Calculus Examples

$f (x) = 2 x^{3} + 12 x^{2} - 7$ , $(0, 4)$

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 $[0, 4]$ .

$f (x)$ 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.

$A (x) = \frac{1}{4 - 0} (\int_{0}^{4} 2 x^{3} + 12 x^{2} - 7 d x)$

Step 5

Split the single integral into multiple integrals.

$A (x) = \frac{1}{4 - 0} (\int_{0}^{4} 2 x^{3} d x + \int_{0}^{4} 12 x^{2} d x + \int_{0}^{4} - 7 d x)$

Step 6

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

$A (x) = \frac{1}{4 - 0} (2 \int_{0}^{4} x^{3} d x + \int_{0}^{4} 12 x^{2} d x + \int_{0}^{4} - 7 d x)$

Step 7

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

$A (x) = \frac{1}{4 - 0} (2 (\frac{1}{4} x^{4}]_{0}^{4}) + \int_{0}^{4} 12 x^{2} d x + \int_{0}^{4} - 7 d x)$

Step 8

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

$A (x) = \frac{1}{4 - 0} (2 (\frac{x^{4}}{4}]_{0}^{4}) + \int_{0}^{4} 12 x^{2} d x + \int_{0}^{4} - 7 d x)$

Step 9

Since $12$ is constant with respect to $x$ , move $12$ out of the integral.

$A (x) = \frac{1}{4 - 0} (2 (\frac{x^{4}}{4}]_{0}^{4}) + 12 \int_{0}^{4} x^{2} d x + \int_{0}^{4} - 7 d x)$

Step 10

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

$A (x) = \frac{1}{4 - 0} (2 (\frac{x^{4}}{4}]_{0}^{4}) + 12 (\frac{1}{3} x^{3}]_{0}^{4}) + \int_{0}^{4} - 7 d x)$

Step 11

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

$A (x) = \frac{1}{4 - 0} (2 (\frac{x^{4}}{4}]_{0}^{4}) + 12 (\frac{x^{3}}{3}]_{0}^{4}) + \int_{0}^{4} - 7 d x)$

Step 12

Apply the constant rule.

$A (x) = \frac{1}{4 - 0} (2 (\frac{x^{4}}{4}]_{0}^{4}) + 12 (\frac{x^{3}}{3}]_{0}^{4}) + - 7 x]_{0}^{4})$

Step 13

Substitute and simplify.

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Step 13.1

Evaluate $\frac{x^{4}}{4}$ at $4$ and at $0$ .

$A (x) = \frac{1}{4 - 0} (2 ((\frac{4^{4}}{4}) - \frac{0^{4}}{4}) + 12 (\frac{x^{3}}{3}]_{0}^{4}) + - 7 x]_{0}^{4})$

Step 13.2

Evaluate $\frac{x^{3}}{3}$ at $4$ and at $0$ .

$A (x) = \frac{1}{4 - 0} (2 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 12 (\frac{4^{3}}{3} - \frac{0^{3}}{3}) + - 7 x]_{0}^{4})$

Step 13.3

Evaluate $- 7 x$ at $4$ and at $0$ .

$A (x) = \frac{1}{4 - 0} (2 (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 12 (\frac{4^{3}}{3} - \frac{0^{3}}{3}) - 7 \cdot 4 + 7 \cdot 0)$

Step 13.4

Simplify.

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Step 13.4.1

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

$A (x) = \frac{1}{4 - 0} (2 (\frac{256}{4} - \frac{0^{4}}{4}) + 12 (\frac{4^{3}}{3} - \frac{0^{3}}{3}) - 7 \cdot 4 + 7 \cdot 0)$

Step 13.4.2

Cancel the common factor of $256$ and $4$ .

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Step 13.4.2.1

Factor $4$ out of $256$ .

$A (x) = \frac{1}{4 - 0} (2 (\frac{4 \cdot 64}{4} - \frac{0^{4}}{4}) + 12 (\frac{4^{3}}{3} - \frac{0^{3}}{3}) - 7 \cdot 4 + 7 \cdot 0)$

Step 13.4.2.2

Cancel the common factors.

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Step 13.4.2.2.1

Factor $4$ out of $4$ .

$A (x) = \frac{1}{4 - 0} (2 (\frac{4 \cdot 64}{4 (1)} - \frac{0^{4}}{4}) + 12 (\frac{4^{3}}{3} - \frac{0^{3}}{3}) - 7 \cdot 4 + 7 \cdot 0)$

Step 13.4.2.2.2

Cancel the common factor.

$A (x) = \frac{1}{4 - 0} (2 (\frac{4 \cdot 64}{4 \cdot 1} - \frac{0^{4}}{4}) + 12 (\frac{4^{3}}{3} - \frac{0^{3}}{3}) - 7 \cdot 4 + 7 \cdot 0)$

Step 13.4.2.2.3

Rewrite the expression.

$A (x) = \frac{1}{4 - 0} (2 (\frac{64}{1} - \frac{0^{4}}{4}) + 12 (\frac{4^{3}}{3} - \frac{0^{3}}{3}) - 7 \cdot 4 + 7 \cdot 0)$

Step 13.4.2.2.4

Divide $64$ by $1$ .

