Calculus Examples

$f (x) = \frac{3}{x}$ , $[1, 7]$

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 $[1, 7]$ or not, find the domain of $f (x) = \frac{3}{x}$ .

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Step 1.1

Set the denominator in $\frac{3}{x}$ 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 $[1, 7]$ .

$f (x)$ 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.

$A (x) = \frac{1}{7 - 1} (\int_{1}^{7} \frac{3}{x} d x)$

Step 5

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$A (x) = \frac{1}{7 - 1} (3 \int_{1}^{7} \frac{1}{x} d x)$

Step 6

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

$A (x) = \frac{1}{7 - 1} (3 (\ln (| x |)]_{1}^{7}))$

Step 7

Simplify the answer.

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Step 7.1

Evaluate $\ln (| x |)$ at $7$ and at $1$ .

$A (x) = \frac{1}{7 - 1} (3 (\ln (| 7 |) - \ln (| 1 |)))$

Step 7.2

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

$A (x) = \frac{1}{7 - 1} (3 \ln (\frac{| 7 |}{| 1 |}))$

Step 7.3

Simplify.

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Step 7.3.1

The absolute value is the distance between a number and zero. The distance between $0$ and $7$ is $7$ .

$A (x) = \frac{1}{7 - 1} (3 \ln (\frac{7}{| 1 |}))$

Step 7.3.2

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$A (x) = \frac{1}{7 - 1} (3 \ln (\frac{7}{1}))$

Step 7.3.3

Divide $7$ by $1$ .

$A (x) = \frac{1}{7 - 1} (3 \ln (7))$

Step 8

Subtract $1$ from $7$ .

$A (x) = \frac{1}{6} \cdot (3 \ln (7))$

Step 9

Cancel the common factor of $3$ .

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Step 9.1

Factor $3$ out of $6$ .

$A (x) = \frac{1}{3 (2)} \cdot (3 \ln (7))$

Step 9.2

Factor $3$ out of $3 \ln (7)$ .

$A (x) = \frac{1}{3 (2)} \cdot (3 (\ln (7)))$

Step 9.3

Cancel the common factor.

$A (x) = \frac{1}{3 \cdot 2} \cdot (3 \ln (7))$

Step 9.4

Rewrite the expression.

$A (x) = \frac{1}{2} \cdot \ln (7)$

Step 10

Simplify $\frac{1}{2} \cdot \ln (7)$ by moving $\frac{1}{2}$ inside the logarithm.

$A (x) = \ln (7^{\frac{1}{2}})$

Step 11