Calculus Examples

$f (x) = x^{2} + x - 7$ ; $[0, 14]$

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 $[0, 14]$ .

$f (x)$ 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.

$A (x) = \frac{1}{14 - 0} (\int_{0}^{14} x^{2} + x - 7 d x)$

Step 5

Split the single integral into multiple integrals.

$A (x) = \frac{1}{14 - 0} (\int_{0}^{14} x^{2} d x + \int_{0}^{14} x d x + \int_{0}^{14} - 7 d x)$

Step 6

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

$A (x) = \frac{1}{14 - 0} (\frac{1}{3} x^{3}]_{0}^{14} + \int_{0}^{14} x d x + \int_{0}^{14} - 7 d x)$

Step 7

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

$A (x) = \frac{1}{14 - 0} (\frac{1}{3} x^{3}]_{0}^{14} + \frac{1}{2} x^{2}]_{0}^{14} + \int_{0}^{14} - 7 d x)$

Step 8

Apply the constant rule.

$A (x) = \frac{1}{14 - 0} (\frac{1}{3} x^{3}]_{0}^{14} + \frac{1}{2} x^{2}]_{0}^{14} + - 7 x]_{0}^{14})$

Step 9

Simplify the answer.

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Step 9.1

Simplify.

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Step 9.1.1

Combine $\frac{1}{3} x^{3}]_{0}^{14}$ and $\frac{1}{2} x^{2}]_{0}^{14}$ .

$A (x) = \frac{1}{14 - 0} (\frac{1}{3} x^{3} + \frac{1}{2} x^{2}]_{0}^{14} + - 7 x]_{0}^{14})$

Step 9.1.2

Combine $\frac{1}{3} x^{3} + \frac{1}{2} x^{2}]_{0}^{14}$ and $- 7 x]_{0}^{14}$ .

$A (x) = \frac{1}{14 - 0} (\frac{1}{3} x^{3} + \frac{1}{2} x^{2} - 7 x]_{0}^{14})$

Step 9.2

Substitute and simplify.

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Step 9.2.1

Evaluate $\frac{1}{3} x^{3} + \frac{1}{2} x^{2} - 7 x$ at $14$ and at $0$ .

$A (x) = \frac{1}{14 - 0} ((\frac{1}{3} \cdot 14^{3} + \frac{1}{2} \cdot 14^{2} - 7 \cdot 14) - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} - 7 \cdot 0))$

Step 9.2.2

Simplify.

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Step 9.2.2.1

Raise $14$ to the power of $3$ .

$A (x) = \frac{1}{14 - 0} (\frac{1}{3} \cdot 2744 + \frac{1}{2} \cdot 14^{2} - 7 \cdot 14 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} - 7 \cdot 0))$

Step 9.2.2.2

Combine $\frac{1}{3}$ and $2744$ .

$A (x) = \frac{1}{14 - 0} (\frac{2744}{3} + \frac{1}{2} \cdot 14^{2} - 7 \cdot 14 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} - 7 \cdot 0))$

Step 9.2.2.3

Raise $14$ to the power of $2$ .

$A (x) = \frac{1}{14 - 0} (\frac{2744}{3} + \frac{1}{2} \cdot 196 - 7 \cdot 14 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} - 7 \cdot 0))$

Step 9.2.2.4

Combine $\frac{1}{2}$ and $196$ .

$A (x) = \frac{1}{14 - 0} (\frac{2744}{3} + \frac{196}{2} - 7 \cdot 14 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} - 7 \cdot 0))$

Step 9.2.2.5

Cancel the common factor of $196$ and $2$ .

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Step 9.2.2.5.1

Factor $2$ out of $196$ .

$A (x) = \frac{1}{14 - 0} (\frac{2744}{3} + \frac{2 \cdot 98}{2} - 7 \cdot 14 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} - 7 \cdot 0))$

Step 9.2.2.5.2

Cancel the common factors.

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Step 9.2.2.5.2.1

Factor $2$ out of $2$ .

$A (x) = \frac{1}{14 - 0} (\frac{2744}{3} + \frac{2 \cdot 98}{2 (1)} - 7 \cdot 14 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} - 7 \cdot 0))$

Step 9.2.2.5.2.2

Cancel the common factor.

$A (x) = \frac{1}{14 - 0} (\frac{2744}{3} + \frac{2 \cdot 98}{2 \cdot 1} - 7 \cdot 14 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} - 7 \cdot 0))$

Step 9.2.2.5.2.3

Rewrite the expression.

$A (x) = \frac{1}{14 - 0} (\frac{2744}{3} + \frac{98}{1} - 7 \cdot 14 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} - 7 \cdot 0))$

Step 9.2.2.5.2.4

Divide $98$ by $1$ .

$A (x) = \frac{1}{14 - 0} (\frac{2744}{3} + 98 - 7 \cdot 14 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} - 7 \cdot 0))$

Step 9.2.2.6

To write $98$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$A (x) = \frac{1}{14 - 0} (\frac{2744}{3} + 98 \cdot \frac{3}{3} - 7 \cdot 14 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} - 7 \cdot 0))$

Step 9.2.2.7

Combine $98$ and $\frac{3}{3}$ .

