Calculus Examples

$f (x) = 29 + 8 {(2.71828182)}^{- 0.04 x}$ , $[0, 5]$

Step 1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

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.

$A (x) = \frac{1}{5 - 0} (\int_{0}^{5} 29 + 8 {(2.71828182)}^{- 0.04 x} d x)$

Step 5

Split the single integral into multiple integrals.

$A (x) = \frac{1}{5 - 0} (\int_{0}^{5} 29 d x + \int_{0}^{5} 8 \cdot {2.71828182}^{- 0.04 x} d x)$

Step 6

Apply the constant rule.

$A (x) = \frac{1}{5 - 0} (29 x]_{0}^{5} + \int_{0}^{5} 8 \cdot {2.71828182}^{- 0.04 x} d x)$

Step 7

Since $8$ is constant with respect to $x$ , move $8$ out of the integral.

$A (x) = \frac{1}{5 - 0} (29 x]_{0}^{5} + 8 \int_{0}^{5} {2.71828182}^{- 0.04 x} d x)$

Step 8

Let $u = - 0.04 x$ . Then $d u = - 0.04 d x$ , so $\frac{1}{- 0.04} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 8.1

Let $u = - 0.04 x$ . Find $\frac{d u}{d x}$ .

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Step 8.1.1

Differentiate $- 0.04 x$ .

$\frac{d}{d x} [- 0.04 x]$

Step 8.1.2

Since $- 0.04$ is constant with respect to $x$ , the derivative of $- 0.04 x$ with respect to $x$ is $- 0.04 \frac{d}{d x} [x]$ .

$- 0.04 \frac{d}{d x} [x]$

Step 8.1.3

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

$- 0.04 \cdot 1$

Step 8.1.4

Multiply $- 0.04$ by $1$ .

$- 0.04$

Step 8.2

Substitute the lower limit in for $x$ in $u = - 0.04 x$ .

$u_{lower} = - 0.04 \cdot 0$

Step 8.3

Multiply $- 0.04$ by $0$ .

$u_{lower} = 0$

Step 8.4

Substitute the upper limit in for $x$ in $u = - 0.04 x$ .

$u_{upper} = - 0.04 \cdot 5$

Step 8.5

Multiply $- 0.04$ by $5$ .

$u_{upper} = - 0.2$

Step 8.6

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

$u_{lower} = 0$

$u_{upper} = - 0.2$

Step 8.7

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

$A (x) = \frac{1}{5 - 0} (29 x]_{0}^{5} + 8 \int_{0}^{- 0.2} {2.71828182}^{u} \cdot \frac{1}{- 0.04} d u)$

Step 9

Simplify.

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Step 9.1

Move the negative in front of the fraction.

$A (x) = \frac{1}{5 - 0} (29 x]_{0}^{5} + 8 \int_{0}^{- 0.2} {2.71828182}^{u} \cdot (- \frac{1}{0.04}) d u)$

Step 9.2

Combine ${2.71828182}^{u}$ and $\frac{1}{0.04}$ .

$A (x) = \frac{1}{5 - 0} (29 x]_{0}^{5} + 8 \int_{0}^{- 0.2} - \frac{{2.71828182}^{u}}{0.04} d u)$

Step 10

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$A (x) = \frac{1}{5 - 0} (29 x]_{0}^{5} + 8 (- \int_{0}^{- 0.2} \frac{{2.71828182}^{u}}{0.04} d u))$

Step 11

Multiply $- 1$ by $8$ .

$A (x) = \frac{1}{5 - 0} (29 x]_{0}^{5} - 8 \int_{0}^{- 0.2} \frac{{2.71828182}^{u}}{0.04} d u)$

Step 12

Since $\frac{1}{0.04}$ is constant with respect to $u$ , move $\frac{1}{0.04}$ out of the integral.

$A (x) = \frac{1}{5 - 0} (29 x]_{0}^{5} - 8 (\frac{1}{0.04} \cdot \int_{0}^{- 0.2} {2.71828182}^{u} d u))$

Step 13

Simplify.

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Step 13.1

Combine $\frac{1}{0.04}$ and $- 8$ .

$A (x) = \frac{1}{5 - 0} (29 x]_{0}^{5} + \frac{- 8}{0.04} \cdot \int_{0}^{- 0.2} {2.71828182}^{u} d u)$

Step 13.2

Move the negative in front of the fraction.

