

$\sin (x) = \frac{5 \cdot - 1}{5 \cdot 2}$

Step 3.2.3.1.2.3

Rewrite the expression.

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

Step 3.2.3.2

Move the negative in front of the fraction.

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

Step 3.3

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

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

Step 3.4

Simplify the right side.

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Step 3.4.1

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

$x = - \frac{π}{6}$

Step 3.5

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 3.6

Simplify the expression to find the second solution.

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Step 3.6.1

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

$x = 2 π + \frac{π}{6} + π - 2 π$

Step 3.6.2

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 3.7

Find the period of $\sin (x)$ .

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Step 3.7.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 3.7.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 3.7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 3.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 3.8

Add $2 π$ to every negative angle to get positive angles.

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Step 3.8.1

Add $2 π$ to $- \frac{π}{6}$ to find the positive angle.

$- \frac{π}{6} + 2 π$

Step 3.8.2

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$2 π \cdot \frac{6}{6} - \frac{π}{6}$

Step 3.8.3

Combine fractions.

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Step 3.8.3.1

Combine $2 π$ and $\frac{6}{6}$ .

$\frac{2 π \cdot 6}{6} - \frac{π}{6}$

Step 3.8.3.2

Combine the numerators over the common denominator.

$\frac{2 π \cdot 6 - π}{6}$

Step 3.8.4

Simplify the numerator.

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Step 3.8.4.1

Multiply $6$ by $2$ .

$\frac{12 π - π}{6}$

Step 3.8.4.2

Subtract $π$ from $12 π$ .

$\frac{11 π}{6}$

Step 3.8.5

List the new angles.

$x = \frac{11 π}{6}$

Step 3.9

The period of the $\sin (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = \frac{7 π}{6} + 2 π n, \frac{11 π}{6} + 2 π n$ , for any integer $n$

Step 4

Solve the original function $f (x) = 5 x - 10 \cos (x)$ at $x = \frac{7 π}{6}$ .

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Step 4.1

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

$f (\frac{7 π}{6}) = 5 (\frac{7 π}{6}) - 10 \cos (\frac{7 π}{6})$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply $5 (\frac{7 π}{6})$ .

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Step 4.2.1.1.1

Combine $5$ and $\frac{7 π}{6}$ .

$f (\frac{7 π}{6}) = \frac{5 (7 π)}{6} - 10 \cos (\frac{7 π}{6})$

Step 4.2.1.1.2

Multiply $7$ by $5$ .

$f (\frac{7 π}{6}) = \frac{35 π}{6} - 10 \cos (\frac{7 π}{6})$

Step 4.2.1.2

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.

$f (\frac{7 π}{6}) = \frac{35 π}{6} - 10 (- \cos (\frac{π}{6}))$

Step 4.2.1.3

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

$f (\frac{7 π}{6}) = \frac{35 π}{6} - 10 (- \frac{\sqrt{3}}{2})$

Step 4.2.1.4

Cancel the common factor of $2$ .

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Step 4.2.1.4.1

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

$f (\frac{7 π}{6}) = \frac{35 π}{6} - 10 \frac{- \sqrt{3}}{2}$

Step 4.2.1.4.2

Factor $2$ out of $- 10$ .

$f (\frac{7 π}{6}) = \frac{35 π}{6} + 2 (- 5) (\frac{- \sqrt{3}}{2})$

Step 4.2.1.4.3

Cancel the common factor.

$f (\frac{7 π}{6}) = \frac{35 π}{6} + 2 \cdot (- 5 \frac{- \sqrt{3}}{2})$

Step 4.2.1.4.4

Rewrite the expression.

$f (\frac{7 π}{6}) = \frac{35 π}{6} - 5 (- \sqrt{3})$

Step 4.2.1.5

Multiply $- 1$ by $- 5$ .

$f (\frac{7 π}{6}) = \frac{35 π}{6} + 5 \sqrt{3}$

Step 4.2.2

The final answer is $\frac{35 π}{6} + 5 \sqrt{3}$ .

$\frac{35 π}{6} + 5 \sqrt{3}$

Step 5

Solve the original function $f (x) = 5 x - 10 \cos (x)$ at $x = \frac{11 π}{6}$ .

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Step 5.1

Replace the variable $x$ with $\frac{11 π}{6}$ in the expression.

$f (\frac{11 π}{6}) = 5 (\frac{11 π}{6}) - 10 \cos (\frac{11 π}{6})$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Multiply $5 (\frac{11 π}{6})$ .

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Step 5.2.1.1.1

Combine $5$ and $\frac{11 π}{6}$ .

$f (\frac{11 π}{6}) = \frac{5 (11 π)}{6} - 10 \cos (\frac{11 π}{6})$

Step 5.2.1.1.2

Multiply $11$ by $5$ .

$f (\frac{11 π}{6}) = \frac{55 π}{6} - 10 \cos (\frac{11 π}{6})$

Step 5.2.1.2

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

$f (\frac{11 π}{6}) = \frac{55 π}{6} - 10 \cos (\frac{π}{6})$

Step 5.2.1.3

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

$f (\frac{11 π}{6}) = \frac{55 π}{6} - 10 \frac{\sqrt{3}}{2}$

Step 5.2.1.4

Cancel the common factor of $2$ .

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Step 5.2.1.4.1

Factor $2$ out of $- 10$ .

$f (\frac{11 π}{6}) = \frac{55 π}{6} + 2 (- 5) (\frac{\sqrt{3}}{2})$

Step 5.2.1.4.2

Cancel the common factor.

$f (\frac{11 π}{6}) = \frac{55 π}{6} + 2 \cdot (- 5 \frac{\sqrt{3}}{2})$

Step 5.2.1.4.3

Rewrite the expression.

$f (\frac{11 π}{6}) = \frac{55 π}{6} - 5 \sqrt{3}$

Step 5.2.2

The final answer is $\frac{55 π}{6} - 5 \sqrt{3}$ .

$\frac{55 π}{6} - 5 \sqrt{3}$

Step 6

The horizontal tangent lines on function $f (x) = 5 x - 10 \cos (x)$ are $y = \frac{35 π}{6} + 5 \sqrt{3}, y = \frac{55 π}{6} - 5 \sqrt{3}$ .

$y = \frac{35 π}{6} + 5 \sqrt{3}, y = \frac{55 π}{6} - 5 \sqrt{3}$

Step 7