Calculus Examples

$y = 7 x + 7 \sin (x)$ , $0 \leq x \leq 2 π$

Step 1

Find the critical points.

Tap for more steps...

Step 1.1

Find the first derivative.

Tap for more steps...

Step 1.1.1

Find the first derivative.

Tap for more steps...

Step 1.1.1.1

By the Sum Rule, the derivative of $7 x + 7 \sin (x)$ with respect to $x$ is $\frac{d}{d x} [7 x] + \frac{d}{d x} [7 \sin (x)]$ .

$\frac{d}{d x} [7 x] + \frac{d}{d x} [7 \sin (x)]$

Step 1.1.1.2

Evaluate $\frac{d}{d x} [7 x]$ .

Tap for more steps...

Step 1.1.1.2.1

Since $7$ is constant with respect to $x$ , the derivative of $7 x$ with respect to $x$ is $7 \frac{d}{d x} [x]$ .

$7 \frac{d}{d x} [x] + \frac{d}{d x} [7 \sin (x)]$

Step 1.1.1.2.2

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

$7 \cdot 1 + \frac{d}{d x} [7 \sin (x)]$

Step 1.1.1.2.3

Multiply $7$ by $1$ .

$7 + \frac{d}{d x} [7 \sin (x)]$

Step 1.1.1.3

Evaluate $\frac{d}{d x} [7 \sin (x)]$ .

Tap for more steps...

Step 1.1.1.3.1

Since $7$ is constant with respect to $x$ , the derivative of $7 \sin (x)$ with respect to $x$ is $7 \frac{d}{d x} [\sin (x)]$ .

$7 + 7 \frac{d}{d x} [\sin (x)]$

Step 1.1.1.3.2

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

$f' (x) = 7 + 7 \cos (x)$

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $7 + 7 \cos (x)$ .

$7 + 7 \cos (x)$

Step 1.2

Set the first derivative equal to $0$ then solve the equation $7 + 7 \cos (x) = 0$ .

Tap for more steps...

Step 1.2.1

Set the first derivative equal to $0$ .

$7 + 7 \cos (x) = 0$

Step 1.2.2

Subtract $7$ from both sides of the equation.

$7 \cos (x) = - 7$

Step 1.2.3

Divide each term in $7 \cos (x) = - 7$ by $7$ and simplify.

Tap for more steps...

Step 1.2.3.1

Divide each term in $7 \cos (x) = - 7$ by $7$ .

$\frac{7 \cos (x)}{7} = \frac{- 7}{7}$

Step 1.2.3.2

Simplify the left side.

Tap for more steps...

Step 1.2.3.2.1

Cancel the common factor of $7$ .

Tap for more steps...

Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{7 \cos (x)}{7} = \frac{- 7}{7}$

Step 1.2.3.2.1.2

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

$\cos (x) = \frac{- 7}{7}$

Step 1.2.3.3

Simplify the right side.

Tap for more steps...

Step 1.2.3.3.1

Divide $- 7$ by $7$ .

$\cos (x) = - 1$

Step 1.2.4

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

$x = \arccos (- 1)$

Step 1.2.5

Simplify the right side.

Tap for more steps...

Step 1.2.5.1

The exact value of $\arccos (- 1)$ is $π$ .

$x = π$

Step 1.2.6

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

$x = 2 π - π$

Step 1.2.7

Subtract $π$ from $2 π$ .

$x = π$

Step 1.2.8

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

Tap for more steps...

Step 1.2.8.1

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

$\frac{2 π}{| b |}$

Step 1.2.8.2

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

$\frac{2 π}{| 1 |}$

Step 1.2.8.3

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

$\frac{2 π}{1}$

Step 1.2.8.4

Divide $2 π$ by $1$ .

$2 π$

Step 1.2.9

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

$x = π + 2 π n$ , for any integer $n$

Step 1.3

Find the values where the derivative is undefined.

Tap for more steps...

Step 1.3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 1.4

Evaluate $7 x + 7 \sin (x)$ at each $x$ value where the derivative is $0$ or undefined.

Tap for more steps...

Step 1.4.1

Evaluate at $x = π$ .

Tap for more steps...

Step 1.4.1.1

Substitute $π$ for $x$ .

$7 (π) + 7 \sin (π)$

Step 1.4.1.2

Simplify.

Tap for more steps...

Step 1.4.1.2.1

Simplify each term.

Tap for more steps...

Step 1.4.1.2.1.1

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

$7 π + 7 \sin (0)$

Step 1.4.1.2.1.2

The exact value of $\sin (0)$ is $0$ .

$7 π + 7 \cdot 0$

Step 1.4.1.2.1.3

Multiply $7$ by $0$ .

$7 π + 0$

Step 1.4.1.2.2

Add $7 π$ and $0$ .

$7 π$

Step 1.4.2

Evaluate at $x = 3 π$ .

Tap for more steps...

Step 1.4.2.1

Substitute $3 π$ for $x$ .

$7 (3 π) + 7 \sin (3 π)$

Step 1.4.2.2

Simplify.

Tap for more steps...

Step 1.4.2.2.1

Simplify each term.

Tap for more steps...

Step 1.4.2.2.1.1

Multiply $3$ by $7$ .

$21 π + 7 \sin (3 π)$

Step 1.4.2.2.1.2

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

$21 π + 7 \sin (π)$

Step 1.4.2.2.1.3

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

$21 π + 7 \sin (0)$

Step 1.4.2.2.1.4

The exact value of $\sin (0)$ is $0$ .

