Calculus Examples

$f (x) = \frac{1}{{(x - 6)}^{2}}$ , $[0, 5]$

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 $[0, 5]$ or not, find the domain of $f (x) = \frac{1}{{(x - 6)}^{2}}$ .

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Step 1.1

Set the denominator in $\frac{1}{{(x - 6)}^{2}}$ equal to $0$ to find where the expression is undefined.

${(x - 6)}^{2} = 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Set the $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 1.2.2

Add $6$ to both sides of the equation.

$x = 6$

Step 1.3

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

Interval Notation:

$(- \infty, 6) \cup (6, \infty)$

Set-Builder Notation:

${x | x \neq 6}$

Interval Notation:

$(- \infty, 6) \cup (6, \infty)$

Set-Builder Notation:

${x | x \neq 6}$

Step 2

$f (x)$ is continuous on $[0, 5]$ .

$f (x)$ 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.

$A (x) = \frac{1}{5 - 0} (\int_{0}^{5} \frac{1}{{(x - 6)}^{2}} d x)$

Step 5

Let $u = x - 6$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 5.1

Let $u = x - 6$ . Find $\frac{d u}{d x}$ .

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Step 5.1.1

Differentiate $x - 6$ .

$\frac{d}{d x} [x - 6]$

Step 5.1.2

By the Sum Rule, the derivative of $x - 6$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 6]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 6]$

Step 5.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d x} [- 6]$

Step 5.1.4

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$1 + 0$

Step 5.1.5

Add $1$ and $0$ .

$1$

Step 5.2

Substitute the lower limit in for $x$ in $u = x - 6$ .

$u_{lower} = 0 - 6$

Step 5.3

Subtract $6$ from $0$ .

$u_{lower} = - 6$

Step 5.4

Substitute the upper limit in for $x$ in $u = x - 6$ .

$u_{upper} = 5 - 6$

Step 5.5

Subtract $6$ from $5$ .

$u_{upper} = - 1$

Step 5.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = - 6$

$u_{upper} = - 1$

Step 5.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$A (x) = \frac{1}{5 - 0} (\int_{- 6}^{- 1} \frac{1}{u^{2}} d u)$

Step 6

Apply basic rules of exponents.

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Step 6.1

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

$A (x) = \frac{1}{5 - 0} (\int_{- 6}^{- 1} {(u^{2})}^{- 1} d u)$

Step 6.2

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

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Step 6.2.1

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

$A (x) = \frac{1}{5 - 0} (\int_{- 6}^{- 1} u^{2 \cdot - 1} d u)$

Step 6.2.2

Multiply $2$ by $- 1$ .

$A (x) = \frac{1}{5 - 0} (\int_{- 6}^{- 1} u^{- 2} d u)$

Step 7

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

$A (x) = \frac{1}{5 - 0} (- u^{- 1}]_{- 6}^{- 1})$

Step 8

Substitute and simplify.

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Step 8.1

Evaluate $- u^{- 1}$ at $- 1$ and at $- 6$ .

$A (x) = \frac{1}{5 - 0} ((- {(- 1)}^{- 1}) + {(- 6)}^{- 1})$

Step 8.2

Simplify.

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Step 8.2.1

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

$A (x) = \frac{1}{5 - 0} (- \frac{1}{- 1} + {(- 6)}^{- 1})$

Step 8.2.2

Move the negative one from the denominator of $\frac{1}{- 1}$ .

$A (x) = \frac{1}{5 - 0} (- (- 1 \cdot 1) + {(- 6)}^{- 1})$

Step 8.2.3

Multiply $- 1$ by $1$ .

$A (x) = \frac{1}{5 - 0} (1 + {(- 6)}^{- 1})$

Step 8.2.4

Multiply $- 1$ by $- 1$ .

$A (x) = \frac{1}{5 - 0} (1 + {(- 6)}^{- 1})$

Step 8.2.5

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

$A (x) = \frac{1}{5 - 0} (1 + \frac{1}{- 6})$

Step 8.2.6

Move the negative in front of the fraction.

$A (x) = \frac{1}{5 - 0} (1 - \frac{1}{6})$

Step 8.2.7

Write $1$ as a fraction with a common denominator.

$A (x) = \frac{1}{5 - 0} (\frac{6}{6} - \frac{1}{6})$

Step 8.2.8

Combine the numerators over the common denominator.

$A (x) = \frac{1}{5 - 0} (\frac{6 - 1}{6})$

Step 8.2.9

Subtract $1$ from $6$ .

$A (x) = \frac{1}{5 - 0} (\frac{5}{6})$

Step 9

Simplify the denominator.

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Step 9.1

Multiply $- 1$ by $0$ .

$A (x) = \frac{1}{5 + 0} \cdot \frac{5}{6}$

Step 9.2

Add $5$ and $0$ .

$A (x) = \frac{1}{5} \cdot \frac{5}{6}$

Step 10

Cancel the common factor of $5$ .

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Step 10.1

Cancel the common factor.

$A (x) = \frac{1}{5} \cdot \frac{5}{6}$

Step 10.2

Rewrite the expression.

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

Step 11