Calculus Examples

$f (x) = \frac{1}{x}$ , $[4, 6]$

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 $[4, 6]$ or not, find the domain of $f (x) = \frac{1}{x}$ .

Tap for more steps...

Step 1.1

Set the denominator in $\frac{1}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 1.2

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

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0}$

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0}$

Step 2

$f (x)$ is continuous on $[4, 6]$ .

$f (x)$ 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.

$A (x) = \frac{1}{6 - 4} (\int_{4}^{6} \frac{1}{x} d x)$

Step 5

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

$A (x) = \frac{1}{6 - 4} (\ln (| x |)]_{4}^{6})$

Step 6

Simplify the answer.

Tap for more steps...

Step 6.1

Evaluate $\ln (| x |)$ at $6$ and at $4$ .

$A (x) = \frac{1}{6 - 4} (\ln (| 6 |) - \ln (| 4 |))$

Step 6.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$A (x) = \frac{1}{6 - 4} (\ln (\frac{| 6 |}{| 4 |}))$

Step 6.3

Simplify.

Tap for more steps...

Step 6.3.1

The absolute value is the distance between a number and zero. The distance between $0$ and $6$ is $6$ .

$A (x) = \frac{1}{6 - 4} (\ln (\frac{6}{| 4 |}))$

Step 6.3.2

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$A (x) = \frac{1}{6 - 4} (\ln (\frac{6}{4}))$

Step 6.3.3

Cancel the common factor of $6$ and $4$ .

Tap for more steps...

Step 6.3.3.1

Factor $2$ out of $6$ .

$A (x) = \frac{1}{6 - 4} (\ln (\frac{2 (3)}{4}))$

Step 6.3.3.2

Cancel the common factors.

Tap for more steps...

Step 6.3.3.2.1

Factor $2$ out of $4$ .

$A (x) = \frac{1}{6 - 4} (\ln (\frac{2 \cdot 3}{2 \cdot 2}))$

Step 6.3.3.2.2

Cancel the common factor.

$A (x) = \frac{1}{6 - 4} (\ln (\frac{2 \cdot 3}{2 \cdot 2}))$

Step 6.3.3.2.3

Rewrite the expression.

$A (x) = \frac{1}{6 - 4} (\ln (\frac{3}{2}))$

Step 7

Subtract $4$ from $6$ .

$A (x) = \frac{1}{2} \cdot \ln (\frac{3}{2})$

Step 8

Simplify $\frac{1}{2} \ln (\frac{3}{2})$ by moving $\frac{1}{2}$ inside the logarithm.

$A (x) = \ln ({(\frac{3}{2})}^{\frac{1}{2}})$

Step 9

Apply the product rule to $\frac{3}{2}$ .

$A (x) = \ln (\frac{3^{\frac{1}{2}}}{2^{\frac{1}{2}}})$

Step 10