Calculus Examples

$y = x \sqrt{8 - x^{2}}$

Step 1

Find the domain for $y = x \sqrt{8 - x^{2}}$ 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 $\sqrt{8 - x^{2}}$ greater than or equal to $0$ to find where the expression is defined.

$8 - x^{2} \geq 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Subtract $8$ from both sides of the inequality.

$- x^{2} \geq - 8$

Step 1.2.2

Divide each term in $- x^{2} \geq - 8$ by $- 1$ and simplify.

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Step 1.2.2.1

Divide each term in $- x^{2} \geq - 8$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x^{2}}{- 1} \leq \frac{- 8}{- 1}$

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.

$\frac{x^{2}}{1} \leq \frac{- 8}{- 1}$

Step 1.2.2.2.2

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

$x^{2} \leq \frac{- 8}{- 1}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $- 8$ by $- 1$ .

$x^{2} \leq 8$

Step 1.2.3

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

$\sqrt{x^{2}} \leq \sqrt{8}$

Step 1.2.4

Simplify the equation.

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Step 1.2.4.1

Simplify the left side.

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Step 1.2.4.1.1

Pull terms out from under the radical.

$| x | \leq \sqrt{8}$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Simplify $\sqrt{8}$ .

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Step 1.2.4.2.1.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 1.2.4.2.1.1.1

Factor $4$ out of $8$ .

$| x | \leq \sqrt{4 (2)}$

Step 1.2.4.2.1.1.2

Rewrite $4$ as $2^{2}$ .

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

Step 1.2.4.2.1.2

Pull terms out from under the radical.

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

Step 1.2.4.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

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

Step 1.2.5

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

$x \leq 2 \sqrt{2}$

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 $- 1$ .

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

Step 1.2.5.5

Write as a piecewise.

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

Step 1.2.6

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

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

Step 1.2.7

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

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Step 1.2.7.1

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

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Step 1.2.7.1.1

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

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

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.

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

Step 1.2.7.1.2.2

Divide $x$ by $1$ .

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

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 $\frac{2 \sqrt{2}}{- 1}$ .

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

Step 1.2.7.1.3.2

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

$x \geq - (2 \sqrt{2})$

Step 1.2.7.1.3.3

Multiply $2$ by $- 1$ .

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

Step 1.2.7.2

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

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

Step 1.2.8

Find the union of the solutions.

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

Step 1.3

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

Interval Notation:

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

Set-Builder Notation:

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

Interval Notation:

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

Set-Builder Notation:

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

Step 2

To find the end points, substitute the bounds of the $x$ values from the domain into $f (x) = x \sqrt{8 - x^{2}}$ .

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Step 2.1

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

$f (- 2 \sqrt{2}) = (- 2 \sqrt{2}) \sqrt{8 - {(- 2 \sqrt{2})}^{2}}$

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify the expression.

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Step 2.2.1.1

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

$f (- 2 \sqrt{2}) = - 2 \sqrt{2} \sqrt{8 - ({(- 2)}^{2} {\sqrt{2}}^{2})}$

Step 2.2.1.2

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

$f (- 2 \sqrt{2}) = - 2 \sqrt{2} \sqrt{8 - (4 {\sqrt{2}}^{2})}$

Step 2.2.2

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

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Step 2.2.2.1

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

$f (- 2 \sqrt{2}) = - 2 \sqrt{2} \sqrt{8 - (4 {(2^{\frac{1}{2}})}^{2})}$

Step 2.2.2.2

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

$f (- 2 \sqrt{2}) = - 2 \sqrt{2} \sqrt{8 - (4 \cdot 2^{\frac{1}{2} \cdot 2})}$

Step 2.2.2.3

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

$f (- 2 \sqrt{2}) = - 2 \sqrt{2} \sqrt{8 - (4 \cdot 2^{\frac{2}{2}})}$

Step 2.2.2.4

Cancel the common factor of $2$ .

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Step 2.2.2.4.1

Cancel the common factor.

$f (- 2 \sqrt{2}) = - 2 \sqrt{2} \sqrt{8 - (4 \cdot 2^{\frac{2}{2}})}$

Step 2.2.2.4.2

Rewrite the expression.

$f (- 2 \sqrt{2}) = - 2 \sqrt{2} \sqrt{8 - (4 \cdot 2)}$

Step 2.2.2.5

Evaluate the exponent.

$f (- 2 \sqrt{2}) = - 2 \sqrt{2} \sqrt{8 - (4 \cdot 2)}$

Step 2.2.3

Multiply $- (4 \cdot 2)$ .

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Step 2.2.3.1

Multiply $4$ by $2$ .

$f (- 2 \sqrt{2}) = - 2 \sqrt{2} \sqrt{8 - 1 \cdot 8}$

Step 2.2.3.2

Multiply $- 1$ by $8$ .

$f (- 2 \sqrt{2}) = - 2 \sqrt{2} \sqrt{8 - 8}$

Step 2.2.4

Subtract $8$ from $8$ .