$A (x) = \frac{1}{5 - 0} (29 x]_{0}^{5} - \frac{8}{0.04} \cdot \int_{0}^{- 0.2} {2.71828182}^{u} d u)$

Step 14

The integral of ${2.71828182}^{u}$ with respect to $u$ is $\frac{{2.71828182}^{u}}{\ln (2.71828182)}$ .

$A (x) = \frac{1}{5 - 0} (29 x]_{0}^{5} - \frac{8}{0.04} \cdot \frac{{2.71828182}^{u}}{\ln (2.71828182)}]_{0}^{- 0.2})$

Step 15

Simplify.

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Step 15.1

Combine $\frac{{2.71828182}^{u}}{\ln (2.71828182)}]_{0}^{- 0.2}$ and $\frac{8}{0.04}$ .

$A (x) = \frac{1}{5 - 0} (29 x]_{0}^{5} - \frac{\frac{{2.71828182}^{u}}{\ln (2.71828182)}]_{0}^{- 0.2} \cdot 8}{0.04})$

Step 15.2

Move $8$ to the left of $\frac{{2.71828182}^{u}}{\ln (2.71828182)}]_{0}^{- 0.2}$ .

$A (x) = \frac{1}{5 - 0} (29 x]_{0}^{5} - \frac{8 (\frac{{2.71828182}^{u}}{\ln (2.71828182)}]_{0}^{- 0.2})}{0.04})$

Step 16

Substitute and simplify.

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Step 16.1

Evaluate $29 x$ at $5$ and at $0$ .

$A (x) = \frac{1}{5 - 0} ((29 \cdot 5) - 29 \cdot 0 - \frac{8 (\frac{{2.71828182}^{u}}{\ln (2.71828182)}]_{0}^{- 0.2})}{0.04})$

Step 16.2

Evaluate $\frac{{2.71828182}^{u}}{\ln (2.71828182)}$ at $- 0.2$ and at $0$ .

$A (x) = \frac{1}{5 - 0} ((29 \cdot 5) - 29 \cdot 0 - \frac{8 ((\frac{{2.71828182}^{- 0.2}}{\ln (2.71828182)}) - \frac{{2.71828182}^{0}}{\ln (2.71828182)})}{0.04})$

Step 16.3

Simplify.

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Step 16.3.1

Multiply $29$ by $5$ .

$A (x) = \frac{1}{5 - 0} (145 - 29 \cdot 0 - \frac{8 ((\frac{{2.71828182}^{- 0.2}}{\ln (2.71828182)}) - \frac{{2.71828182}^{0}}{\ln (2.71828182)})}{0.04})$

Step 16.3.2

Multiply $- 29$ by $0$ .

$A (x) = \frac{1}{5 - 0} (145 + 0 - \frac{8 ((\frac{{2.71828182}^{- 0.2}}{\ln (2.71828182)}) - \frac{{2.71828182}^{0}}{\ln (2.71828182)})}{0.04})$

Step 16.3.3

Add $145$ and $0$ .

$A (x) = \frac{1}{5 - 0} (145 - \frac{8 ((\frac{{2.71828182}^{- 0.2}}{\ln (2.71828182)}) - \frac{{2.71828182}^{0}}{\ln (2.71828182)})}{0.04})$

Step 16.3.4

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

$A (x) = \frac{1}{5 - 0} (145 - \frac{8 (\frac{\frac{1}{{2.71828182}^{0.2}}}{\ln (2.71828182)} - \frac{{2.71828182}^{0}}{\ln (2.71828182)})}{0.04})$

Step 16.3.5

Raise $2.71828182$ to the power of $0.2$ .

$A (x) = \frac{1}{5 - 0} (145 - \frac{8 (\frac{\frac{1}{1.22140275}}{\ln (2.71828182)} - \frac{{2.71828182}^{0}}{\ln (2.71828182)})}{0.04})$

Step 16.3.6

Rewrite $\frac{\frac{1}{1.22140275}}{\ln (2.71828182)}$ as a product.

$A (x) = \frac{1}{5 - 0} (145 - \frac{8 (\frac{1}{1.22140275} \cdot \frac{1}{\ln (2.71828182)} - \frac{{2.71828182}^{0}}{\ln (2.71828182)})}{0.04})$

Step 16.3.7

Multiply $\frac{1}{1.22140275}$ by $\frac{1}{\ln (2.71828182)}$ .

$A (x) = \frac{1}{5 - 0} (145 - \frac{8 (\frac{1}{1.22140275 \ln (2.71828182)} - \frac{{2.71828182}^{0}}{\ln (2.71828182)})}{0.04})$

Step 16.3.8

Multiply $1.22140275$ by $\ln (2.71828182)$ .

