Calculus Examples

$f (x) = 7 \cos (x) + 2 x$

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 $7 \cos (x) + 2 x$ with respect to $x$ is $\frac{d}{d x} [7 \cos (x)] + \frac{d}{d x} [2 x]$ .

$\frac{d}{d x} [7 \cos (x)] + \frac{d}{d x} [2 x]$

Step 1.1.2

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

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Step 1.1.2.1

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

$7 \frac{d}{d x} [\cos (x)] + \frac{d}{d x} [2 x]$

Step 1.1.2.2

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

$7 (- \sin (x)) + \frac{d}{d x} [2 x]$

Step 1.1.2.3

Multiply $- 1$ by $7$ .

$- 7 \sin (x) + \frac{d}{d x} [2 x]$

Step 1.1.3

Evaluate $\frac{d}{d x} [2 x]$ .

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Step 1.1.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$- 7 \sin (x) + 2 \frac{d}{d x} [x]$

Step 1.1.3.2

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

$- 7 \sin (x) + 2 \cdot 1$

Step 1.1.3.3

Multiply $2$ by $1$ .

$- 7 \sin (x) + 2$

Step 1.1.4

Reorder terms.

$f' (x) = 2 - 7 \sin (x)$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $2 - 7 \sin (x)$ .

$2 - 7 \sin (x)$

Step 2

Set the first derivative equal to $0$ then solve the equation $2 - 7 \sin (x) = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$2 - 7 \sin (x) = 0$

Step 2.2

Subtract $2$ from both sides of the equation.

$- 7 \sin (x) = - 2$

Step 2.3

Divide each term in $- 7 \sin (x) = - 2$ by $- 7$ and simplify.

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Step 2.3.1

Divide each term in $- 7 \sin (x) = - 2$ by $- 7$ .

$\frac{- 7 \sin (x)}{- 7} = \frac{- 2}{- 7}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $- 7$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{- 7 \sin (x)}{- 7} = \frac{- 2}{- 7}$

Step 2.3.2.1.2

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

$\sin (x) = \frac{- 2}{- 7}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Dividing two negative values results in a positive value.

$\sin (x) = \frac{2}{7}$

Step 2.4

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

$x = \arcsin (\frac{2}{7})$

Step 2.5

Simplify the right side.

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Step 2.5.1

Evaluate $\arcsin (\frac{2}{7})$ .

$x = 0.2897517$

Step 2.6

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.

$x = (3.14159265) - 0.2897517$

Step 2.7

Solve for $x$ .

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Step 2.7.1

Remove parentheses.

$x = 3.14159265 - 0.2897517$

Step 2.7.2

Remove parentheses.

$x = (3.14159265) - 0.2897517$

Step 2.7.3

Subtract $0.2897517$ from $3.14159265$ .

$x = 2.85184095$

Step 2.8

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

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Step 2.8.1

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

$\frac{2 π}{| b |}$

Step 2.8.2

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

$\frac{2 π}{| 1 |}$

Step 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 2.8.4

Divide $2 π$ by $1$ .

$2 π$

Step 2.9

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

$x = 0.2897517 + 2 π n, 2.85184095 + 2 π n$ , for any integer $n$

Step 3

Find the values where the derivative is undefined.

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Step 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 4

Evaluate $7 \cos (x) + 2 x$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 0.2897517$ .

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Step 4.1.1

Substitute $0.2897517$ for $x$ .

$7 \cos (0.2897517) + 2 (0.2897517)$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Multiply $2$ by $0.2897517$ .

$7 \cos (0.2897517) + 0.5795034$

Step 4.1.2.2

Add $7 \cos (0.2897517)$ and $0.5795034$ .

$7.28770733$

Step 4.2

Evaluate at $x = 0.2897517 + 2 π$ .

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Step 4.2.1

Substitute $0.2897517 + 2 π$ for $x$ .

$7 \cos (0.2897517 + 2 π) + 2 (0.2897517 + 2 π)$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Add $0.2897517$ and $2 π$ .

