Calculus Examples

$f (x) = 3 - 5 x$ , $(- 1, 8)$

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 $[- 1, 8]$ .

$f (x)$ 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.

$A (x) = \frac{1}{8 + 1} (\int_{- 1}^{8} 3 - 5 x d x)$

Step 5

Split the single integral into multiple integrals.

$A (x) = \frac{1}{8 + 1} (\int_{- 1}^{8} 3 d x + \int_{- 1}^{8} - 5 x d x)$

Step 6

Apply the constant rule.

$A (x) = \frac{1}{8 + 1} (3 x]_{- 1}^{8} + \int_{- 1}^{8} - 5 x d x)$

Step 7

Since $- 5$ is constant with respect to $x$ , move $- 5$ out of the integral.

$A (x) = \frac{1}{8 + 1} (3 x]_{- 1}^{8} - 5 \int_{- 1}^{8} x d x)$

Step 8

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

$A (x) = \frac{1}{8 + 1} (3 x]_{- 1}^{8} - 5 (\frac{1}{2} x^{2}]_{- 1}^{8}))$

Step 9

Simplify the answer.

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Step 9.1

Combine $\frac{1}{2}$ and $x^{2}$ .

$A (x) = \frac{1}{8 + 1} (3 x]_{- 1}^{8} - 5 (\frac{x^{2}}{2}]_{- 1}^{8}))$

Step 9.2

Substitute and simplify.

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Step 9.2.1

Evaluate $3 x$ at $8$ and at $- 1$ .

$A (x) = \frac{1}{8 + 1} ((3 \cdot 8) - 3 \cdot - 1 - 5 (\frac{x^{2}}{2}]_{- 1}^{8}))$

Step 9.2.2

Evaluate $\frac{x^{2}}{2}$ at $8$ and at $- 1$ .

$A (x) = \frac{1}{8 + 1} (3 \cdot 8 - 3 \cdot - 1 - 5 (\frac{8^{2}}{2} - \frac{{(- 1)}^{2}}{2}))$

Step 9.2.3

Simplify.

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Step 9.2.3.1

Multiply $3$ by $8$ .

$A (x) = \frac{1}{8 + 1} (24 - 3 \cdot - 1 - 5 (\frac{8^{2}}{2} - \frac{{(- 1)}^{2}}{2}))$

Step 9.2.3.2

Multiply $- 3$ by $- 1$ .

$A (x) = \frac{1}{8 + 1} (24 + 3 - 5 (\frac{8^{2}}{2} - \frac{{(- 1)}^{2}}{2}))$

Step 9.2.3.3

Add $24$ and $3$ .

$A (x) = \frac{1}{8 + 1} (27 - 5 (\frac{8^{2}}{2} - \frac{{(- 1)}^{2}}{2}))$

Step 9.2.3.4

Raise $8$ to the power of $2$ .

$A (x) = \frac{1}{8 + 1} (27 - 5 (\frac{64}{2} - \frac{{(- 1)}^{2}}{2}))$

Step 9.2.3.5

Cancel the common factor of $64$ and $2$ .

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Step 9.2.3.5.1

Factor $2$ out of $64$ .

$A (x) = \frac{1}{8 + 1} (27 - 5 (\frac{2 \cdot 32}{2} - \frac{{(- 1)}^{2}}{2}))$

Step 9.2.3.5.2

Cancel the common factors.

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Step 9.2.3.5.2.1

Factor $2$ out of $2$ .

$A (x) = \frac{1}{8 + 1} (27 - 5 (\frac{2 \cdot 32}{2 (1)} - \frac{{(- 1)}^{2}}{2}))$

Step 9.2.3.5.2.2

Cancel the common factor.

$A (x) = \frac{1}{8 + 1} (27 - 5 (\frac{2 \cdot 32}{2 \cdot 1} - \frac{{(- 1)}^{2}}{2}))$

Step 9.2.3.5.2.3

Rewrite the expression.

