Calculus Examples

$f (x) = 2 x^{2} + 5 x + 3$ , $- 1 < x < 2$

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 $[- 1, 2]$ .

$f (x)$ 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.

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

Apply the constant rule.

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

Step 13

Substitute and simplify.

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Step 13.1

Evaluate $\frac{x^{3}}{3}$ at $2$ and at $- 1$ .

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

Step 13.2

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

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

Step 13.3

Evaluate $3 x$ at $2$ and at $- 1$ .

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

Step 13.4

Simplify.

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Step 13.4.1

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

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

Step 13.4.2

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

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

Step 13.4.3

Move the negative in front of the fraction.

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

Step 13.4.4

Multiply $- 1$ by $- 1$ .

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

Step 13.4.5

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

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

Step 13.4.6

Combine the numerators over the common denominator.

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

Step 13.4.7

Add $8$ and $1$ .

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

Step 13.4.8

Cancel the common factor of $9$ and $3$ .

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Step 13.4.8.1

Factor $3$ out of $9$ .

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

Step 13.4.8.2

Cancel the common factors.

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Step 13.4.8.2.1

Factor $3$ out of $3$ .

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

Step 13.4.8.2.2

Cancel the common factor.

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

Step 13.4.8.2.3

Rewrite the expression.

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

Step 13.4.8.2.4

Divide $3$ by $1$ .

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

Step 13.4.9

Multiply $2$ by $3$ .

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

Step 13.4.10

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

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

Step 13.4.11

Cancel the common factor of $4$ and $2$ .

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Step 13.4.11.1

Factor $2$ out of $4$ .

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

Step 13.4.11.2

Cancel the common factors.

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Step 13.4.11.2.1

Factor $2$ out of $2$ .

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

Step 13.4.11.2.2

Cancel the common factor.

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

Step 13.4.11.2.3

Rewrite the expression.

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

Step 13.4.11.2.4

Divide $2$ by $1$ .

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

Step 13.4.12

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

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

Step 13.4.13

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

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

Step 13.4.14

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

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

Step 13.4.15

Combine the numerators over the common denominator.

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

Step 13.4.16

Simplify the numerator.

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Step 13.4.16.1

Multiply $2$ by $2$ .

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

Step 13.4.16.2

Subtract $1$ from $4$ .

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

Step 13.4.17

Combine $5$ and $\frac{3}{2}$ .

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

Step 13.4.18

Multiply $5$ by $3$ .

$A (x) = \frac{1}{2 + 1} (6 + \frac{15}{2} + 3 \cdot 2 - 3 \cdot - 1)$

Step 13.4.19

To write $6$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$A (x) = \frac{1}{2 + 1} (6 \cdot \frac{2}{2} + \frac{15}{2} + 3 \cdot 2 - 3 \cdot - 1)$

Step 13.4.20

Combine $6$ and $\frac{2}{2}$ .

$A (x) = \frac{1}{2 + 1} (\frac{6 \cdot 2}{2} + \frac{15}{2} + 3 \cdot 2 - 3 \cdot - 1)$

Step 13.4.21

Combine the numerators over the common denominator.

$A (x) = \frac{1}{2 + 1} (\frac{6 \cdot 2 + 15}{2} + 3 \cdot 2 - 3 \cdot - 1)$

Step 13.4.22

Simplify the numerator.

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Step 13.4.22.1

Multiply $6$ by $2$ .

$A (x) = \frac{1}{2 + 1} (\frac{12 + 15}{2} + 3 \cdot 2 - 3 \cdot - 1)$

Step 13.4.22.2

Add $12$ and $15$ .

$A (x) = \frac{1}{2 + 1} (\frac{27}{2} + 3 \cdot 2 - 3 \cdot - 1)$

Step 13.4.23

Multiply $3$ by $2$ .

$A (x) = \frac{1}{2 + 1} (\frac{27}{2} + 6 - 3 \cdot - 1)$

Step 13.4.24

Multiply $- 3$ by $- 1$ .

$A (x) = \frac{1}{2 + 1} (\frac{27}{2} + 6 + 3)$

Step 13.4.25

Add $6$ and $3$ .

$A (x) = \frac{1}{2 + 1} (\frac{27}{2} + 9)$

Step 13.4.26

To write $9$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$A (x) = \frac{1}{2 + 1} (\frac{27}{2} + 9 \cdot \frac{2}{2})$

Step 13.4.27

Combine $9$ and $\frac{2}{2}$ .

$A (x) = \frac{1}{2 + 1} (\frac{27}{2} + \frac{9 \cdot 2}{2})$

Step 13.4.28

Combine the numerators over the common denominator.

$A (x) = \frac{1}{2 + 1} (\frac{27 + 9 \cdot 2}{2})$

Step 13.4.29

Simplify the numerator.

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Step 13.4.29.1

Multiply $9$ by $2$ .

$A (x) = \frac{1}{2 + 1} (\frac{27 + 18}{2})$

Step 13.4.29.2

Add $27$ and $18$ .

$A (x) = \frac{1}{2 + 1} (\frac{45}{2})$

Step 14

Add $2$ and $1$ .

$A (x) = \frac{1}{3} \cdot \frac{45}{2}$

Step 15

Cancel the common factor of $3$ .

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Step 15.1

Factor $3$ out of $45$ .

$A (x) = \frac{1}{3} \cdot \frac{3 (15)}{2}$

Step 15.2

Cancel the common factor.

$A (x) = \frac{1}{3} \cdot \frac{3 \cdot 15}{2}$

Step 15.3

Rewrite the expression.

$A (x) = \frac{15}{2}$

Step 16