$7 \cos (6.572937) + 2 (0.2897517 + 2 π)$

Step 4.2.2.1.2

Add $0.2897517$ and $2 π$ .

$7 \cos (6.572937) + 2 \cdot 6.572937$

Step 4.2.2.1.3

Multiply $2$ by $6.572937$ .

$7 \cos (6.572937) + 13.14587401$

Step 4.2.2.2

Add $7 \cos (6.572937)$ and $13.14587401$ .

$19.85407794$

Step 4.3

Evaluate at $x = 0.2897517 + 4 π$ .

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Step 4.3.1

Substitute $0.2897517 + 4 π$ for $x$ .

$7 \cos (0.2897517 + 4 π) + 2 (0.2897517 + 4 π)$

Step 4.3.2

Simplify.

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Step 4.3.2.1

Simplify each term.

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Step 4.3.2.1.1

Add $0.2897517$ and $4 π$ .

$7 \cos (12.85612231) + 2 (0.2897517 + 4 π)$

Step 4.3.2.1.2

Add $0.2897517$ and $4 π$ .

$7 \cos (12.85612231) + 2 \cdot 12.85612231$

Step 4.3.2.1.3

Multiply $2$ by $12.85612231$ .

$7 \cos (12.85612231) + 25.71224463$

Step 4.3.2.2

Add $7 \cos (12.85612231)$ and $25.71224463$ .

$32.42044856$

Step 4.4

Evaluate at $x = 0.2897517 + 6 π$ .

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Step 4.4.1

Substitute $0.2897517 + 6 π$ for $x$ .

$7 \cos (0.2897517 + 6 π) + 2 (0.2897517 + 6 π)$

Step 4.4.2

Simplify.

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Step 4.4.2.1

Simplify each term.

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Step 4.4.2.1.1

Add $0.2897517$ and $6 π$ .

$7 \cos (19.13930762) + 2 (0.2897517 + 6 π)$

Step 4.4.2.1.2

Add $0.2897517$ and $6 π$ .

$7 \cos (19.13930762) + 2 \cdot 19.13930762$

Step 4.4.2.1.3

Multiply $2$ by $19.13930762$ .

$7 \cos (19.13930762) + 38.27861524$

Step 4.4.2.2

Add $7 \cos (19.13930762)$ and $38.27861524$ .

$44.98681917$

Step 4.5

Evaluate at $x = 0.2897517 + 8 π$ .

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Step 4.5.1

Substitute $0.2897517 + 8 π$ for $x$ .

$7 \cos (0.2897517 + 8 π) + 2 (0.2897517 + 8 π)$

Step 4.5.2

Simplify.

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Step 4.5.2.1

Simplify each term.

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Step 4.5.2.1.1

Add $0.2897517$ and $8 π$ .

$7 \cos (25.42249293) + 2 (0.2897517 + 8 π)$

Step 4.5.2.1.2

Add $0.2897517$ and $8 π$ .

$7 \cos (25.42249293) + 2 \cdot 25.42249293$

Step 4.5.2.1.3

Multiply $2$ by $25.42249293$ .

$7 \cos (25.42249293) + 50.84498586$

Step 4.5.2.2

Add $7 \cos (25.42249293)$ and $50.84498586$ .

$57.55318979$

Step 4.6

Evaluate at $x = 2.85184095$ .

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Step 4.6.1

Substitute $2.85184095$ for $x$ .

$7 \cos (2.85184095) + 2 (2.85184095)$

Step 4.6.2

Simplify.

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Step 4.6.2.1

Multiply $2$ by $2.85184095$ .

$7 \cos (2.85184095) + 5.7036819$

Step 4.6.2.2

Add $7 \cos (2.85184095)$ and $5.7036819$ .

$- 1.00452202$

Step 4.7

Evaluate at $x = 2.85184095 + 2 π$ .

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Step 4.7.1

Substitute $2.85184095 + 2 π$ for $x$ .

$7 \cos (2.85184095 + 2 π) + 2 (2.85184095 + 2 π)$

Step 4.7.2

Simplify.