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

Step 2.2.5

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

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

Step 2.2.6

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

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

Step 2.2.7

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

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Step 2.2.7.1

Multiply $0$ by $- 2$ .

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

Step 2.2.7.2

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

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

Step 2.2.8

The final answer is $0$ .

$0$

Step 2.3

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

$f (2 \sqrt{2}) = (2 \sqrt{2}) \sqrt{8 - {(2 \sqrt{2})}^{2}}$

Step 2.4

Simplify the result.

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Step 2.4.1

Simplify the expression.

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Step 2.4.1.1

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

$f (2 \sqrt{2}) = 2 \sqrt{2} \sqrt{8 - (2^{2} {\sqrt{2}}^{2})}$

Step 2.4.1.2

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

$f (2 \sqrt{2}) = 2 \sqrt{2} \sqrt{8 - (4 {\sqrt{2}}^{2})}$

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 $2^{\frac{1}{2}}$ .

$f (2 \sqrt{2}) = 2 \sqrt{2} \sqrt{8 - (4 {(2^{\frac{1}{2}})}^{2})}$

Step 2.4.2.2

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

$f (2 \sqrt{2}) = 2 \sqrt{2} \sqrt{8 - (4 \cdot 2^{\frac{1}{2} \cdot 2})}$

Step 2.4.2.3

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

$f (2 \sqrt{2}) = 2 \sqrt{2} \sqrt{8 - (4 \cdot 2^{\frac{2}{2}})}$

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.

$f (2 \sqrt{2}) = 2 \sqrt{2} \sqrt{8 - (4 \cdot 2^{\frac{2}{2}})}$

Step 2.4.2.4.2

Rewrite the expression.

$f (2 \sqrt{2}) = 2 \sqrt{2} \sqrt{8 - (4 \cdot 2)}$

Step 2.4.2.5

Evaluate the exponent.

$f (2 \sqrt{2}) = 2 \sqrt{2} \sqrt{8 - (4 \cdot 2)}$

Step 2.4.3

Multiply $- (4 \cdot 2)$ .

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Step 2.4.3.1

Multiply $4$ by $2$ .

$f (2 \sqrt{2}) = 2 \sqrt{2} \sqrt{8 - 1 \cdot 8}$

Step 2.4.3.2

Multiply $- 1$ by $8$ .

$f (2 \sqrt{2}) = 2 \sqrt{2} \sqrt{8 - 8}$

Step 2.4.4

Subtract $8$ from $8$ .

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

Step 2.4.5

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

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

Step 2.4.6

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

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

Step 2.4.7

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

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Step 2.4.7.1

Multiply $0$ by $2$ .

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

Step 2.4.7.2

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

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

Step 2.4.8

The final answer is $0$ .

$0$

Step 3

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

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

Step 4

Select a few $x$ values from the domain. It would be more useful to select the values so that they are next to the $x$ value of the radical expression end point.

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Step 4.1

Substitute the $x$ value $- 1.82842712$ into $f (x) = x \sqrt{8 - x^{2}}$ . In this case, the point is $(- 1.82842712, - 3.94569924)$ .

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Step 4.1.1

Replace the variable $x$ with $- 1.82842712$ in the expression.

$f (- 1.82842712) = (- 1.82842712) \sqrt{8 - {(- 1.82842712)}^{2}}$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Simplify the expression.

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Step 4.1.2.1.1

Raise $- 1.82842712$ to the power of $2$ .

$f (- 1.82842712) = - 1.82842712 \sqrt{8 - 1 \cdot 3.34314575}$

Step 4.1.2.1.2

Multiply $- 1$ by $3.34314575$ .

$f (- 1.82842712) = - 1.82842712 \sqrt{8 - 3.34314575}$

Step 4.1.2.1.3

Subtract $3.34314575$ from $8$ .

$f (- 1.82842712) = - 1.82842712 \sqrt{4.65685424}$

Step 4.1.2.2

Multiply $- 1.82842712$ by $\sqrt{4.65685424}$ .

$f (- 1.82842712) = - 3.94569924$

Step 4.1.2.3

The final answer is $- 3.94569924$ .

$y = - 3.94569924$

Step 4.2

Substitute the $x$ value $- 0.82842712$ into $f (x) = x \sqrt{8 - x^{2}}$ . In this case, the point is $(- 0.82842712, - 2.24038746)$ .

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Step 4.2.1

Replace the variable $x$ with $- 0.82842712$ in the expression.

$f (- 0.82842712) = (- 0.82842712) \sqrt{8 - {(- 0.82842712)}^{2}}$

Step 4.2.2

Simplify the result.

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Step 4.2.2.1

Simplify the expression.

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Step 4.2.2.1.1

Raise $- 0.82842712$ to the power of $2$ .

$f (- 0.82842712) = - 0.82842712 \sqrt{8 - 1 \cdot 0.6862915}$

Step 4.2.2.1.2

Multiply $- 1$ by $0.6862915$ .

