Calculus Examples

$f (x) = \frac{4 (x + 1)}{x^{2}}$ , $[2, 4]$

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 $[2, 4]$ or not, find the domain of $f (x) = \frac{4 (x + 1)}{x^{2}}$ .

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Step 1.1

Set the denominator in $\frac{4 (x + 1)}{x^{2}}$ equal to $0$ to find where the expression is undefined.

$x^{2} = 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

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

$x = \pm \sqrt{0}$

Step 1.2.2

Simplify $\pm \sqrt{0}$ .

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Step 1.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 1.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 1.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 1.3

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 $[2, 4]$ .

$f (x)$ 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.

$A (x) = \frac{1}{4 - 2} (\int_{2}^{4} \frac{4 (x + 1)}{x^{2}} d x)$

Step 5

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$A (x) = \frac{1}{4 - 2} (4 \int_{2}^{4} \frac{x + 1}{x^{2}} d x)$

Step 6

Apply basic rules of exponents.

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Step 6.1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

$A (x) = \frac{1}{4 - 2} (4 \int_{2}^{4} (x + 1) {(x^{2})}^{- 1} d x)$

Step 6.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 6.2.1

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

$A (x) = \frac{1}{4 - 2} (4 \int_{2}^{4} (x + 1) x^{2 \cdot - 1} d x)$

Step 6.2.2

Multiply $2$ by $- 1$ .

$A (x) = \frac{1}{4 - 2} (4 \int_{2}^{4} (x + 1) x^{- 2} d x)$

Step 7

Multiply $(x + 1) x^{- 2}$ .

$A (x) = \frac{1}{4 - 2} (4 \int_{2}^{4} x \cdot x^{- 2} + 1 x^{- 2} d x)$

Step 8

Simplify.

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Step 8.1

Multiply $x$ by $x^{- 2}$ by adding the exponents.

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Step 8.1.1

Multiply $x$ by $x^{- 2}$ .

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Step 8.1.1.1

Raise $x$ to the power of $1$ .

$A (x) = \frac{1}{4 - 2} (4 \int_{2}^{4} x \cdot x^{- 2} + 1 x^{- 2} d x)$

Step 8.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$A (x) = \frac{1}{4 - 2} (4 \int_{2}^{4} x^{1 - 2} + 1 x^{- 2} d x)$

Step 8.1.2

Subtract $2$ from $1$ .

$A (x) = \frac{1}{4 - 2} (4 \int_{2}^{4} x^{- 1} + 1 x^{- 2} d x)$

Step 8.2

Multiply $x^{- 2}$ by $1$ .

$A (x) = \frac{1}{4 - 2} (4 \int_{2}^{4} x^{- 1} + x^{- 2} d x)$

Step 9

Split the single integral into multiple integrals.

$A (x) = \frac{1}{4 - 2} (4 (\int_{2}^{4} x^{- 1} d x + \int_{2}^{4} x^{- 2} d x))$

Step 10

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

$A (x) = \frac{1}{4 - 2} (4 (\ln (| x |)]_{2}^{4} + \int_{2}^{4} x^{- 2} d x))$

Step 11

By the Power Rule, the integral of $x^{- 2}$ with respect to $x$ is $- x^{- 1}$ .

$A (x) = \frac{1}{4 - 2} (4 (\ln (| x |)]_{2}^{4} + - x^{- 1}]_{2}^{4}))$

Step 12

Simplify the answer.

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Step 12.1

Combine $\ln (| x |)]_{2}^{4}$ and $- x^{- 1}]_{2}^{4}$ .

$A (x) = \frac{1}{4 - 2} (4 (\ln (| x |) - x^{- 1}]_{2}^{4}))$

Step 12.2

Substitute and simplify.

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Step 12.2.1

Evaluate $\ln (| x |) - x^{- 1}$ at $4$ and at $2$ .

$A (x) = \frac{1}{4 - 2} (4 ((\ln (| 4 |) - 4^{- 1}) - (\ln (| 2 |) - 2^{- 1})))$

Step 12.2.2

Simplify.

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Step 12.2.2.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$A (x) = \frac{1}{4 - 2} (4 (\ln (| 4 |) - \frac{1}{4} - (\ln (| 2 |) - 2^{- 1})))$

Step 12.2.2.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 12.2.2.3

To write $- (\ln (| 2 |) - \frac{1}{2})$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 12.2.2.4

Combine $- (\ln (| 2 |) - \frac{1}{2})$ and $\frac{4}{4}$ .

