Calculus Examples

$y = x + \cos (x)$

Step 1

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

$f (x) = x + \cos (x)$

Step 2

Find the derivative.

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Step 2.1

Differentiate.

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Step 2.1.1

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

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

Step 2.1.2

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

$1 + \frac{d}{d x} [\cos (x)]$

Step 2.2

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

$1 - \sin (x)$

Step 3

Set the derivative equal to $0$ then solve the equation $1 - \sin (x) = 0$ .

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Step 3.1

Subtract $1$ from both sides of the equation.

$- \sin (x) = - 1$

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Dividing two negative values results in a positive value.

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

Step 3.2.2.2

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

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

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $- 1$ by $- 1$ .

$\sin (x) = 1$

Step 3.3

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

$x = \arcsin (1)$

Step 3.4

Simplify the right side.

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Step 3.4.1

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

$x = \frac{π}{2}$

Step 3.5

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.

$x = π - \frac{π}{2}$

Step 3.6

Simplify $π - \frac{π}{2}$ .

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Step 3.6.1

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

$x = π \cdot \frac{2}{2} - \frac{π}{2}$

Step 3.6.2

Combine fractions.

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Step 3.6.2.1

Combine $π$ and $\frac{2}{2}$ .

$x = \frac{π \cdot 2}{2} - \frac{π}{2}$

Step 3.6.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 2 - π}{2}$

Step 3.6.3

Simplify the numerator.

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Step 3.6.3.1

Move $2$ to the left of $π$ .

$x = \frac{2 \cdot π - π}{2}$

Step 3.6.3.2

Subtract $π$ from $2 π$ .

$x = \frac{π}{2}$

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

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

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

Step 4

Solve the original function $f (x) = x + \cos (x)$ at $x = \frac{π}{2}$ .

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Step 4.1

Replace the variable $x$ with $\frac{π}{2}$ in the expression.

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

Step 4.2

Simplify the result.

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Step 4.2.1

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$f (\frac{π}{2}) = \frac{π}{2} + 0$

Step 4.2.2

Add $\frac{π}{2}$ and $0$ .

$f (\frac{π}{2}) = \frac{π}{2}$

Step 4.2.3

The final answer is $\frac{π}{2}$ .

$\frac{π}{2}$

Step 5

Solve the original function $f (x) = x + \cos (x)$ at $x = \frac{π}{2} + 2 π$ .

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Step 5.1

Replace the variable $x$ with $\frac{π}{2} + 2 π$ in the expression.

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

Step 5.2

Simplify the result.

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Step 5.2.1

Find the common denominator.

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Step 5.2.1.1

Write $2 π$ as a fraction with denominator $1$ .

$f (\frac{π}{2} + 2 π) = \frac{π}{2} + \frac{2 π}{1} + \cos (\frac{π}{2} + 2 π)$

Step 5.2.1.2

Multiply $\frac{2 π}{1}$ by $\frac{2}{2}$ .

$f (\frac{π}{2} + 2 π) = \frac{π}{2} + \frac{2 π}{1} \cdot \frac{2}{2} + \cos (\frac{π}{2} + 2 π)$

Step 5.2.1.3

Multiply $\frac{2 π}{1}$ by $\frac{2}{2}$ .

$f (\frac{π}{2} + 2 π) = \frac{π}{2} + \frac{2 π \cdot 2}{2} + \cos (\frac{π}{2} + 2 π)$

Step 5.2.1.4

Write $\cos (\frac{π}{2} + 2 π)$ as a fraction with denominator $1$ .

$f (\frac{π}{2} + 2 π) = \frac{π}{2} + \frac{2 π \cdot 2}{2} + \frac{\cos (\frac{π}{2} + 2 π)}{1}$

Step 5.2.1.5

Multiply $\frac{\cos (\frac{π}{2} + 2 π)}{1}$ by $\frac{2}{2}$ .

$f (\frac{π}{2} + 2 π) = \frac{π}{2} + \frac{2 π \cdot 2}{2} + \frac{\cos (\frac{π}{2} + 2 π)}{1} \cdot \frac{2}{2}$

Step 5.2.1.6

Multiply $\frac{\cos (\frac{π}{2} + 2 π)}{1}$ by $\frac{2}{2}$ .

$f (\frac{π}{2} + 2 π) = \frac{π}{2} + \frac{2 π \cdot 2}{2} + \frac{\cos (\frac{π}{2} + 2 π) \cdot 2}{2}$

Step 5.2.2

Combine the numerators over the common denominator.

$f (\frac{π}{2} + 2 π) = \frac{π + 2 π \cdot 2 + \cos (\frac{π}{2} + 2 π) \cdot 2}{2}$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Multiply $2$ by $2$ .

$f (\frac{π}{2} + 2 π) = \frac{π + 4 π + \cos (\frac{π}{2} + 2 π) \cdot 2}{2}$

Step 5.2.3.2

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

$f (\frac{π}{2} + 2 π) = \frac{π + 4 π + \cos (\frac{π}{2} + 2 π \cdot \frac{2}{2}) \cdot 2}{2}$

Step 5.2.3.3

Combine $2 π$ and $\frac{2}{2}$ .

$f (\frac{π}{2} + 2 π) = \frac{π + 4 π + \cos (\frac{π}{2} + \frac{2 π \cdot 2}{2}) \cdot 2}{2}$

Step 5.2.3.4

Combine the numerators over the common denominator.

$f (\frac{π}{2} + 2 π) = \frac{π + 4 π + \cos (\frac{π + 2 π \cdot 2}{2}) \cdot 2}{2}$

Step 5.2.3.5

Simplify the numerator.

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Step 5.2.3.5.1

Multiply $2$ by $2$ .

$f (\frac{π}{2} + 2 π) = \frac{π + 4 π + \cos (\frac{π + 4 π}{2}) \cdot 2}{2}$

Step 5.2.3.5.2

Add $π$ and $4 π$ .

$f (\frac{π}{2} + 2 π) = \frac{π + 4 π + \cos (\frac{5 π}{2}) \cdot 2}{2}$

Step 5.2.3.6

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

$f (\frac{π}{2} + 2 π) = \frac{π + 4 π + \cos (\frac{π}{2}) \cdot 2}{2}$

Step 5.2.3.7

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$f (\frac{π}{2} + 2 π) = \frac{π + 4 π + 0 \cdot 2}{2}$

Step 5.2.3.8

Multiply $0$ by $2$ .

$f (\frac{π}{2} + 2 π) = \frac{π + 4 π + 0}{2}$

Step 5.2.4

Simplify by adding terms.

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Step 5.2.4.1

Add $π$ and $4 π$ .

$f (\frac{π}{2} + 2 π) = \frac{5 π + 0}{2}$

Step 5.2.4.2

Add $5 π$ and $0$ .

$f (\frac{π}{2} + 2 π) = \frac{5 π}{2}$

Step 5.2.5

The final answer is $\frac{5 π}{2}$ .

$\frac{5 π}{2}$

Step 6

The horizontal tangent line on function $f (x) = x + \cos (x)$ is $y = \frac{π}{2}, y = \frac{5 π}{2}$ .

$y = \frac{π}{2}, y = \frac{5 π}{2}$

Step 7