Calculus Examples

$f (x) = 2 x^{3} - 6 x^{2} - 48 x + 9$ , $(- 3, 5)$

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 $[- 3, 5]$ .

$f (x)$ 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.

$A (x) = \frac{1}{5 + 3} (\int_{- 3}^{5} 2 x^{3} - 6 x^{2} - 48 x + 9 d x)$

Step 5

Split the single integral into multiple integrals.

$A (x) = \frac{1}{5 + 3} (\int_{- 3}^{5} 2 x^{3} d x + \int_{- 3}^{5} - 6 x^{2} d x + \int_{- 3}^{5} - 48 x d x + \int_{- 3}^{5} 9 d x)$

Step 6

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$A (x) = \frac{1}{5 + 3} (2 \int_{- 3}^{5} x^{3} d x + \int_{- 3}^{5} - 6 x^{2} d x + \int_{- 3}^{5} - 48 x d x + \int_{- 3}^{5} 9 d x)$

Step 7

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$A (x) = \frac{1}{5 + 3} (2 (\frac{1}{4} x^{4}]_{- 3}^{5}) + \int_{- 3}^{5} - 6 x^{2} d x + \int_{- 3}^{5} - 48 x d x + \int_{- 3}^{5} 9 d x)$

Step 8

Combine $\frac{1}{4}$ and $x^{4}$ .

$A (x) = \frac{1}{5 + 3} (2 (\frac{x^{4}}{4}]_{- 3}^{5}) + \int_{- 3}^{5} - 6 x^{2} d x + \int_{- 3}^{5} - 48 x d x + \int_{- 3}^{5} 9 d x)$

Step 9

Since $- 6$ is constant with respect to $x$ , move $- 6$ out of the integral.

$A (x) = \frac{1}{5 + 3} (2 (\frac{x^{4}}{4}]_{- 3}^{5}) - 6 \int_{- 3}^{5} x^{2} d x + \int_{- 3}^{5} - 48 x d x + \int_{- 3}^{5} 9 d x)$

Step 10

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$A (x) = \frac{1}{5 + 3} (2 (\frac{x^{4}}{4}]_{- 3}^{5}) - 6 (\frac{1}{3} x^{3}]_{- 3}^{5}) + \int_{- 3}^{5} - 48 x d x + \int_{- 3}^{5} 9 d x)$

Step 11

Combine $\frac{1}{3}$ and $x^{3}$ .

$A (x) = \frac{1}{5 + 3} (2 (\frac{x^{4}}{4}]_{- 3}^{5}) - 6 (\frac{x^{3}}{3}]_{- 3}^{5}) + \int_{- 3}^{5} - 48 x d x + \int_{- 3}^{5} 9 d x)$

Step 12

Since $- 48$ is constant with respect to $x$ , move $- 48$ out of the integral.

$A (x) = \frac{1}{5 + 3} (2 (\frac{x^{4}}{4}]_{- 3}^{5}) - 6 (\frac{x^{3}}{3}]_{- 3}^{5}) - 48 \int_{- 3}^{5} x d x + \int_{- 3}^{5} 9 d x)$

Step 13

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

$A (x) = \frac{1}{5 + 3} (2 (\frac{x^{4}}{4}]_{- 3}^{5}) - 6 (\frac{x^{3}}{3}]_{- 3}^{5}) - 48 (\frac{1}{2} x^{2}]_{- 3}^{5}) + \int_{- 3}^{5} 9 d x)$

Step 14

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

$A (x) = \frac{1}{5 + 3} (2 (\frac{x^{4}}{4}]_{- 3}^{5}) - 6 (\frac{x^{3}}{3}]_{- 3}^{5}) - 48 (\frac{x^{2}}{2}]_{- 3}^{5}) + \int_{- 3}^{5} 9 d x)$

Step 15

Apply the constant rule.