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

Step 12.2.2.5

Combine the numerators over the common denominator.

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

Step 12.2.2.6

Multiply $4$ by $- 1$ .

$A (x) = \frac{1}{4 - 2} (4 (\ln (| 4 |) + \frac{- 1 - 4 (\ln (| 2 |) - \frac{1}{2})}{4}))$

Step 13

Simplify.

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Step 13.1

Simplify each term.

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Step 13.1.1

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

$A (x) = \frac{1}{4 - 2} (4 (\ln (4) + \frac{- 1 - 4 (\ln (| 2 |) - \frac{1}{2})}{4}))$

Step 13.1.2

Rewrite $\ln (4)$ as $\ln (2^{2})$ .

$A (x) = \frac{1}{4 - 2} (4 (\ln (2^{2}) + \frac{- 1 - 4 (\ln (| 2 |) - \frac{1}{2})}{4}))$

Step 13.1.3

Expand $\ln (2^{2})$ by moving $2$ outside the logarithm.

$A (x) = \frac{1}{4 - 2} (4 (2 \ln (2) + \frac{- 1 - 4 (\ln (| 2 |) - \frac{1}{2})}{4}))$

Step 13.1.4

Simplify the numerator.

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Step 13.1.4.1

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

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

Step 13.1.4.2

Apply the distributive property.

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

Step 13.1.4.3

Cancel the common factor of $2$ .

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Step 13.1.4.3.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

Step 13.1.4.3.2

Factor $2$ out of $- 4$ .

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

Step 13.1.4.3.3

Cancel the common factor.

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

Step 13.1.4.3.4

Rewrite the expression.

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

Step 13.1.4.4

Multiply $- 2$ by $- 1$ .

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

Step 13.1.4.5

Add $- 1$ and $2$ .

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

Step 13.2

To write $2 \ln (2)$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 13.3

Combine $2 \ln (2)$ and $\frac{4}{4}$ .

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

Step 13.4

Combine the numerators over the common denominator.

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

Step 13.5

Cancel the common factor of $4$ .

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Step 13.5.1

Cancel the common factor.

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

Step 13.5.2

Rewrite the expression.

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

Step 13.6

Multiply $4$ by $2$ .

$A (x) = \frac{1}{4 - 2} (8 \ln (2) + 1 - 4 \ln (2))$

Step 13.7

Subtract $4 \ln (2)$ from $8 \ln (2)$ .

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

Step 14

Subtract $2$ from $4$ .

$A (x) = \frac{1}{2} \cdot (4 \ln (2) + 1)$

Step 15

Simplify each term.

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Step 15.1

Simplify $4 \ln (2)$ by moving $4$ inside the logarithm.

$A (x) = \frac{1}{2} \cdot (\ln (2^{4}) + 1)$

Step 15.2

Raise $2$ to the power of $4$ .

$A (x) = \frac{1}{2} \cdot (\ln (16) + 1)$

Step 16

Apply the distributive property.

$A (x) = \frac{1}{2} \cdot \ln (16) + \frac{1}{2} \cdot 1$

Step 17

Simplify $\frac{1}{2} \ln (16)$ by moving $\frac{1}{2}$ inside the logarithm.

$A (x) = \ln (16^{\frac{1}{2}}) + \frac{1}{2} \cdot 1$

Step 18

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

$A (x) = \ln (16^{\frac{1}{2}}) + \frac{1}{2}$

Step 19

Simplify each term.

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Step 19.1

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

$A (x) = \ln ({(4^{2})}^{\frac{1}{2}}) + \frac{1}{2}$

Step 19.2

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

$A (x) = \ln (4^{2 (\frac{1}{2})}) + \frac{1}{2}$

Step 19.3

Cancel the common factor of $2$ .

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Step 19.3.1

Cancel the common factor.

$A (x) = \ln (4^{2 (\frac{1}{2})}) + \frac{1}{2}$

Step 19.3.2

Rewrite the expression.

$A (x) = \ln (4) + \frac{1}{2}$

Step 19.4

Evaluate the exponent.

$A (x) = \ln (4) + \frac{1}{2}$

Step 20