Calculus Examples

$f (x) = \sqrt[4]{x}$ , $[1, 16]$

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 $[1, 16]$ or not, find the domain of $f (x) = \sqrt[4]{x}$ .

Tap for more steps...

Step 1.1

Set the radicand in $\sqrt[4]{x}$ greater than or equal to $0$ to find where the expression is defined.

$x \geq 0$

Step 1.2

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 continuous on $[1, 16]$ .

$f (x)$ 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.

$A (x) = \frac{1}{16 - 1} (\int_{1}^{16} \sqrt[4]{x} d x)$

Step 5

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x}$ as $x^{\frac{1}{4}}$ .

$A (x) = \frac{1}{16 - 1} (\int_{1}^{16} x^{\frac{1}{4}} d x)$

Step 6

By the Power Rule, the integral of $x^{\frac{1}{4}}$ with respect to $x$ is $\frac{4}{5} x^{\frac{5}{4}}$ .

$A (x) = \frac{1}{16 - 1} (\frac{4}{5} x^{\frac{5}{4}}]_{1}^{16})$

Step 7

Substitute and simplify.

Tap for more steps...

Step 7.1

Evaluate $\frac{4}{5} x^{\frac{5}{4}}$ at $16$ and at $1$ .

$A (x) = \frac{1}{16 - 1} ((\frac{4}{5} \cdot 16^{\frac{5}{4}}) - \frac{4}{5} \cdot 1^{\frac{5}{4}})$

Step 7.2

Simplify.

Tap for more steps...

Step 7.2.1

Rewrite $16$ as $2^{4}$ .

$A (x) = \frac{1}{16 - 1} (\frac{4}{5} \cdot {(2^{4})}^{\frac{5}{4}} - \frac{4}{5} \cdot 1^{\frac{5}{4}})$

Step 7.2.2

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

$A (x) = \frac{1}{16 - 1} (\frac{4}{5} \cdot 2^{4 (\frac{5}{4})} - \frac{4}{5} \cdot 1^{\frac{5}{4}})$

Step 7.2.3

Cancel the common factor of $4$ .

Tap for more steps...

Step 7.2.3.1

Cancel the common factor.

$A (x) = \frac{1}{16 - 1} (\frac{4}{5} \cdot 2^{4 (\frac{5}{4})} - \frac{4}{5} \cdot 1^{\frac{5}{4}})$

Step 7.2.3.2

Rewrite the expression.

$A (x) = \frac{1}{16 - 1} (\frac{4}{5} \cdot 2^{5} - \frac{4}{5} \cdot 1^{\frac{5}{4}})$

Step 7.2.4

Raise $2$ to the power of $5$ .

$A (x) = \frac{1}{16 - 1} (\frac{4}{5} \cdot 32 - \frac{4}{5} \cdot 1^{\frac{5}{4}})$

Step 7.2.5

Combine $\frac{4}{5}$ and $32$ .

$A (x) = \frac{1}{16 - 1} (\frac{4 \cdot 32}{5} - \frac{4}{5} \cdot 1^{\frac{5}{4}})$

Step 7.2.6

Multiply $4$ by $32$ .

$A (x) = \frac{1}{16 - 1} (\frac{128}{5} - \frac{4}{5} \cdot 1^{\frac{5}{4}})$

Step 7.2.7

One to any power is one.

$A (x) = \frac{1}{16 - 1} (\frac{128}{5} - \frac{4}{5} \cdot 1)$

Step 7.2.8

Multiply $- 1$ by $1$ .

$A (x) = \frac{1}{16 - 1} (\frac{128}{5} - \frac{4}{5})$

Step 7.2.9

Combine the numerators over the common denominator.

$A (x) = \frac{1}{16 - 1} (\frac{128 - 4}{5})$

Step 7.2.10

Subtract $4$ from $128$ .

$A (x) = \frac{1}{16 - 1} (\frac{124}{5})$

Step 8

Subtract $1$ from $16$ .

$A (x) = \frac{1}{15} \cdot \frac{124}{5}$

Step 9

Multiply $\frac{1}{15} \cdot \frac{124}{5}$ .

Tap for more steps...

Step 9.1

Multiply $\frac{1}{15}$ by $\frac{124}{5}$ .

$A (x) = \frac{124}{15 \cdot 5}$

Step 9.2

Multiply $15$ by $5$ .

$A (x) = \frac{124}{75}$

Step 10