$A (x) = \frac{1}{5 + 3} (2 (\frac{x^{4}}{4}]_{- 3}^{5}) - 6 (\frac{x^{3}}{3}]_{- 3}^{5}) - 48 (\frac{x^{2}}{2}]_{- 3}^{5}) + 9 x]_{- 3}^{5})$

Step 16

Substitute and simplify.

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Step 16.1

Evaluate $\frac{x^{4}}{4}$ at $5$ and at $- 3$ .

$A (x) = \frac{1}{5 + 3} (2 ((\frac{5^{4}}{4}) - \frac{{(- 3)}^{4}}{4}) - 6 (\frac{x^{3}}{3}]_{- 3}^{5}) - 48 (\frac{x^{2}}{2}]_{- 3}^{5}) + 9 x]_{- 3}^{5})$

Step 16.2

Evaluate $\frac{x^{3}}{3}$ at $5$ and at $- 3$ .

$A (x) = \frac{1}{5 + 3} (2 (\frac{5^{4}}{4} - \frac{{(- 3)}^{4}}{4}) - 6 (\frac{5^{3}}{3} - \frac{{(- 3)}^{3}}{3}) - 48 (\frac{x^{2}}{2}]_{- 3}^{5}) + 9 x]_{- 3}^{5})$

Step 16.3

Evaluate $\frac{x^{2}}{2}$ at $5$ and at $- 3$ .

$A (x) = \frac{1}{5 + 3} (2 (\frac{5^{4}}{4} - \frac{{(- 3)}^{4}}{4}) - 6 (\frac{5^{3}}{3} - \frac{{(- 3)}^{3}}{3}) - 48 ((\frac{5^{2}}{2}) - \frac{{(- 3)}^{2}}{2}) + 9 x]_{- 3}^{5})$

Step 16.4

Evaluate $9 x$ at $5$ and at $- 3$ .

$A (x) = \frac{1}{5 + 3} (2 (\frac{5^{4}}{4} - \frac{{(- 3)}^{4}}{4}) - 6 (\frac{5^{3}}{3} - \frac{{(- 3)}^{3}}{3}) - 48 (\frac{5^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + 9 \cdot 5 - 9 \cdot - 3)$

Step 16.5

Simplify.

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Step 16.5.1

Raise $5$ to the power of $4$ .

$A (x) = \frac{1}{5 + 3} (2 (\frac{625}{4} - \frac{{(- 3)}^{4}}{4}) - 6 (\frac{5^{3}}{3} - \frac{{(- 3)}^{3}}{3}) - 48 (\frac{5^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + 9 \cdot 5 - 9 \cdot - 3)$

Step 16.5.2

Raise $- 3$ to the power of $4$ .

$A (x) = \frac{1}{5 + 3} (2 (\frac{625}{4} - \frac{81}{4}) - 6 (\frac{5^{3}}{3} - \frac{{(- 3)}^{3}}{3}) - 48 (\frac{5^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + 9 \cdot 5 - 9 \cdot - 3)$

Step 16.5.3

Combine the numerators over the common denominator.

$A (x) = \frac{1}{5 + 3} (2 (\frac{625 - 81}{4}) - 6 (\frac{5^{3}}{3} - \frac{{(- 3)}^{3}}{3}) - 48 (\frac{5^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + 9 \cdot 5 - 9 \cdot - 3)$

Step 16.5.4

Subtract $81$ from $625$ .

$A (x) = \frac{1}{5 + 3} (2 (\frac{544}{4}) - 6 (\frac{5^{3}}{3} - \frac{{(- 3)}^{3}}{3}) - 48 (\frac{5^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + 9 \cdot 5 - 9 \cdot - 3)$

Step 16.5.5

Cancel the common factor of $544$ and $4$ .

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Step 16.5.5.1

Factor $4$ out of $544$ .

