Calculus Examples

$f (x) = x \sqrt{400 - x^{2}}$

Step 1

Set the radicand in $\sqrt{400 - x^{2}}$ greater than or equal to $0$ to find where the expression is defined.

$400 - x^{2} \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $400$ from both sides of the inequality.

$- x^{2} \geq - 400$

Step 2.2

Divide each term in $- x^{2} \geq - 400$ by $- 1$ and simplify.

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Step 2.2.1

Divide each term in $- x^{2} \geq - 400$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x^{2}}{- 1} \leq \frac{- 400}{- 1}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} \leq \frac{- 400}{- 1}$

Step 2.2.2.2

Divide $x^{2}$ by $1$ .

$x^{2} \leq \frac{- 400}{- 1}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $- 400$ by $- 1$ .

$x^{2} \leq 400$

Step 2.3

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

$\sqrt{x^{2}} \leq \sqrt{400}$

Step 2.4

Simplify the equation.

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Step 2.4.1

Simplify the left side.

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Step 2.4.1.1

Pull terms out from under the radical.

$| x | \leq \sqrt{400}$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify $\sqrt{400}$ .

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Step 2.4.2.1.1

Rewrite $400$ as $20^{2}$ .

$| x | \leq \sqrt{20^{2}}$

Step 2.4.2.1.2

Pull terms out from under the radical.

$| x | \leq | 20 |$

Step 2.4.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $20$ is $20$ .

$| x | \leq 20$

Step 2.5

Write $| x | \leq 20$ as a piecewise.

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Step 2.5.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

Step 2.5.2

In the piece where $x$ is non-negative, remove the absolute value.

$x \leq 20$

Step 2.5.3

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x < 0$

Step 2.5.4

In the piece where $x$ is negative, remove the absolute value and multiply by $- 1$ .

$- x \leq 20$

Step 2.5.5

Write as a piecewise.

${\begin{cases} x \leq 20 & x \geq 0 \\ - x \leq 20 & x < 0 \end{cases}$

Step 2.6

Find the intersection of $x \leq 20$ and $x \geq 0$ .

$0 \leq x \leq 20$

Step 2.7

Solve $- x \leq 20$ when $x < 0$ .

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Step 2.7.1

Divide each term in $- x \leq 20$ by $- 1$ and simplify.

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Step 2.7.1.1

Divide each term in $- x \leq 20$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \geq \frac{20}{- 1}$

Step 2.7.1.2

Simplify the left side.

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Step 2.7.1.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{20}{- 1}$

Step 2.7.1.2.2

Divide $x$ by $1$ .

$x \geq \frac{20}{- 1}$

Step 2.7.1.3

Simplify the right side.

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Step 2.7.1.3.1

Divide $20$ by $- 1$ .

$x \geq - 20$

Step 2.7.2

Find the intersection of $x \geq - 20$ and $x < 0$ .

$- 20 \leq x < 0$

Step 2.8

Find the union of the solutions.

$- 20 \leq x \leq 20$

Step 3

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$[- 20, 20]$

Set-Builder Notation:

${x | - 20 \leq x \leq 20}$

Step 4

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

Interval Notation:

$[- 200, 200]$

Set-Builder Notation:

${y | - 200 \leq y \leq 200}$

Step 5

Determine the domain and range.

Domain: $[- 20, 20], {x | - 20 \leq x \leq 20}$

Range: $[- 200, 200], {y | - 200 \leq y \leq 200}$

Step 6