Calculus Examples

$x \sqrt{2 - x^{2}}$

Step 1

Find the domain for $y = x \sqrt{2 - x^{2}}$ so that a list of $x$ values can be picked to find a list of points, which will help graphing the radical.

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Step 1.1

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

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

Step 1.2

Solve for $x$ .

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Step 1.2.1

Subtract $2$ from both sides of the inequality.

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

Step 1.2.2

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

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Step 1.2.2.1

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

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

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Dividing two negative values results in a positive value.

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

Step 1.2.2.2.2

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

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

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $- 2$ by $- 1$ .

$x^{2} \leq 2$

Step 1.2.3

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

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

Step 1.2.4

Simplify the left side.

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Step 1.2.4.1

Pull terms out from under the radical.

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

Step 1.2.5

Write $| x | \leq \sqrt{2}$ as a piecewise.

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Step 1.2.5.1

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

$x \geq 0$

Step 1.2.5.2

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

$x \leq \sqrt{2}$

Step 1.2.5.3

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

$x < 0$

Step 1.2.5.4

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

$- x \leq \sqrt{2}$

Step 1.2.5.5

Write as a piecewise.

${\begin{cases} x \leq \sqrt{2} & x \geq 0 \\ - x \leq \sqrt{2} & x < 0 \end{cases}$

Step 1.2.6

Find the intersection of $x \leq \sqrt{2}$ and $x \geq 0$ .

$0 \leq x \leq \sqrt{2}$

Step 1.2.7

Solve $- x \leq \sqrt{2}$ when $x < 0$ .

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Step 1.2.7.1

Divide each term in $- x \leq \sqrt{2}$ by $- 1$ and simplify.

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Step 1.2.7.1.1

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

$\frac{- x}{- 1} \geq \frac{\sqrt{2}}{- 1}$

Step 1.2.7.1.2

Simplify the left side.

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Step 1.2.7.1.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{\sqrt{2}}{- 1}$

Step 1.2.7.1.2.2

Divide $x$ by $1$ .

$x \geq \frac{\sqrt{2}}{- 1}$

Step 1.2.7.1.3

Simplify the right side.

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Step 1.2.7.1.3.1

Move the negative one from the denominator of $\frac{\sqrt{2}}{- 1}$ .

$x \geq - 1 \cdot \sqrt{2}$

Step 1.2.7.1.3.2

Rewrite $- 1 \cdot \sqrt{2}$ as $- \sqrt{2}$ .

$x \geq - \sqrt{2}$

Step 1.2.7.2

Find the intersection of $x \geq - \sqrt{2}$ and $x < 0$ .

$- \sqrt{2} \leq x < 0$

Step 1.2.8

Find the union of the solutions.

$- \sqrt{2} \leq x \leq \sqrt{2}$

Step 1.3

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

Interval Notation:

$[- \sqrt{2}, \sqrt{2}]$

Set-Builder Notation:

${x | - \sqrt{2} \leq x \leq \sqrt{2}}$

Interval Notation:

$[- \sqrt{2}, \sqrt{2}]$

Set-Builder Notation:

${x | - \sqrt{2} \leq x \leq \sqrt{2}}$

Step 2

To find the end points, substitute the bounds of the $x$ values from the domain into $f (x) = x \sqrt{2 - x^{2}}$ .

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Step 2.1

Replace the variable $x$ with $- \sqrt{2}$ in the expression.

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

Step 2.2

Simplify the result.

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Step 2.2.1

Apply the product rule to $- \sqrt{2}$ .

$f (- \sqrt{2}) = - \sqrt{2} \sqrt{2 - ({(- 1)}^{2} {\sqrt{2}}^{2})}$

Step 2.2.2

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 2.2.2.1

Move ${(- 1)}^{2}$ .

$f (- \sqrt{2}) = - \sqrt{2} \sqrt{2 + {(- 1)}^{2} \cdot (- 1 {\sqrt{2}}^{2})}$

Step 2.2.2.2

Multiply ${(- 1)}^{2}$ by $- 1$ .

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Step 2.2.2.2.1

Raise $- 1$ to the power of $1$ .

$f (- \sqrt{2}) = - \sqrt{2} \sqrt{2 + {(- 1)}^{2} \cdot ((- 1) {\sqrt{2}}^{2})}$

Step 2.2.2.2.2

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

$f (- \sqrt{2}) = - \sqrt{2} \sqrt{2 + {(- 1)}^{2 + 1} {\sqrt{2}}^{2}}$

Step 2.2.2.3

Add $2$ and $1$ .