$A (x) = \frac{1}{5 + 3} (2 (\frac{4 \cdot 136}{4}) - 6 (\frac{5^{3}}{3} - \frac{{(- 3)}^{3}}{3}) - 48 (\frac{5^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + 9 \cdot 5 - 9 \cdot - 3)$

Step 16.5.5.2

Cancel the common factors.

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Step 16.5.5.2.1

Factor $4$ out of $4$ .

$A (x) = \frac{1}{5 + 3} (2 (\frac{4 \cdot 136}{4 (1)}) - 6 (\frac{5^{3}}{3} - \frac{{(- 3)}^{3}}{3}) - 48 (\frac{5^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + 9 \cdot 5 - 9 \cdot - 3)$

Step 16.5.5.2.2

Cancel the common factor.

$A (x) = \frac{1}{5 + 3} (2 (\frac{4 \cdot 136}{4 \cdot 1}) - 6 (\frac{5^{3}}{3} - \frac{{(- 3)}^{3}}{3}) - 48 (\frac{5^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + 9 \cdot 5 - 9 \cdot - 3)$

Step 16.5.5.2.3

Rewrite the expression.

$A (x) = \frac{1}{5 + 3} (2 (\frac{136}{1}) - 6 (\frac{5^{3}}{3} - \frac{{(- 3)}^{3}}{3}) - 48 (\frac{5^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + 9 \cdot 5 - 9 \cdot - 3)$

Step 16.5.5.2.4

Divide $136$ by $1$ .

$A (x) = \frac{1}{5 + 3} (2 \cdot 136 - 6 (\frac{5^{3}}{3} - \frac{{(- 3)}^{3}}{3}) - 48 (\frac{5^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + 9 \cdot 5 - 9 \cdot - 3)$

Step 16.5.6

Multiply $2$ by $136$ .

$A (x) = \frac{1}{5 + 3} (272 - 6 (\frac{5^{3}}{3} - \frac{{(- 3)}^{3}}{3}) - 48 (\frac{5^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + 9 \cdot 5 - 9 \cdot - 3)$

Step 16.5.7

Raise $5$ to the power of $3$ .

$A (x) = \frac{1}{5 + 3} (272 - 6 (\frac{125}{3} - \frac{{(- 3)}^{3}}{3}) - 48 (\frac{5^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + 9 \cdot 5 - 9 \cdot - 3)$

Step 16.5.8

Raise $- 3$ to the power of $3$ .

$A (x) = \frac{1}{5 + 3} (272 - 6 (\frac{125}{3} - \frac{- 27}{3}) - 48 (\frac{5^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + 9 \cdot 5 - 9 \cdot - 3)$

Step 16.5.9

Cancel the common factor of $- 27$ and $3$ .

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Step 16.5.9.1

Factor $3$ out of $- 27$ .

$A (x) = \frac{1}{5 + 3} (272 - 6 (\frac{125}{3} - \frac{3 \cdot - 9}{3}) - 48 (\frac{5^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + 9 \cdot 5 - 9 \cdot - 3)$

Step 16.5.9.2

Cancel the common factors.

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Step 16.5.9.2.1

Factor $3$ out of $3$ .

$A (x) = \frac{1}{5 + 3} (272 - 6 (\frac{125}{3} - \frac{3 \cdot - 9}{3 (1)}) - 48 (\frac{5^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + 9 \cdot 5 - 9 \cdot - 3)$

Step 16.5.9.2.2

Cancel the common factor.

$A (x) = \frac{1}{5 + 3} (272 - 6 (\frac{125}{3} - \frac{3 \cdot - 9}{3 \cdot 1}) - 48 (\frac{5^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + 9 \cdot 5 - 9 \cdot - 3)$

Step 16.5.9.2.3

Rewrite the expression.

$A (x) = \frac{1}{5 + 3} (272 - 6 (\frac{125}{3} - \frac{- 9}{1}) - 48 (\frac{5^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + 9 \cdot 5 - 9 \cdot - 3)$

Step 16.5.9.2.4

Divide $- 9$ by $1$ .

