Calculus Examples

$f (x) = \frac{4 (x^{2} + 1)}{x^{2}}$ , $[1, 3]$

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 $[1, 3]$ or not, find the domain of $f (x) = \frac{4 (x^{2} + 1)}{x^{2}}$ .

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Step 1.1

Set the denominator in $\frac{4 (x^{2} + 1)}{x^{2}}$ 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 $[1, 3]$ .

$f (x)$ 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.

$A (x) = \frac{1}{3 - 1} (\int_{1}^{3} \frac{4 (x^{2} + 1)}{x^{2}} d x)$

Step 5

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

$A (x) = \frac{1}{3 - 1} (4 \int_{1}^{3} \frac{x^{2} + 1}{x^{2}} d x)$

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.

$A (x) = \frac{1}{3 - 1} (4 \int_{1}^{3} (x^{2} + 1) {(x^{2})}^{- 1} d x)$

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, ${(a^{m})}^{n} = a^{m n}$ .

$A (x) = \frac{1}{3 - 1} (4 \int_{1}^{3} (x^{2} + 1) x^{2 \cdot - 1} d x)$

Step 6.2.2

Multiply $2$ by $- 1$ .

$A (x) = \frac{1}{3 - 1} (4 \int_{1}^{3} (x^{2} + 1) x^{- 2} d x)$

Step 7

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

$A (x) = \frac{1}{3 - 1} (4 \int_{1}^{3} x^{2} x^{- 2} + 1 x^{- 2} d x)$

Step 8

Simplify.

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Step 8.1

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

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Step 8.1.1

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

$A (x) = \frac{1}{3 - 1} (4 \int_{1}^{3} x^{2 - 2} + 1 x^{- 2} d x)$

Step 8.1.2

Subtract $2$ from $2$ .

$A (x) = \frac{1}{3 - 1} (4 \int_{1}^{3} x^{0} + 1 x^{- 2} d x)$

Step 8.2

Simplify $x^{0}$ .

$A (x) = \frac{1}{3 - 1} (4 \int_{1}^{3} 1 + 1 x^{- 2} d x)$

Step 8.3

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

$A (x) = \frac{1}{3 - 1} (4 \int_{1}^{3} 1 + x^{- 2} d x)$

Step 9

Split the single integral into multiple integrals.

$A (x) = \frac{1}{3 - 1} (4 (\int_{1}^{3} d x + \int_{1}^{3} x^{- 2} d x))$

Step 10

Apply the constant rule.

$A (x) = \frac{1}{3 - 1} (4 (x]_{1}^{3} + \int_{1}^{3} x^{- 2} d x))$

Step 11

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

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

Step 12

Simplify the answer.

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Step 12.1

Combine $x]_{1}^{3}$ and $- x^{- 1}]_{1}^{3}$ .

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

Step 12.2

Substitute and simplify.

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Step 12.2.1

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

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

Step 12.2.2

Simplify.

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Step 12.2.2.1

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

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

Step 12.2.2.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 12.2.2.3

Combine $3$ and $\frac{3}{3}$ .

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

Step 12.2.2.4

Combine the numerators over the common denominator.

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

Step 12.2.2.5

Simplify the numerator.

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Step 12.2.2.5.1

Multiply $3$ by $3$ .

$A (x) = \frac{1}{3 - 1} (4 (\frac{9 - 1}{3} - (1 - 1^{- 1})))$

Step 12.2.2.5.2

Subtract $1$ from $9$ .

$A (x) = \frac{1}{3 - 1} (4 (\frac{8}{3} - (1 - 1^{- 1})))$

Step 12.2.2.6

One to any power is one.

$A (x) = \frac{1}{3 - 1} (4 (\frac{8}{3} - (1 - 1 \cdot 1)))$

Step 12.2.2.7

Multiply $- 1$ by $1$ .

$A (x) = \frac{1}{3 - 1} (4 (\frac{8}{3} - (1 - 1)))$

Step 12.2.2.8

Subtract $1$ from $1$ .

$A (x) = \frac{1}{3 - 1} (4 (\frac{8}{3} - 0))$

Step 12.2.2.9

Multiply $- 1$ by $0$ .

$A (x) = \frac{1}{3 - 1} (4 (\frac{8}{3} + 0))$

Step 12.2.2.10

Add $\frac{8}{3}$ and $0$ .

$A (x) = \frac{1}{3 - 1} (4 (\frac{8}{3}))$

Step 12.2.2.11

Combine $4$ and $\frac{8}{3}$ .

$A (x) = \frac{1}{3 - 1} (\frac{4 \cdot 8}{3})$

Step 12.2.2.12

Multiply $4$ by $8$ .

$A (x) = \frac{1}{3 - 1} (\frac{32}{3})$

Step 13

Subtract $1$ from $3$ .

$A (x) = \frac{1}{2} \cdot \frac{32}{3}$

Step 14

Cancel the common factor of $2$ .

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Step 14.1

Factor $2$ out of $32$ .

$A (x) = \frac{1}{2} \cdot \frac{2 (16)}{3}$

Step 14.2

Cancel the common factor.

$A (x) = \frac{1}{2} \cdot \frac{2 \cdot 16}{3}$

Step 14.3

Rewrite the expression.

$A (x) = \frac{16}{3}$

Step 15