$f (- \sqrt{2}) = - \sqrt{2} \sqrt{2 + {(- 1)}^{3} {\sqrt{2}}^{2}}$

Step 2.2.3

Raise $- 1$ to the power of $3$ .

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

Step 2.2.4

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

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Step 2.2.4.1

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

$f (- \sqrt{2}) = - \sqrt{2} \sqrt{2 - {(2^{\frac{1}{2}})}^{2}}$

Step 2.2.4.2

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

$f (- \sqrt{2}) = - \sqrt{2} \sqrt{2 - 2^{\frac{1}{2} \cdot 2}}$

Step 2.2.4.3

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

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

Step 2.2.4.4

Cancel the common factor of $2$ .

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Step 2.2.4.4.1

Cancel the common factor.

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

Step 2.2.4.4.2

Rewrite the expression.

$f (- \sqrt{2}) = - \sqrt{2} \sqrt{2 - 2}$

Step 2.2.4.5

Evaluate the exponent.

$f (- \sqrt{2}) = - \sqrt{2} \sqrt{2 - 1 \cdot 2}$

Step 2.2.5

Simplify the expression.

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Step 2.2.5.1

Multiply $- 1$ by $2$ .

$f (- \sqrt{2}) = - \sqrt{2} \sqrt{2 - 2}$

Step 2.2.5.2

Subtract $2$ from $2$ .

$f (- \sqrt{2}) = - \sqrt{2} \sqrt{0}$

Step 2.2.5.3

Rewrite $0$ as $0^{2}$ .

$f (- \sqrt{2}) = - \sqrt{2} \sqrt{0^{2}}$

Step 2.2.5.4

Pull terms out from under the radical, assuming positive real numbers.

$f (- \sqrt{2}) = - \sqrt{2} \cdot 0$

Step 2.2.6

Multiply $- \sqrt{2} \cdot 0$ .

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Step 2.2.6.1

Multiply $0$ by $- 1$ .

$f (- \sqrt{2}) = 0 \sqrt{2}$

Step 2.2.6.2

Multiply $0$ by $\sqrt{2}$ .

$f (- \sqrt{2}) = 0$

Step 2.2.7

The final answer is $0$ .

$0$

Step 2.3

Replace the variable $x$ with $\sqrt{2}$ in the expression.

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

Step 2.4

Simplify the result.

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Step 2.4.1

Combine using the product rule for radicals.

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

Step 2.4.2

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

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Step 2.4.2.1

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

$f (\sqrt{2}) = \sqrt{2 (2 - {(2^{\frac{1}{2}})}^{2})}$

Step 2.4.2.2

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

$f (\sqrt{2}) = \sqrt{2 (2 - 2^{\frac{1}{2} \cdot 2})}$

Step 2.4.2.3

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

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

Step 2.4.2.4

Cancel the common factor of $2$ .

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Step 2.4.2.4.1

Cancel the common factor.

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

Step 2.4.2.4.2

Rewrite the expression.

$f (\sqrt{2}) = \sqrt{2 (2 - 2)}$

Step 2.4.2.5

Evaluate the exponent.

$f (\sqrt{2}) = \sqrt{2 (2 - 1 \cdot 2)}$

Step 2.4.3

Simplify the expression.

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Step 2.4.3.1

Multiply $- 1$ by $2$ .

$f (\sqrt{2}) = \sqrt{2 (2 - 2)}$

Step 2.4.3.2

Subtract $2$ from $2$ .

$f (\sqrt{2}) = \sqrt{2 \cdot 0}$

Step 2.4.3.3

Multiply $2$ by $0$ .

$f (\sqrt{2}) = \sqrt{0}$

Step 2.4.3.4

Rewrite $0$ as $0^{2}$ .

$f (\sqrt{2}) = \sqrt{0^{2}}$

Step 2.4.3.5

Pull terms out from under the radical, assuming positive real numbers.

$f (\sqrt{2}) = 0$

Step 2.4.4

The final answer is $0$ .

$0$

Step 3

The end points are $(- \sqrt{2}, 0), (\sqrt{2}, 0)$ .

$(- \sqrt{2}, 0), (\sqrt{2}, 0)$

Step 4

The square root can be graphed using the points around the vertex $(- \sqrt{2}, 0), (\sqrt{2}, 0),$

$\begin{array}{c} x & y \\ - \sqrt{2} & 0 \\ \sqrt{2} & 0 \end{array}$

Step 5