$A (x) = \frac{1}{5 + 3} (272 - 6 (\frac{125}{3} + 9) - 48 (\frac{5^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + 9 \cdot 5 - 9 \cdot - 3)$

Step 16.5.10

Multiply $- 1$ by $- 9$ .

$A (x) = \frac{1}{5 + 3} (272 - 6 (\frac{125}{3} + 9) - 48 (\frac{5^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + 9 \cdot 5 - 9 \cdot - 3)$

Step 16.5.11

To write $9$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$A (x) = \frac{1}{5 + 3} (272 - 6 (\frac{125}{3} + 9 \cdot \frac{3}{3}) - 48 (\frac{5^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + 9 \cdot 5 - 9 \cdot - 3)$

Step 16.5.12

Combine $9$ and $\frac{3}{3}$ .

$A (x) = \frac{1}{5 + 3} (272 - 6 (\frac{125}{3} + \frac{9 \cdot 3}{3}) - 48 (\frac{5^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + 9 \cdot 5 - 9 \cdot - 3)$

Step 16.5.13

Combine the numerators over the common denominator.

$A (x) = \frac{1}{5 + 3} (272 - 6 \frac{125 + 9 \cdot 3}{3} - 48 (\frac{5^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + 9 \cdot 5 - 9 \cdot - 3)$

Step 16.5.14

Simplify the numerator.

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Step 16.5.14.1

Multiply $9$ by $3$ .

$A (x) = \frac{1}{5 + 3} (272 - 6 \frac{125 + 27}{3} - 48 (\frac{5^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + 9 \cdot 5 - 9 \cdot - 3)$

Step 16.5.14.2

Add $125$ and $27$ .

$A (x) = \frac{1}{5 + 3} (272 - 6 (\frac{152}{3}) - 48 (\frac{5^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + 9 \cdot 5 - 9 \cdot - 3)$

Step 16.5.15

Combine $- 6$ and $\frac{152}{3}$ .

$A (x) = \frac{1}{5 + 3} (272 + \frac{- 6 \cdot 152}{3} - 48 (\frac{5^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + 9 \cdot 5 - 9 \cdot - 3)$

Step 16.5.16

Multiply $- 6$ by $152$ .

$A (x) = \frac{1}{5 + 3} (272 + \frac{- 912}{3} - 48 (\frac{5^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + 9 \cdot 5 - 9 \cdot - 3)$

Step 16.5.17

Cancel the common factor of $- 912$ and $3$ .

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Step 16.5.17.1

Factor $3$ out of $- 912$ .

$A (x) = \frac{1}{5 + 3} (272 + \frac{3 \cdot - 304}{3} - 48 (\frac{5^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + 9 \cdot 5 - 9 \cdot - 3)$

Step 16.5.17.2

Cancel the common factors.

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Step 16.5.17.2.1

Factor $3$ out of $3$ .

$A (x) = \frac{1}{5 + 3} (272 + \frac{3 \cdot - 304}{3 (1)} - 48 (\frac{5^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + 9 \cdot 5 - 9 \cdot - 3)$

Step 16.5.17.2.2

Cancel the common factor.

$A (x) = \frac{1}{5 + 3} (272 + \frac{3 \cdot - 304}{3 \cdot 1} - 48 (\frac{5^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + 9 \cdot 5 - 9 \cdot - 3)$

Step 16.5.17.2.3

Rewrite the expression.

$A (x) = \frac{1}{5 + 3} (272 + \frac{- 304}{1} - 48 (\frac{5^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + 9 \cdot 5 - 9 \cdot - 3)$

Step 16.5.17.2.4

Divide $- 304$ by $1$ .

$A (x) = \frac{1}{5 + 3} (272 - 304 - 48 (\frac{5^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + 9 \cdot 5 - 9 \cdot - 3)$

Step 16.5.18

Subtract $304$ from $272$ .

