Calculus Examples

$f (x) = \frac{x^{2}}{7} \cdot \sqrt{16 - \frac{x^{3}}{7}}$

Step 1

Set the radicand in $\sqrt{16 - \frac{x^{3}}{7}}$ greater than or equal to $0$ to find where the expression is defined.

$16 - \frac{x^{3}}{7} \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $16$ from both sides of the inequality.

$- \frac{x^{3}}{7} \geq - 16$

Step 2.2

Divide each term in $- \frac{x^{3}}{7} \geq - 16$ by $- 1$ and simplify.

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Step 2.2.1

Divide each term in $- \frac{x^{3}}{7} \geq - 16$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{x^{3}}{7}}{- 1} \leq \frac{- 16}{- 1}$

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.

$\frac{\frac{x^{3}}{7}}{1} \leq \frac{- 16}{- 1}$

Step 2.2.2.2

Divide $\frac{x^{3}}{7}$ by $1$ .

$\frac{x^{3}}{7} \leq \frac{- 16}{- 1}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $- 16$ by $- 1$ .

$\frac{x^{3}}{7} \leq 16$

Step 2.3

Multiply both sides by $7$ .

$\frac{x^{3}}{7} \cdot 7 \leq 16 \cdot 7$

Step 2.4

Simplify.

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Step 2.4.1

Simplify the left side.

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Step 2.4.1.1

Cancel the common factor of $7$ .

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Step 2.4.1.1.1

Cancel the common factor.

$\frac{x^{3}}{7} \cdot 7 \leq 16 \cdot 7$

Step 2.4.1.1.2

Rewrite the expression.

$x^{3} \leq 16 \cdot 7$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Multiply $16$ by $7$ .

$x^{3} \leq 112$

Step 2.5

Solve for $x$ .

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Step 2.5.1

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

$\sqrt[3]{x^{3}} \leq \sqrt[3]{112}$

Step 2.5.2

Simplify the equation.

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Step 2.5.2.1

Simplify the left side.

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Step 2.5.2.1.1

Pull terms out from under the radical.

$x \leq \sqrt[3]{112}$

Step 2.5.2.2

Simplify the right side.

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Step 2.5.2.2.1

Simplify $\sqrt[3]{112}$ .

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Step 2.5.2.2.1.1

Rewrite $112$ as $2^{3} \cdot 14$ .

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Step 2.5.2.2.1.1.1

Factor $8$ out of $112$ .

$x \leq \sqrt[3]{8 (14)}$

Step 2.5.2.2.1.1.2

Rewrite $8$ as $2^{3}$ .

$x \leq \sqrt[3]{2^{3} \cdot 14}$

Step 2.5.2.2.1.2

Pull terms out from under the radical.

$x \leq 2 \sqrt[3]{14}$

Step 3

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

Interval Notation:

$(- \infty, 2 \sqrt[3]{14}]$

Set-Builder Notation:

${x | x \leq 2 \sqrt[3]{14}}$

Step 4

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

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${y | y \geq 0}$

Step 5

Determine the domain and range.

Domain: $(- \infty, 2 \sqrt[3]{14}], {x | x \leq 2 \sqrt[3]{14}}$

Range: $[0, \infty), {y | y \geq 0}$

Step 6