Calculus Examples

$f (x) = \sin (x) + 8$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

By the Sum Rule, the derivative of $\sin (x) + 8$ with respect to $x$ is $\frac{d}{d x} [\sin (x)] + \frac{d}{d x} [8]$ .

$\frac{d}{d x} [\sin (x)] + \frac{d}{d x} [8]$

Step 1.1.2

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

$\cos (x) + \frac{d}{d x} [8]$

Step 1.1.3

Differentiate using the Constant Rule.

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Step 1.1.3.1

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$\cos (x) + 0$

Step 1.1.3.2

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

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

Step 1.2

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

$\cos (x)$

Step 2

Set the first derivative equal to $0$ then solve the equation $\cos (x) = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\cos (x) = 0$

Step 2.2

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

$x = \arccos (0)$

Step 2.3

Simplify the right side.

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Step 2.3.1

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

$x = \frac{π}{2}$

Step 2.4

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$x = 2 π - \frac{π}{2}$

Step 2.5

Simplify $2 π - \frac{π}{2}$ .

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Step 2.5.1

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

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

Step 2.5.2

Combine fractions.

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Step 2.5.2.1

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 2.5.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 2 - π}{2}$

Step 2.5.3

Simplify the numerator.

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Step 2.5.3.1

Multiply $2$ by $2$ .

$x = \frac{4 π - π}{2}$

Step 2.5.3.2

Subtract $π$ from $4 π$ .

$x = \frac{3 π}{2}$

Step 2.6

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

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Step 2.6.1

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

$\frac{2 π}{| b |}$

Step 2.6.2

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

$\frac{2 π}{| 1 |}$

Step 2.6.3

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

$\frac{2 π}{1}$

Step 2.6.4

Divide $2 π$ by $1$ .

$2 π$

Step 2.7

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

$x = \frac{π}{2} + 2 π n, \frac{3 π}{2} + 2 π n$ , for any integer $n$

Step 2.8

Consolidate the answers.

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

Step 3

The values which make the derivative equal to $0$ are $\frac{π}{2} + π n$ .

$\frac{π}{2} + π n$

Step 4

After finding the point that makes the derivative $f' (x) = \cos (x)$ equal to $0$ or undefined, the interval to check where $f (x) = \sin (x) + 8$ is increasing and where it is decreasing is $(- \infty, \frac{π}{2} + π n) \cup (\frac{π}{2} + π n, \infty)$ .

$(- \infty, \frac{π}{2} + π n) \cup (\frac{π}{2} + π n, \infty)$

Step 5

Substitute a value from the interval $(- \infty, \frac{π}{2} + π n)$ into the derivative to determine if the function is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $\frac{π}{2} + π n - 1$ in the expression.

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

Step 5.2

The final answer is $\cos (\frac{π}{2} + π n - 1)$ .

$\cos (\frac{π}{2} + π n - 1)$

Step 5.3

Simplify.

$\cos (\frac{π}{2} + π n - 1)$

Step 5.4

At $x = \frac{π}{2} + π n - 1$ the derivative is $\cos (\frac{π}{2} + π n - 1)$ . Since this is negative, the function is decreasing on $(- \infty, \frac{π}{2} + π n)$ .

Decreasing on $(- \infty, \frac{π}{2} + π n)$ since $f' (x) < 0$

Step 6

Substitute a value from the interval $(\frac{π}{2} + π n, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $\frac{π}{2} + π n + 1$ in the expression.

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

Step 6.2

The final answer is $\cos (\frac{π}{2} + π n + 1)$ .

$\cos (\frac{π}{2} + π n + 1)$

Step 6.3

Simplify.

$\cos (\frac{π}{2} + π n + 1)$

Step 6.4

At $x = \frac{π}{2} + π n + 1$ the derivative is $\cos (\frac{π}{2} + π n + 1)$ . Since this is positive, the function is increasing on $(\frac{π}{2} + π n, \infty)$ .

Increasing on $(\frac{π}{2} + π n, \infty)$ since $f' (x) > 0$

Step 7

List the intervals on which the function is increasing and decreasing.

Increasing on: $(\frac{π}{2} + π n, \infty)$

Decreasing on: $(- \infty, \frac{π}{2} + π n)$

Step 8