Calculus Examples

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

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 $[- 1, 4]$ .

$f (x)$ 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.

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

Simplify the answer.

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Step 11.1

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

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

Step 11.2

Substitute and simplify.

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Step 11.2.1

Evaluate $\frac{x^{3}}{3}$ at $4$ and at $- 1$ .

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

Step 11.2.2

Evaluate $\frac{x^{2}}{2}$ at $4$ and at $- 1$ .

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

Step 11.2.3

Simplify.

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Step 11.2.3.1

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

$A (x) = \frac{1}{4 + 1} (3 (\frac{64}{3} - \frac{{(- 1)}^{3}}{3}) - (\frac{4^{2}}{2} - \frac{{(- 1)}^{2}}{2}))$

Step 11.2.3.2

Raise $- 1$ to the power of $3$ .

$A (x) = \frac{1}{4 + 1} (3 (\frac{64}{3} - \frac{- 1}{3}) - (\frac{4^{2}}{2} - \frac{{(- 1)}^{2}}{2}))$

Step 11.2.3.3

Move the negative in front of the fraction.

$A (x) = \frac{1}{4 + 1} (3 (\frac{64}{3} + \frac{1}{3}) - (\frac{4^{2}}{2} - \frac{{(- 1)}^{2}}{2}))$

Step 11.2.3.4

Multiply $- 1$ by $- 1$ .

$A (x) = \frac{1}{4 + 1} (3 (\frac{64}{3} + 1 (\frac{1}{3})) - (\frac{4^{2}}{2} - \frac{{(- 1)}^{2}}{2}))$

Step 11.2.3.5

Multiply $\frac{1}{3}$ by $1$ .

$A (x) = \frac{1}{4 + 1} (3 (\frac{64}{3} + \frac{1}{3}) - (\frac{4^{2}}{2} - \frac{{(- 1)}^{2}}{2}))$

Step 11.2.3.6

Combine the numerators over the common denominator.

$A (x) = \frac{1}{4 + 1} (3 (\frac{64 + 1}{3}) - (\frac{4^{2}}{2} - \frac{{(- 1)}^{2}}{2}))$

Step 11.2.3.7

Add $64$ and $1$ .

$A (x) = \frac{1}{4 + 1} (3 (\frac{65}{3}) - (\frac{4^{2}}{2} - \frac{{(- 1)}^{2}}{2}))$

Step 11.2.3.8

Combine $3$ and $\frac{65}{3}$ .

$A (x) = \frac{1}{4 + 1} (\frac{3 \cdot 65}{3} - (\frac{4^{2}}{2} - \frac{{(- 1)}^{2}}{2}))$

Step 11.2.3.9

Multiply $3$ by $65$ .

$A (x) = \frac{1}{4 + 1} (\frac{195}{3} - (\frac{4^{2}}{2} - \frac{{(- 1)}^{2}}{2}))$

Step 11.2.3.10

Cancel the common factor of $195$ and $3$ .

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Step 11.2.3.10.1

Factor $3$ out of $195$ .

$A (x) = \frac{1}{4 + 1} (\frac{3 \cdot 65}{3} - (\frac{4^{2}}{2} - \frac{{(- 1)}^{2}}{2}))$

Step 11.2.3.10.2

Cancel the common factors.

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Step 11.2.3.10.2.1

Factor $3$ out of $3$ .

$A (x) = \frac{1}{4 + 1} (\frac{3 \cdot 65}{3 (1)} - (\frac{4^{2}}{2} - \frac{{(- 1)}^{2}}{2}))$

Step 11.2.3.10.2.2

Cancel the common factor.

$A (x) = \frac{1}{4 + 1} (\frac{3 \cdot 65}{3 \cdot 1} - (\frac{4^{2}}{2} - \frac{{(- 1)}^{2}}{2}))$

Step 11.2.3.10.2.3

Rewrite the expression.

$A (x) = \frac{1}{4 + 1} (\frac{65}{1} - (\frac{4^{2}}{2} - \frac{{(- 1)}^{2}}{2}))$

Step 11.2.3.10.2.4

Divide $65$ by $1$ .

