Calculus Examples

$f (x) = x^{\frac{1}{3}}$ , $[- 1, 1]$

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 $[- 1, 1]$ or not, find the domain of $f (x) = x^{\frac{1}{3}}$ .

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Step 1.1

Convert expressions with fractional exponents to radicals.

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Step 1.1.1

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

$\sqrt[3]{x^{1}}$

Step 1.1.2

Anything raised to $1$ is the base itself.

$\sqrt[3]{x}$

Step 1.2

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 2

$f (x)$ is continuous on $[- 1, 1]$ .

$f (x)$ is continuous

Step 3

The average value of function $f$ over the interval $[a, b]$ is defined as $A (x) = \frac{1}{b - a} \int_{a}^{b} f (x) d x$ .

$A (x) = \frac{1}{b - a} \int_{a}^{b} f (x) d x$

Step 4

Substitute the actual values into the formula for the average value of a function.

$A (x) = \frac{1}{1 + 1} (\int_{- 1}^{1} x^{\frac{1}{3}} d x)$

Step 5

By the Power Rule, the integral of $x^{\frac{1}{3}}$ with respect to $x$ is $\frac{3}{4} x^{\frac{4}{3}}$ .

$A (x) = \frac{1}{1 + 1} (\frac{3}{4} x^{\frac{4}{3}}]_{- 1}^{1})$

Step 6

Substitute and simplify.

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Step 6.1

Evaluate $\frac{3}{4} x^{\frac{4}{3}}$ at $1$ and at $- 1$ .

$A (x) = \frac{1}{1 + 1} ((\frac{3}{4} \cdot 1^{\frac{4}{3}}) - \frac{3}{4} \cdot {(- 1)}^{\frac{4}{3}})$

Step 6.2

Simplify.

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Step 6.2.1

One to any power is one.

$A (x) = \frac{1}{1 + 1} (\frac{3}{4} \cdot 1 - \frac{3}{4} \cdot {(- 1)}^{\frac{4}{3}})$

Step 6.2.2

Multiply $\frac{3}{4}$ by $1$ .

$A (x) = \frac{1}{1 + 1} (\frac{3}{4} - \frac{3}{4} \cdot {(- 1)}^{\frac{4}{3}})$

Step 6.2.3

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$A (x) = \frac{1}{1 + 1} (\frac{3}{4} - \frac{3}{4} \cdot {({(- 1)}^{3})}^{\frac{4}{3}})$

Step 6.2.4

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

$A (x) = \frac{1}{1 + 1} (\frac{3}{4} - \frac{3}{4} \cdot {(- 1)}^{3 (\frac{4}{3})})$

Step 6.2.5

Cancel the common factor of $3$ .

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Step 6.2.5.1

Cancel the common factor.

$A (x) = \frac{1}{1 + 1} (\frac{3}{4} - \frac{3}{4} \cdot {(- 1)}^{3 (\frac{4}{3})})$

Step 6.2.5.2

Rewrite the expression.

$A (x) = \frac{1}{1 + 1} (\frac{3}{4} - \frac{3}{4} \cdot {(- 1)}^{4})$

Step 6.2.6

Raise $- 1$ to the power of $4$ .

$A (x) = \frac{1}{1 + 1} (\frac{3}{4} - \frac{3}{4} \cdot 1)$

Step 6.2.7

Multiply $- 1$ by $1$ .

$A (x) = \frac{1}{1 + 1} (\frac{3}{4} - \frac{3}{4})$

Step 6.2.8

Combine the numerators over the common denominator.

$A (x) = \frac{1}{1 + 1} (\frac{3 - 3}{4})$

Step 6.2.9

Subtract $3$ from $3$ .

$A (x) = \frac{1}{1 + 1} (\frac{0}{4})$

Step 6.2.10

Cancel the common factor of $0$ and $4$ .

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Step 6.2.10.1

Factor $4$ out of $0$ .

$A (x) = \frac{1}{1 + 1} (\frac{4 (0)}{4})$

Step 6.2.10.2

Cancel the common factors.

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Step 6.2.10.2.1

Factor $4$ out of $4$ .

$A (x) = \frac{1}{1 + 1} (\frac{4 \cdot 0}{4 \cdot 1})$

Step 6.2.10.2.2

Cancel the common factor.

$A (x) = \frac{1}{1 + 1} (\frac{4 \cdot 0}{4 \cdot 1})$

Step 6.2.10.2.3

Rewrite the expression.

$A (x) = \frac{1}{1 + 1} (\frac{0}{1})$

Step 6.2.10.2.4

Divide $0$ by $1$ .

$A (x) = \frac{1}{1 + 1} (0)$

Step 7

Add $1$ and $1$ .

$A (x) = \frac{1}{2} \cdot 0$

Step 8

Multiply $\frac{1}{2}$ by $0$ .

$A (x) = 0$

Step 9