Calculus Examples

$f (x) = \frac{2 x^{\frac{3}{2}}}{3} + C$ , $0 < x < 4$

Step 1

To find the average value of a function, the function should be continuous on the closed interval $[a, b]$ . To find whether $f (x)$ is continuous on $[0, 4]$ or not, find the domain of $f (x) = \frac{2 x^{\frac{3}{2}}}{3} + C$ .

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Step 1.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\frac{2 \sqrt{x^{3}}}{3} + C$

Step 1.2

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

$x^{3} \geq 0$

Step 1.3

Solve for $x$ .

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Step 1.3.1

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

$\sqrt[3]{x^{3}} \geq \sqrt[3]{0}$

Step 1.3.2

Simplify the equation.

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Step 1.3.2.1

Simplify the left side.

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Step 1.3.2.1.1

Pull terms out from under the radical.

$x \geq \sqrt[3]{0}$

Step 1.3.2.2

Simplify the right side.

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Step 1.3.2.2.1

Simplify $\sqrt[3]{0}$ .

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Step 1.3.2.2.1.1

Rewrite $0$ as $0^{3}$ .

$x \geq \sqrt[3]{0^{3}}$

Step 1.3.2.2.1.2

Pull terms out from under the radical.

$x \geq 0$

Step 1.4

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

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${x | x \geq 0}$

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${x | x \geq 0}$

Step 2

$f (x)$ is not continuous on $[0, 4]$ because $0$ is not in the domain of $f (x)$ . The function should be continuous to find the average value

$f (x)$ is not continuous

Step 3