$21 π + 7 \cdot 0$

Step 1.4.2.2.1.5

Multiply $7$ by $0$ .

$21 π + 0$

Step 1.4.2.2.2

Add $21 π$ and $0$ .

$21 π$

Step 1.4.3

Evaluate at $x = 5 π$ .

Tap for more steps...

Step 1.4.3.1

Substitute $5 π$ for $x$ .

$7 (5 π) + 7 \sin (5 π)$

Step 1.4.3.2

Simplify.

Tap for more steps...

Step 1.4.3.2.1

Simplify each term.

Tap for more steps...

Step 1.4.3.2.1.1

Multiply $5$ by $7$ .

$35 π + 7 \sin (5 π)$

Step 1.4.3.2.1.2

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

$35 π + 7 \sin (π)$

Step 1.4.3.2.1.3

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

$35 π + 7 \sin (0)$

Step 1.4.3.2.1.4

The exact value of $\sin (0)$ is $0$ .

$35 π + 7 \cdot 0$

Step 1.4.3.2.1.5

Multiply $7$ by $0$ .

$35 π + 0$

Step 1.4.3.2.2

Add $35 π$ and $0$ .

$35 π$

Step 1.4.4

Evaluate at $x = 7 π$ .

Tap for more steps...

Step 1.4.4.1

Substitute $7 π$ for $x$ .

$7 (7 π) + 7 \sin (7 π)$

Step 1.4.4.2

Simplify.

Tap for more steps...

Step 1.4.4.2.1

Simplify each term.

Tap for more steps...

Step 1.4.4.2.1.1

Multiply $7$ by $7$ .

$49 π + 7 \sin (7 π)$

Step 1.4.4.2.1.2

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

$49 π + 7 \sin (π)$

Step 1.4.4.2.1.3

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

$49 π + 7 \sin (0)$

Step 1.4.4.2.1.4

The exact value of $\sin (0)$ is $0$ .

$49 π + 7 \cdot 0$

Step 1.4.4.2.1.5

Multiply $7$ by $0$ .

$49 π + 0$

Step 1.4.4.2.2

Add $49 π$ and $0$ .

$49 π$

Step 1.4.5

Evaluate at $x = 9 π$ .

Tap for more steps...

Step 1.4.5.1

Substitute $9 π$ for $x$ .

$7 (9 π) + 7 \sin (9 π)$

Step 1.4.5.2

Simplify.

Tap for more steps...

Step 1.4.5.2.1

Simplify each term.

Tap for more steps...

Step 1.4.5.2.1.1

Multiply $9$ by $7$ .

$63 π + 7 \sin (9 π)$

Step 1.4.5.2.1.2

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

$63 π + 7 \sin (π)$

Step 1.4.5.2.1.3

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

$63 π + 7 \sin (0)$

Step 1.4.5.2.1.4

The exact value of $\sin (0)$ is $0$ .

$63 π + 7 \cdot 0$

Step 1.4.5.2.1.5

Multiply $7$ by $0$ .

$63 π + 0$

Step 1.4.5.2.2

Add $63 π$ and $0$ .

$63 π$

Step 1.4.6

List all of the points.

$(π, 7 π), (3 π, 21 π), (5 π, 35 π), (7 π, 49 π), (9 π, 63 π)$

Step 2

Exclude the points that are not on the interval.

$(π, 7 π)$

Step 3

Evaluate at the included endpoints.

Tap for more steps...

Step 3.1

Evaluate at $x = 0$ .

Tap for more steps...

Step 3.1.1

Substitute $0$ for $x$ .

$7 (0) + 7 \sin (0)$

Step 3.1.2

Simplify.

Tap for more steps...

Step 3.1.2.1

Simplify each term.

Tap for more steps...

Step 3.1.2.1.1

Multiply $7$ by $0$ .

$0 + 7 \sin (0)$

Step 3.1.2.1.2

The exact value of $\sin (0)$ is $0$ .

$0 + 7 \cdot 0$

Step 3.1.2.1.3

Multiply $7$ by $0$ .

$0 + 0$

Step 3.1.2.2

Add $0$ and $0$ .

$0$

Step 3.2

Evaluate at $x = 2 π$ .

Tap for more steps...

Step 3.2.1

Substitute $2 π$ for $x$ .

$7 (2 π) + 7 \sin (2 π)$

Step 3.2.2

Simplify.

Tap for more steps...

Step 3.2.2.1

Simplify each term.

Tap for more steps...

Step 3.2.2.1.1

Multiply $2$ by $7$ .

$14 π + 7 \sin (2 π)$

Step 3.2.2.1.2

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

$14 π + 7 \sin (0)$

Step 3.2.2.1.3

The exact value of $\sin (0)$ is $0$ .

$14 π + 7 \cdot 0$

Step 3.2.2.1.4

Multiply $7$ by $0$ .

$14 π + 0$

Step 3.2.2.2

Add $14 π$ and $0$ .

$14 π$

Step 3.3

List all of the points.

$(0, 0), (2 π, 14 π)$

Step 4

Compare the $f (x)$ values found for each value of $x$ in order to determine the absolute maximum and minimum over the given interval. The maximum will occur at the highest $f (x)$ value and the minimum will occur at the lowest $f (x)$ value.

Absolute Maximum: $(2 π, 14 π)$

Absolute Minimum: $(0, 0)$

Step 5