Calculus Examples

$f (x) = \frac{1}{\sqrt{x} \sec (x)}$

Step 1

Simplify $\frac{1}{\sqrt{x} \sec (x)}$ .

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Step 1.1

Separate fractions.

$y = \frac{1}{\sqrt{x}} \cdot \frac{1}{\sec (x)}$

Step 1.2

Rewrite $\sec (x)$ in terms of sines and cosines.

$y = \frac{1}{\sqrt{x}} \cdot \frac{1}{\frac{1}{\cos (x)}}$

Step 1.3

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (x)}$ .

$y = \frac{1}{\sqrt{x}} (1 \cos (x))$

Step 1.4

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

$y = \frac{1}{\sqrt{x}} \cos (x)$

Step 1.5

Multiply $\frac{1}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$y = \frac{1}{\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}} \cos (x)$

Step 1.6

Combine and simplify the denominator.

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Step 1.6.1

Multiply $\frac{1}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$y = \frac{\sqrt{x}}{\sqrt{x} \sqrt{x}} \cos (x)$

Step 1.6.2

Raise $\sqrt{x}$ to the power of $1$ .

$y = \frac{\sqrt{x}}{{\sqrt{x}}^{1} \sqrt{x}} \cos (x)$

Step 1.6.3

Raise $\sqrt{x}$ to the power of $1$ .

$y = \frac{\sqrt{x}}{{\sqrt{x}}^{1} {\sqrt{x}}^{1}} \cos (x)$

Step 1.6.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{\sqrt{x}}{{\sqrt{x}}^{1 + 1}} \cos (x)$

Step 1.6.5

Add $1$ and $1$ .

$y = \frac{\sqrt{x}}{{\sqrt{x}}^{2}} \cos (x)$

Step 1.6.6

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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Step 1.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$y = \frac{\sqrt{x}}{{(x^{\frac{1}{2}})}^{2}} \cos (x)$

Step 1.6.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = \frac{\sqrt{x}}{x^{\frac{1}{2} \cdot 2}} \cos (x)$

Step 1.6.6.3

Combine $\frac{1}{2}$ and $2$ .

$y = \frac{\sqrt{x}}{x^{\frac{2}{2}}} \cos (x)$

Step 1.6.6.4

Cancel the common factor of $2$ .

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Step 1.6.6.4.1

Cancel the common factor.

$y = \frac{\sqrt{x}}{x^{\frac{2}{2}}} \cos (x)$

Step 1.6.6.4.2

Rewrite the expression.

$y = \frac{\sqrt{x}}{x^{1}} \cos (x)$

Step 1.6.6.5

Simplify.

$y = \frac{\sqrt{x}}{x} \cos (x)$

Step 1.7

Combine $\frac{\sqrt{x}}{x}$ and $\cos (x)$ .

$y = \frac{\sqrt{x} \cos (x)}{x}$

Step 2

For any $y = \sec (x)$ , vertical asymptotes occur at $x = \frac{π}{2} + n π$ , where $n$ is an integer. Use the basic period for $y = s e c (x)$ , $(- \frac{π}{2}, \frac{3 π}{2})$ , to find the vertical asymptotes for $y = \frac{\sqrt{x} \cos (x)}{x}$ . Set the inside of the secant function, $b x + c$ , for $y = a \sec (b x + c) + d$ equal to $- \frac{π}{2}$ to find where the vertical asymptote occurs for $y = \frac{\sqrt{x} \cos (x)}{x}$ .

$x = - \frac{π}{2}$

Step 3

Set the inside of the secant function $x$ equal to $\frac{3 π}{2}$ .

$x = \frac{3 π}{2}$

Step 4

The basic period for $y = \frac{\sqrt{x} \cos (x)}{x}$ will occur at $(- \frac{π}{2}, \frac{3 π}{2})$ , where $- \frac{π}{2}$ and $\frac{3 π}{2}$ are vertical asymptotes.

$(- \frac{π}{2}, \frac{3 π}{2})$

Step 5

Find the period $\frac{2 π}{| b |}$ to find where the vertical asymptotes exist. Vertical asymptotes occur every half period.

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Step 5.1

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

$\frac{2 π}{1}$

Step 5.2

Divide $2 π$ by $1$ .

$2 π$

Step 6

The vertical asymptotes for $y = \frac{\sqrt{x} \cos (x)}{x}$ occur at $- \frac{π}{2}$ , $\frac{3 π}{2}$ , and every $π n$ , where $n$ is an integer. This is half of the period.

$π n$

Step 7

There are only vertical asymptotes for secant and cosecant functions.

Vertical Asymptotes: $x = \frac{3 π}{2} + π n$ for any integer $n$

No Horizontal Asymptotes

No Oblique Asymptotes

Step 8