$A (x) = \frac{1}{14 - 0} (\frac{2744}{3} + \frac{98 \cdot 3}{3} - 7 \cdot 14 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} - 7 \cdot 0))$

Step 9.2.2.8

Combine the numerators over the common denominator.

$A (x) = \frac{1}{14 - 0} (\frac{2744 + 98 \cdot 3}{3} - 7 \cdot 14 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} - 7 \cdot 0))$

Step 9.2.2.9

Simplify the numerator.

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Step 9.2.2.9.1

Multiply $98$ by $3$ .

$A (x) = \frac{1}{14 - 0} (\frac{2744 + 294}{3} - 7 \cdot 14 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} - 7 \cdot 0))$

Step 9.2.2.9.2

Add $2744$ and $294$ .

$A (x) = \frac{1}{14 - 0} (\frac{3038}{3} - 7 \cdot 14 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} - 7 \cdot 0))$

Step 9.2.2.10

Multiply $- 7$ by $14$ .

$A (x) = \frac{1}{14 - 0} (\frac{3038}{3} - 98 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} - 7 \cdot 0))$

Step 9.2.2.11

To write $- 98$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$A (x) = \frac{1}{14 - 0} (\frac{3038}{3} - 98 \cdot \frac{3}{3} - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} - 7 \cdot 0))$

Step 9.2.2.12

Combine $- 98$ and $\frac{3}{3}$ .

$A (x) = \frac{1}{14 - 0} (\frac{3038}{3} + \frac{- 98 \cdot 3}{3} - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} - 7 \cdot 0))$

Step 9.2.2.13

Combine the numerators over the common denominator.

$A (x) = \frac{1}{14 - 0} (\frac{3038 - 98 \cdot 3}{3} - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} - 7 \cdot 0))$

Step 9.2.2.14

Simplify the numerator.

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Step 9.2.2.14.1

Multiply $- 98$ by $3$ .

$A (x) = \frac{1}{14 - 0} (\frac{3038 - 294}{3} - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} - 7 \cdot 0))$

Step 9.2.2.14.2

Subtract $294$ from $3038$ .

$A (x) = \frac{1}{14 - 0} (\frac{2744}{3} - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} - 7 \cdot 0))$

Step 9.2.2.15

Raising $0$ to any positive power yields $0$ .

$A (x) = \frac{1}{14 - 0} (\frac{2744}{3} - (\frac{1}{3} \cdot 0 + \frac{1}{2} \cdot 0^{2} - 7 \cdot 0))$

Step 9.2.2.16

Multiply $\frac{1}{3}$ by $0$ .

$A (x) = \frac{1}{14 - 0} (\frac{2744}{3} - (0 + \frac{1}{2} \cdot 0^{2} - 7 \cdot 0))$

Step 9.2.2.17

Raising $0$ to any positive power yields $0$ .

$A (x) = \frac{1}{14 - 0} (\frac{2744}{3} - (0 + \frac{1}{2} \cdot 0 - 7 \cdot 0))$

Step 9.2.2.18

Multiply $\frac{1}{2}$ by $0$ .

$A (x) = \frac{1}{14 - 0} (\frac{2744}{3} - (0 + 0 - 7 \cdot 0))$

Step 9.2.2.19

Add $0$ and $0$ .

$A (x) = \frac{1}{14 - 0} (\frac{2744}{3} - (0 - 7 \cdot 0))$

Step 9.2.2.20

Multiply $- 7$ by $0$ .

$A (x) = \frac{1}{14 - 0} (\frac{2744}{3} - (0 + 0))$

Step 9.2.2.21

Add $0$ and $0$ .

$A (x) = \frac{1}{14 - 0} (\frac{2744}{3} - 0)$

Step 9.2.2.22

Multiply $- 1$ by $0$ .

$A (x) = \frac{1}{14 - 0} (\frac{2744}{3} + 0)$

Step 9.2.2.23

Add $\frac{2744}{3}$ and $0$ .

$A (x) = \frac{1}{14 - 0} (\frac{2744}{3})$

Step 10

Simplify the denominator.

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Step 10.1

Multiply $- 1$ by $0$ .

$A (x) = \frac{1}{14 + 0} \cdot \frac{2744}{3}$

Step 10.2

Add $14$ and $0$ .

$A (x) = \frac{1}{14} \cdot \frac{2744}{3}$

Step 11

Cancel the common factor of $14$ .

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Step 11.1

Factor $14$ out of $2744$ .

$A (x) = \frac{1}{14} \cdot \frac{14 (196)}{3}$

Step 11.2

Cancel the common factor.

$A (x) = \frac{1}{14} \cdot \frac{14 \cdot 196}{3}$

Step 11.3

Rewrite the expression.

$A (x) = \frac{196}{3}$

Step 12