$A (x) = \frac{1}{5 - 0} (145 - \frac{8 (\frac{1}{1.22140275} - \frac{{2.71828182}^{0}}{\ln (2.71828182)})}{0.04})$

Step 16.3.9

Anything raised to $0$ is $1$ .

$A (x) = \frac{1}{5 - 0} (145 - \frac{8 (\frac{1}{1.22140275} - \frac{1}{\ln (2.71828182)})}{0.04})$

Step 17

Simplify.

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Step 17.1

Simplify each term.

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Step 17.1.1

Simplify the numerator.

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Step 17.1.1.1

Divide $1$ by $1.22140275$ .

$A (x) = \frac{1}{5 - 0} (145 - \frac{8 (0.81873075 - \frac{1}{\ln (2.71828182)})}{0.04})$

Step 17.1.1.2

To write $0.81873075$ as a fraction with a common denominator, multiply by $\frac{\ln (2.71828182)}{\ln (2.71828182)}$ .

$A (x) = \frac{1}{5 - 0} (145 - \frac{8 (\frac{0.81873075 \ln (2.71828182)}{\ln (2.71828182)} - \frac{1}{\ln (2.71828182)})}{0.04})$

Step 17.1.1.3

Combine the numerators over the common denominator.

$A (x) = \frac{1}{5 - 0} (145 - \frac{8 (\frac{0.81873075 \ln (2.71828182) - 1}{\ln (2.71828182)})}{0.04})$

Step 17.1.1.4

Simplify the numerator.

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Step 17.1.1.4.1

Multiply $0.81873075$ by $\ln (2.71828182)$ .

$A (x) = \frac{1}{5 - 0} (145 - \frac{8 (\frac{0.81873075 - 1}{\ln (2.71828182)})}{0.04})$

Step 17.1.1.4.2

Subtract $1$ from $0.81873075$ .

$A (x) = \frac{1}{5 - 0} (145 - \frac{8 (\frac{- 0.18126924}{\ln (2.71828182)})}{0.04})$

Step 17.1.1.5

Move the negative in front of the fraction.

$A (x) = \frac{1}{5 - 0} (145 - \frac{8 (- \frac{0.18126924}{\ln (2.71828182)})}{0.04})$

Step 17.1.1.6

Replace $e$ with an approximation.

$A (x) = \frac{1}{5 - 0} (145 - \frac{8 (- \frac{0.18126924}{\log_{2.71828182} (2.71828182)})}{0.04})$

Step 17.1.1.7

Log base $2.71828182$ of $2.71828182$ is approximately $0.99999999$ .

$A (x) = \frac{1}{5 - 0} (145 - \frac{8 (- \frac{0.18126924}{0.99999999})}{0.04})$

Step 17.1.1.8

Divide $0.18126924$ by $0.99999999$ .

$A (x) = \frac{1}{5 - 0} (145 - \frac{8 (- 1 \cdot 0.18126924)}{0.04})$

Step 17.1.1.9

Multiply $- 1$ by $0.18126924$ .

$A (x) = \frac{1}{5 - 0} (145 - \frac{8 \cdot - 0.18126924}{0.04})$

Step 17.1.2

Multiply $8$ by $- 0.18126924$ .

$A (x) = \frac{1}{5 - 0} (145 - \frac{- 1.45015397}{0.04})$

Step 17.1.3

Divide $- 1.45015397$ by $0.04$ .

$A (x) = \frac{1}{5 - 0} (145 + 36.25384939)$

Step 17.1.4

Multiply $- 1$ by $- 36.25384939$ .

$A (x) = \frac{1}{5 - 0} (145 + 36.25384939)$

Step 17.2

Add $145$ and $36.25384939$ .

$A (x) = \frac{1}{5 - 0} (181.25384939)$

Step 18

Simplify the denominator.

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Step 18.1

Multiply $- 1$ by $0$ .

$A (x) = \frac{1}{5 + 0} \cdot 181.25384939$

Step 18.2

Add $5$ and $0$ .

$A (x) = \frac{1}{5} \cdot 181.25384939$

Step 19

Combine fractions.

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Step 19.1

Combine $\frac{1}{5}$ and $181.25384939$ .

$A (x) = \frac{181.25384939}{5}$

Step 19.2

Divide $181.25384939$ by $5$ .

$A (x) = 36.25076987$

Step 20