Calculus Examples

$F (v) = \frac{22 - 0.02 v^{2}}{{(22 + 0.02 v^{2})}^{2}}$

Step 1

Set the denominator in $\frac{22 - 0.02 v^{2}}{{(22 + 0.02 v^{2})}^{2}}$ equal to $0$ to find where the expression is undefined.

${(22 + 0.02 v^{2})}^{2} = 0$

Step 2

Solve for $v$ .

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Step 2.1

Factor the left side of the equation.

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Step 2.1.1

Factor $0.02$ out of $22 + 0.02 v^{2}$ .

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Step 2.1.1.1

Factor $0.02$ out of $22$ .

${(0.02 \cdot 1100 + 0.02 v^{2})}^{2} = 0$

Step 2.1.1.2

Factor $0.02$ out of $0.02 \cdot 1100 + 0.02 v^{2}$ .

${(0.02 (1100 + v^{2}))}^{2} = 0$

Step 2.1.2

Apply the product rule to $0.02 (1100 + v^{2})$ .

${0.02}^{2} {(1100 + v^{2})}^{2} = 0$

Step 2.2

Divide each term in ${0.02}^{2} {(1100 + v^{2})}^{2} = 0$ by ${0.02}^{2}$ and simplify.

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Step 2.2.1

Divide each term in ${0.02}^{2} {(1100 + v^{2})}^{2} = 0$ by ${0.02}^{2}$ .

$\frac{{0.02}^{2} {(1100 + v^{2})}^{2}}{{0.02}^{2}} = \frac{0}{{0.02}^{2}}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of ${0.02}^{2}$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{{0.02}^{2} {(1100 + v^{2})}^{2}}{{0.02}^{2}} = \frac{0}{{0.02}^{2}}$

Step 2.2.2.1.2

Divide ${(1100 + v^{2})}^{2}$ by $1$ .

${(1100 + v^{2})}^{2} = \frac{0}{{0.02}^{2}}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Raise $0.02$ to the power of $2$ .

${(1100 + v^{2})}^{2} = \frac{0}{0.0004}$

Step 2.2.3.2

Divide $0$ by $0.0004$ .

${(1100 + v^{2})}^{2} = 0$

Step 2.3

Set the $1100 + v^{2}$ equal to $0$ .

$1100 + v^{2} = 0$

Step 2.4

Solve for $v$ .

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Step 2.4.1

Subtract $1100$ from both sides of the equation.

$v^{2} = - 1100$

Step 2.4.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$v = \pm \sqrt{- 1100}$

Step 2.4.3

Simplify $\pm \sqrt{- 1100}$ .

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Step 2.4.3.1

Rewrite $- 1100$ as $- 1 (1100)$ .

$v = \pm \sqrt{- 1 (1100)}$

Step 2.4.3.2

Rewrite $\sqrt{- 1 (1100)}$ as $\sqrt{- 1} \cdot \sqrt{1100}$ .

$v = \pm \sqrt{- 1} \cdot \sqrt{1100}$

Step 2.4.3.3

Rewrite $\sqrt{- 1}$ as $i$ .

$v = \pm i \cdot \sqrt{1100}$

Step 2.4.3.4

Rewrite $1100$ as $10^{2} \cdot 11$ .

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Step 2.4.3.4.1

Factor $100$ out of $1100$ .

$v = \pm i \cdot \sqrt{100 (11)}$

Step 2.4.3.4.2

Rewrite $100$ as $10^{2}$ .

$v = \pm i \cdot \sqrt{10^{2} \cdot 11}$

Step 2.4.3.5

Pull terms out from under the radical.

$v = \pm i \cdot (10 \sqrt{11})$

Step 2.4.3.6

Move $10$ to the left of $i$ .

$v = \pm 10 i \sqrt{11}$

Step 2.4.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.4.4.1

First, use the positive value of the $\pm$ to find the first solution.

$v = 10 i \sqrt{11}$

Step 2.4.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$v = - 10 i \sqrt{11}$

Step 2.4.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$v = 10 i \sqrt{11}, - 10 i \sqrt{11}$

Step 3

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${v | v \in ℝ}$

Step 4

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$[- \frac{1}{176}, \frac{1}{22}]$

Set-Builder Notation:

${y | - \frac{1}{176} \leq y \leq \frac{1}{22}}$

Step 5

Determine the domain and range.

Domain: $(- \infty, \infty), {v | v \in ℝ}$

Range: $[- \frac{1}{176}, \frac{1}{22}], {y | - \frac{1}{176} \leq y \leq \frac{1}{22}}$

Step 6