$A (x) = \frac{1}{4 + 1} (65 - (\frac{4^{2}}{2} - \frac{{(- 1)}^{2}}{2}))$

Step 11.2.3.11

Raise $4$ to the power of $2$ .

$A (x) = \frac{1}{4 + 1} (65 - (\frac{16}{2} - \frac{{(- 1)}^{2}}{2}))$

Step 11.2.3.12

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

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Step 11.2.3.12.1

Factor $2$ out of $16$ .

$A (x) = \frac{1}{4 + 1} (65 - (\frac{2 \cdot 8}{2} - \frac{{(- 1)}^{2}}{2}))$

Step 11.2.3.12.2

Cancel the common factors.

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Step 11.2.3.12.2.1

Factor $2$ out of $2$ .

$A (x) = \frac{1}{4 + 1} (65 - (\frac{2 \cdot 8}{2 (1)} - \frac{{(- 1)}^{2}}{2}))$

Step 11.2.3.12.2.2

Cancel the common factor.

$A (x) = \frac{1}{4 + 1} (65 - (\frac{2 \cdot 8}{2 \cdot 1} - \frac{{(- 1)}^{2}}{2}))$

Step 11.2.3.12.2.3

Rewrite the expression.

$A (x) = \frac{1}{4 + 1} (65 - (\frac{8}{1} - \frac{{(- 1)}^{2}}{2}))$

Step 11.2.3.12.2.4

Divide $8$ by $1$ .

$A (x) = \frac{1}{4 + 1} (65 - (8 - \frac{{(- 1)}^{2}}{2}))$

Step 11.2.3.13

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

$A (x) = \frac{1}{4 + 1} (65 - (8 - \frac{1}{2}))$

Step 11.2.3.14

To write $8$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$A (x) = \frac{1}{4 + 1} (65 - (8 \cdot \frac{2}{2} - \frac{1}{2}))$

Step 11.2.3.15

Combine $8$ and $\frac{2}{2}$ .

$A (x) = \frac{1}{4 + 1} (65 - (\frac{8 \cdot 2}{2} - \frac{1}{2}))$

Step 11.2.3.16

Combine the numerators over the common denominator.

$A (x) = \frac{1}{4 + 1} (65 - \frac{8 \cdot 2 - 1}{2})$

Step 11.2.3.17

Simplify the numerator.

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Step 11.2.3.17.1

Multiply $8$ by $2$ .

$A (x) = \frac{1}{4 + 1} (65 - \frac{16 - 1}{2})$

Step 11.2.3.17.2

Subtract $1$ from $16$ .

$A (x) = \frac{1}{4 + 1} (65 - \frac{15}{2})$

Step 11.2.3.18

To write $65$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$A (x) = \frac{1}{4 + 1} (65 \cdot \frac{2}{2} - \frac{15}{2})$

Step 11.2.3.19

Combine $65$ and $\frac{2}{2}$ .

$A (x) = \frac{1}{4 + 1} (\frac{65 \cdot 2}{2} - \frac{15}{2})$

Step 11.2.3.20

Combine the numerators over the common denominator.

$A (x) = \frac{1}{4 + 1} (\frac{65 \cdot 2 - 15}{2})$

Step 11.2.3.21

Simplify the numerator.

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Step 11.2.3.21.1

Multiply $65$ by $2$ .

$A (x) = \frac{1}{4 + 1} (\frac{130 - 15}{2})$

Step 11.2.3.21.2

Subtract $15$ from $130$ .

$A (x) = \frac{1}{4 + 1} (\frac{115}{2})$

Step 12

Add $4$ and $1$ .

$A (x) = \frac{1}{5} \cdot \frac{115}{2}$

Step 13

Cancel the common factor of $5$ .

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Step 13.1

Factor $5$ out of $115$ .

$A (x) = \frac{1}{5} \cdot \frac{5 (23)}{2}$

Step 13.2

Cancel the common factor.

$A (x) = \frac{1}{5} \cdot \frac{5 \cdot 23}{2}$

Step 13.3

Rewrite the expression.

$A (x) = \frac{23}{2}$

Step 14