Calculus Examples

$f (x) = \frac{4 x + 1}{5 \cos (\frac{x}{2}) + 1}$

Step 1

Set the denominator in $\frac{4 x + 1}{5 \cos (\frac{x}{2}) + 1}$ equal to $0$ to find where the expression is undefined.

$5 \cos (\frac{x}{2}) + 1 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $1$ from both sides of the equation.

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

Step 2.2

Divide each term in $5 \cos (\frac{x}{2}) = - 1$ by $5$ and simplify.

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Step 2.2.1

Divide each term in $5 \cos (\frac{x}{2}) = - 1$ by $5$ .

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

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $5$ .

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Step 2.2.2.1.1

Cancel the common factor.

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

Step 2.2.2.1.2

Divide $\cos (\frac{x}{2})$ by $1$ .

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

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Move the negative in front of the fraction.

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

Step 2.3

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

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

Step 2.4

Simplify the right side.

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Step 2.4.1

Evaluate $\arccos (- \frac{1}{5})$ .

$\frac{x}{2} = 1.77215424$

Step 2.5

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 \cdot 1.77215424$

Step 2.6

Simplify both sides of the equation.

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Step 2.6.1

Simplify the left side.

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Step 2.6.1.1

Cancel the common factor of $2$ .

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Step 2.6.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 \cdot 1.77215424$

Step 2.6.1.1.2

Rewrite the expression.

$x = 2 \cdot 1.77215424$

Step 2.6.2

Simplify the right side.

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Step 2.6.2.1

Multiply $2$ by $1.77215424$ .

$x = 3.54430849$

Step 2.7

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.

$\frac{x}{2} = 2 (3.14159265) - 1.77215424$

Step 2.8

Solve for $x$ .

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Step 2.8.1

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 (2 (3.14159265) - 1.77215424)$

Step 2.8.2

Simplify both sides of the equation.

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Step 2.8.2.1

Simplify the left side.

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Step 2.8.2.1.1

Cancel the common factor of $2$ .

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Step 2.8.2.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 (2 (3.14159265) - 1.77215424)$

Step 2.8.2.1.1.2

Rewrite the expression.

$x = 2 (2 (3.14159265) - 1.77215424)$

Step 2.8.2.2

Simplify the right side.

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Step 2.8.2.2.1

Simplify $2 (2 (3.14159265) - 1.77215424)$ .

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Step 2.8.2.2.1.1

Multiply $2$ by $3.14159265$ .

$x = 2 (6.2831853 - 1.77215424)$

Step 2.8.2.2.1.2

Subtract $1.77215424$ from $6.2831853$ .

$x = 2 \cdot 4.51103105$

Step 2.8.2.2.1.3

Multiply $2$ by $4.51103105$ .

$x = 9.02206211$

Step 2.9

Find the period of $\cos (\frac{x}{2})$ .

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Step 2.9.1

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

$\frac{2 π}{| b |}$

Step 2.9.2

Replace $b$ with $\frac{1}{2}$ in the formula for period.

$\frac{2 π}{| \frac{1}{2} |}$

Step 2.9.3

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{2}}$

Step 2.9.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 2$

Step 2.9.5

Multiply $2$ by $2$ .

$4 π$

Step 2.10

The period of the $\cos (\frac{x}{2})$ function is $4 π$ so values will repeat every $4 π$ radians in both directions.

$x = 3.54430849 + 4 π n, 9.02206211 + 4 π n$ , for any integer $n$

Step 3

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

${x | x = 3.54430849 + 4 π n, x = 9.02206211 + 4 π n}$ $n$ , for any integer $n$

Step 4