$f (- 0.82842712) = - 0.82842712 \sqrt{8 - 0.6862915}$

Step 4.2.2.1.3

Subtract $0.6862915$ from $8$ .

$f (- 0.82842712) = - 0.82842712 \sqrt{7.31370849}$

Step 4.2.2.2

Multiply $- 0.82842712$ by $\sqrt{7.31370849}$ .

$f (- 0.82842712) = - 2.24038746$

Step 4.2.2.3

The final answer is $- 2.24038746$ .

$y = - 2.24038746$

Step 4.3

Substitute the $x$ value $0.17157287$ into $f (x) = x \sqrt{8 - x^{2}}$ . In this case, the point is $(0.17157287, 0.48438771)$ .

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Step 4.3.1

Replace the variable $x$ with $0.17157287$ in the expression.

$f (0.17157287) = (0.17157287) \sqrt{8 - {(0.17157287)}^{2}}$

Step 4.3.2

Simplify the result.

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Step 4.3.2.1

Simplify the expression.

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Step 4.3.2.1.1

Raise $0.17157287$ to the power of $2$ .

$f (0.17157287) = 0.17157287 \sqrt{8 - 1 \cdot 0.02943725}$

Step 4.3.2.1.2

Multiply $- 1$ by $0.02943725$ .

$f (0.17157287) = 0.17157287 \sqrt{8 - 0.02943725}$

Step 4.3.2.1.3

Subtract $0.02943725$ from $8$ .

$f (0.17157287) = 0.17157287 \sqrt{7.97056274}$

Step 4.3.2.2

Multiply $0.17157287$ by $\sqrt{7.97056274}$ .

$f (0.17157287) = 0.48438771$

Step 4.3.2.3

The final answer is $0.48438771$ .

$y = 0.48438771$

Step 4.4

Substitute the $x$ value $1.17157287$ into $f (x) = x \sqrt{8 - x^{2}}$ . In this case, the point is $(1.17157287, 3.01607027)$ .

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Step 4.4.1

Replace the variable $x$ with $1.17157287$ in the expression.

$f (1.17157287) = (1.17157287) \sqrt{8 - {(1.17157287)}^{2}}$

Step 4.4.2

Simplify the result.

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Step 4.4.2.1

Simplify the expression.

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Step 4.4.2.1.1

Raise $1.17157287$ to the power of $2$ .

$f (1.17157287) = 1.17157287 \sqrt{8 - 1 \cdot 1.372583}$

Step 4.4.2.1.2

Multiply $- 1$ by $1.372583$ .

$f (1.17157287) = 1.17157287 \sqrt{8 - 1.372583}$

Step 4.4.2.1.3

Subtract $1.372583$ from $8$ .

$f (1.17157287) = 1.17157287 \sqrt{6.62741699}$

Step 4.4.2.2

Multiply $1.17157287$ by $\sqrt{6.62741699}$ .

$f (1.17157287) = 3.01607027$

Step 4.4.2.3

The final answer is $3.01607027$ .

$y = 3.01607027$

Step 4.5

Substitute the $x$ value $2.17157287$ into $f (x) = x \sqrt{8 - x^{2}}$ . In this case, the point is $(2.17157287, 3.93544563)$ .

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Step 4.5.1

Replace the variable $x$ with $2.17157287$ in the expression.

$f (2.17157287) = (2.17157287) \sqrt{8 - {(2.17157287)}^{2}}$

Step 4.5.2

Simplify the result.

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Step 4.5.2.1

Simplify the expression.

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Step 4.5.2.1.1

Raise $2.17157287$ to the power of $2$ .

$f (2.17157287) = 2.17157287 \sqrt{8 - 1 \cdot 4.71572875}$

Step 4.5.2.1.2

Multiply $- 1$ by $4.71572875$ .

$f (2.17157287) = 2.17157287 \sqrt{8 - 4.71572875}$

Step 4.5.2.1.3

Subtract $4.71572875$ from $8$ .

$f (2.17157287) = 2.17157287 \sqrt{3.28427124}$

Step 4.5.2.2

Multiply $2.17157287$ by $\sqrt{3.28427124}$ .

$f (2.17157287) = 3.93544563$

Step 4.5.2.3

The final answer is $3.93544563$ .

$y = 3.93544563$

Step 4.6

The square root can be graphed using the points around the vertex $(- 2 \sqrt{2}, 0), (2 \sqrt{2}, 0), (- 1.82842712, - 3.95), (- 0.82842712, - 2.24), (0.17157287, 0.48), (1.17157287, 3.02), (2.17157287, 3.94)$

$\begin{array}{c} x & y \\ - 2 \sqrt{2} & 0 \\ - 1.828 & - 3.95 \\ - 0.828 & - 2.24 \\ 0.172 & 0.48 \\ 1.172 & 3.02 \\ 2.172 & 3.94 \\ 2 \sqrt{2} & 0 \end{array}$

Step 5