$A (x) = \frac{1}{5 + 3} (- 32 - 48 (\frac{5^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + 9 \cdot 5 - 9 \cdot - 3)$

Step 16.5.19

Raise $5$ to the power of $2$ .

$A (x) = \frac{1}{5 + 3} (- 32 - 48 (\frac{25}{2} - \frac{{(- 3)}^{2}}{2}) + 9 \cdot 5 - 9 \cdot - 3)$

Step 16.5.20

Raise $- 3$ to the power of $2$ .

$A (x) = \frac{1}{5 + 3} (- 32 - 48 (\frac{25}{2} - \frac{9}{2}) + 9 \cdot 5 - 9 \cdot - 3)$

Step 16.5.21

Combine the numerators over the common denominator.

$A (x) = \frac{1}{5 + 3} (- 32 - 48 \frac{25 - 9}{2} + 9 \cdot 5 - 9 \cdot - 3)$

Step 16.5.22

Subtract $9$ from $25$ .

$A (x) = \frac{1}{5 + 3} (- 32 - 48 (\frac{16}{2}) + 9 \cdot 5 - 9 \cdot - 3)$

Step 16.5.23

Cancel the common factor of $16$ and $2$ .

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Step 16.5.23.1

Factor $2$ out of $16$ .

$A (x) = \frac{1}{5 + 3} (- 32 - 48 \frac{2 \cdot 8}{2} + 9 \cdot 5 - 9 \cdot - 3)$

Step 16.5.23.2

Cancel the common factors.

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Step 16.5.23.2.1

Factor $2$ out of $2$ .

$A (x) = \frac{1}{5 + 3} (- 32 - 48 \frac{2 \cdot 8}{2 (1)} + 9 \cdot 5 - 9 \cdot - 3)$

Step 16.5.23.2.2

Cancel the common factor.

$A (x) = \frac{1}{5 + 3} (- 32 - 48 \frac{2 \cdot 8}{2 \cdot 1} + 9 \cdot 5 - 9 \cdot - 3)$

Step 16.5.23.2.3

Rewrite the expression.

$A (x) = \frac{1}{5 + 3} (- 32 - 48 (\frac{8}{1}) + 9 \cdot 5 - 9 \cdot - 3)$

Step 16.5.23.2.4

Divide $8$ by $1$ .

$A (x) = \frac{1}{5 + 3} (- 32 - 48 \cdot 8 + 9 \cdot 5 - 9 \cdot - 3)$

Step 16.5.24

Multiply $- 48$ by $8$ .

$A (x) = \frac{1}{5 + 3} (- 32 - 384 + 9 \cdot 5 - 9 \cdot - 3)$

Step 16.5.25

Subtract $384$ from $- 32$ .

$A (x) = \frac{1}{5 + 3} (- 416 + 9 \cdot 5 - 9 \cdot - 3)$

Step 16.5.26

Multiply $9$ by $5$ .

$A (x) = \frac{1}{5 + 3} (- 416 + 45 - 9 \cdot - 3)$

Step 16.5.27

Multiply $- 9$ by $- 3$ .

$A (x) = \frac{1}{5 + 3} (- 416 + 45 + 27)$

Step 16.5.28

Add $45$ and $27$ .

$A (x) = \frac{1}{5 + 3} (- 416 + 72)$

Step 16.5.29

Add $- 416$ and $72$ .

$A (x) = \frac{1}{5 + 3} (- 344)$

Step 17

Add $5$ and $3$ .

$A (x) = \frac{1}{8} \cdot - 344$

Step 18

Cancel the common factor of $8$ .

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Step 18.1

Factor $8$ out of $- 344$ .

$A (x) = \frac{1}{8} \cdot (8 (- 43))$

Step 18.2

Cancel the common factor.

$A (x) = \frac{1}{8} \cdot (8 \cdot - 43)$

Step 18.3

Rewrite the expression.

$A (x) = - 43$

Step 19