$A (x) = \frac{1}{8 + 1} (27 - 5 (\frac{32}{1} - \frac{{(- 1)}^{2}}{2}))$

Step 9.2.3.5.2.4

Divide $32$ by $1$ .

$A (x) = \frac{1}{8 + 1} (27 - 5 (32 - \frac{{(- 1)}^{2}}{2}))$

Step 9.2.3.6

Raise $- 1$ to the power of $2$ .

$A (x) = \frac{1}{8 + 1} (27 - 5 (32 - \frac{1}{2}))$

Step 9.2.3.7

To write $32$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$A (x) = \frac{1}{8 + 1} (27 - 5 (32 \cdot \frac{2}{2} - \frac{1}{2}))$

Step 9.2.3.8

Combine $32$ and $\frac{2}{2}$ .

$A (x) = \frac{1}{8 + 1} (27 - 5 (\frac{32 \cdot 2}{2} - \frac{1}{2}))$

Step 9.2.3.9

Combine the numerators over the common denominator.

$A (x) = \frac{1}{8 + 1} (27 - 5 \frac{32 \cdot 2 - 1}{2})$

Step 9.2.3.10

Simplify the numerator.

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Step 9.2.3.10.1

Multiply $32$ by $2$ .

$A (x) = \frac{1}{8 + 1} (27 - 5 \frac{64 - 1}{2})$

Step 9.2.3.10.2

Subtract $1$ from $64$ .

$A (x) = \frac{1}{8 + 1} (27 - 5 (\frac{63}{2}))$

Step 9.2.3.11

Combine $- 5$ and $\frac{63}{2}$ .

$A (x) = \frac{1}{8 + 1} (27 + \frac{- 5 \cdot 63}{2})$

Step 9.2.3.12

Multiply $- 5$ by $63$ .

$A (x) = \frac{1}{8 + 1} (27 + \frac{- 315}{2})$

Step 9.2.3.13

Move the negative in front of the fraction.

$A (x) = \frac{1}{8 + 1} (27 - \frac{315}{2})$

Step 9.2.3.14

To write $27$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$A (x) = \frac{1}{8 + 1} (27 \cdot \frac{2}{2} - \frac{315}{2})$

Step 9.2.3.15

Combine $27$ and $\frac{2}{2}$ .

$A (x) = \frac{1}{8 + 1} (\frac{27 \cdot 2}{2} - \frac{315}{2})$

Step 9.2.3.16

Combine the numerators over the common denominator.

$A (x) = \frac{1}{8 + 1} (\frac{27 \cdot 2 - 315}{2})$

Step 9.2.3.17

Simplify the numerator.

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Step 9.2.3.17.1

Multiply $27$ by $2$ .

$A (x) = \frac{1}{8 + 1} (\frac{54 - 315}{2})$

Step 9.2.3.17.2

Subtract $315$ from $54$ .

$A (x) = \frac{1}{8 + 1} (\frac{- 261}{2})$

Step 9.2.3.18

Move the negative in front of the fraction.

$A (x) = \frac{1}{8 + 1} (- \frac{261}{2})$

Step 10

Add $8$ and $1$ .

$A (x) = \frac{1}{9} \cdot (- \frac{261}{2})$

Step 11

Cancel the common factor of $9$ .

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Step 11.1

Move the leading negative in $- \frac{261}{2}$ into the numerator.

$A (x) = \frac{1}{9} \cdot \frac{- 261}{2}$

Step 11.2

Factor $9$ out of $- 261$ .

$A (x) = \frac{1}{9} \cdot \frac{9 (- 29)}{2}$

Step 11.3

Cancel the common factor.

$A (x) = \frac{1}{9} \cdot \frac{9 \cdot - 29}{2}$

Step 11.4

Rewrite the expression.

$A (x) = \frac{- 29}{2}$

Step 12

Move the negative in front of the fraction.

$A (x) = - \frac{29}{2}$

Step 13