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Step 4.7.2.1

Simplify each term.

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Step 4.7.2.1.1

Add $2.85184095$ and $2 π$ .

$7 \cos (9.13502625) + 2 (2.85184095 + 2 π)$

Step 4.7.2.1.2

Add $2.85184095$ and $2 π$ .

$7 \cos (9.13502625) + 2 \cdot 9.13502625$

Step 4.7.2.1.3

Multiply $2$ by $9.13502625$ .

$7 \cos (9.13502625) + 18.27005251$

Step 4.7.2.2

Add $7 \cos (9.13502625)$ and $18.27005251$ .

$11.56184858$

Step 4.8

Evaluate at $x = 2.85184095 + 4 π$ .

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Step 4.8.1

Substitute $2.85184095 + 4 π$ for $x$ .

$7 \cos (2.85184095 + 4 π) + 2 (2.85184095 + 4 π)$

Step 4.8.2

Simplify.

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Step 4.8.2.1

Simplify each term.

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Step 4.8.2.1.1

Add $2.85184095$ and $4 π$ .

$7 \cos (15.41821156) + 2 (2.85184095 + 4 π)$

Step 4.8.2.1.2

Add $2.85184095$ and $4 π$ .

$7 \cos (15.41821156) + 2 \cdot 15.41821156$

Step 4.8.2.1.3

Multiply $2$ by $15.41821156$ .

$7 \cos (15.41821156) + 30.83642313$

Step 4.8.2.2

Add $7 \cos (15.41821156)$ and $30.83642313$ .

$24.1282192$

Step 4.9

Evaluate at $x = 2.85184095 + 6 π$ .

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Step 4.9.1

Substitute $2.85184095 + 6 π$ for $x$ .

$7 \cos (2.85184095 + 6 π) + 2 (2.85184095 + 6 π)$

Step 4.9.2

Simplify.

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Step 4.9.2.1

Simplify each term.

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Step 4.9.2.1.1

Add $2.85184095$ and $6 π$ .

$7 \cos (21.70139687) + 2 (2.85184095 + 6 π)$

Step 4.9.2.1.2

Add $2.85184095$ and $6 π$ .

$7 \cos (21.70139687) + 2 \cdot 21.70139687$

Step 4.9.2.1.3

Multiply $2$ by $21.70139687$ .

$7 \cos (21.70139687) + 43.40279374$

Step 4.9.2.2

Add $7 \cos (21.70139687)$ and $43.40279374$ .

$36.69458981$

Step 4.10

Evaluate at $x = 2.85184095 + 8 π$ .

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Step 4.10.1

Substitute $2.85184095 + 8 π$ for $x$ .

$7 \cos (2.85184095 + 8 π) + 2 (2.85184095 + 8 π)$

Step 4.10.2

Simplify.

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Step 4.10.2.1

Simplify each term.

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Step 4.10.2.1.1

Add $2.85184095$ and $8 π$ .

$7 \cos (27.98458218) + 2 (2.85184095 + 8 π)$

Step 4.10.2.1.2

Add $2.85184095$ and $8 π$ .

$7 \cos (27.98458218) + 2 \cdot 27.98458218$

Step 4.10.2.1.3

Multiply $2$ by $27.98458218$ .

$7 \cos (27.98458218) + 55.96916436$

Step 4.10.2.2

Add $7 \cos (27.98458218)$ and $55.96916436$ .

$49.26096042$

Step 4.11

List all of the points.

$(0.2897517, 7.28770733), (0.2897517 + 2 π, 19.85407794), (0.2897517 + 4 π, 32.42044856), (0.2897517 + 6 π, 44.98681917), (0.2897517 + 8 π, 57.55318979), (2.85184095, - 1.00452202), (2.85184095 + 2 π, 11.56184858), (2.85184095 + 4 π, 24.1282192), (2.85184095 + 6 π, 36.69458981), (2.85184095 + 8 π, 49.26096042)$

Step 5