$A (x) = \frac{1}{4 - 0} (2 (64 - \frac{0^{4}}{4}) + 12 (\frac{4^{3}}{3} - \frac{0^{3}}{3}) - 7 \cdot 4 + 7 \cdot 0)$

Step 13.4.3

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

$A (x) = \frac{1}{4 - 0} (2 (64 - \frac{0}{4}) + 12 (\frac{4^{3}}{3} - \frac{0^{3}}{3}) - 7 \cdot 4 + 7 \cdot 0)$

Step 13.4.4

Cancel the common factor of $0$ and $4$ .

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Step 13.4.4.1

Factor $4$ out of $0$ .

$A (x) = \frac{1}{4 - 0} (2 (64 - \frac{4 (0)}{4}) + 12 (\frac{4^{3}}{3} - \frac{0^{3}}{3}) - 7 \cdot 4 + 7 \cdot 0)$

Step 13.4.4.2

Cancel the common factors.

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Step 13.4.4.2.1

Factor $4$ out of $4$ .

$A (x) = \frac{1}{4 - 0} (2 (64 - \frac{4 \cdot 0}{4 \cdot 1}) + 12 (\frac{4^{3}}{3} - \frac{0^{3}}{3}) - 7 \cdot 4 + 7 \cdot 0)$

Step 13.4.4.2.2

Cancel the common factor.

$A (x) = \frac{1}{4 - 0} (2 (64 - \frac{4 \cdot 0}{4 \cdot 1}) + 12 (\frac{4^{3}}{3} - \frac{0^{3}}{3}) - 7 \cdot 4 + 7 \cdot 0)$

Step 13.4.4.2.3

Rewrite the expression.

$A (x) = \frac{1}{4 - 0} (2 (64 - \frac{0}{1}) + 12 (\frac{4^{3}}{3} - \frac{0^{3}}{3}) - 7 \cdot 4 + 7 \cdot 0)$

Step 13.4.4.2.4

Divide $0$ by $1$ .

$A (x) = \frac{1}{4 - 0} (2 (64 - 0) + 12 (\frac{4^{3}}{3} - \frac{0^{3}}{3}) - 7 \cdot 4 + 7 \cdot 0)$

Step 13.4.5

Multiply $- 1$ by $0$ .

$A (x) = \frac{1}{4 - 0} (2 (64 + 0) + 12 (\frac{4^{3}}{3} - \frac{0^{3}}{3}) - 7 \cdot 4 + 7 \cdot 0)$

Step 13.4.6

Add $64$ and $0$ .

$A (x) = \frac{1}{4 - 0} (2 \cdot 64 + 12 (\frac{4^{3}}{3} - \frac{0^{3}}{3}) - 7 \cdot 4 + 7 \cdot 0)$

Step 13.4.7

Multiply $2$ by $64$ .

$A (x) = \frac{1}{4 - 0} (128 + 12 (\frac{4^{3}}{3} - \frac{0^{3}}{3}) - 7 \cdot 4 + 7 \cdot 0)$

Step 13.4.8

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

$A (x) = \frac{1}{4 - 0} (128 + 12 (\frac{64}{3} - \frac{0^{3}}{3}) - 7 \cdot 4 + 7 \cdot 0)$

Step 13.4.9

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

$A (x) = \frac{1}{4 - 0} (128 + 12 (\frac{64}{3} - \frac{0}{3}) - 7 \cdot 4 + 7 \cdot 0)$

Step 13.4.10

Cancel the common factor of $0$ and $3$ .

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Step 13.4.10.1

Factor $3$ out of $0$ .

$A (x) = \frac{1}{4 - 0} (128 + 12 (\frac{64}{3} - \frac{3 (0)}{3}) - 7 \cdot 4 + 7 \cdot 0)$

Step 13.4.10.2

Cancel the common factors.

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Step 13.4.10.2.1

Factor $3$ out of $3$ .

$A (x) = \frac{1}{4 - 0} (128 + 12 (\frac{64}{3} - \frac{3 \cdot 0}{3 \cdot 1}) - 7 \cdot 4 + 7 \cdot 0)$

Step 13.4.10.2.2

Cancel the common factor.

$A (x) = \frac{1}{4 - 0} (128 + 12 (\frac{64}{3} - \frac{3 \cdot 0}{3 \cdot 1}) - 7 \cdot 4 + 7 \cdot 0)$

Step 13.4.10.2.3

Rewrite the expression.

$A (x) = \frac{1}{4 - 0} (128 + 12 (\frac{64}{3} - \frac{0}{1}) - 7 \cdot 4 + 7 \cdot 0)$

Step 13.4.10.2.4

Divide $0$ by $1$ .

$A (x) = \frac{1}{4 - 0} (128 + 12 (\frac{64}{3} - 0) - 7 \cdot 4 + 7 \cdot 0)$

Step 13.4.11

Multiply $- 1$ by $0$ .

$A (x) = \frac{1}{4 - 0} (128 + 12 (\frac{64}{3} + 0) - 7 \cdot 4 + 7 \cdot 0)$

Step 13.4.12

Add $\frac{64}{3}$ and $0$ .

$A (x) = \frac{1}{4 - 0} (128 + 12 (\frac{64}{3}) - 7 \cdot 4 + 7 \cdot 0)$

Step 13.4.13

Combine $12$ and $\frac{64}{3}$ .

$A (x) = \frac{1}{4 - 0} (128 + \frac{12 \cdot 64}{3} - 7 \cdot 4 + 7 \cdot 0)$

Step 13.4.14

Multiply $12$ by $64$ .

$A (x) = \frac{1}{4 - 0} (128 + \frac{768}{3} - 7 \cdot 4 + 7 \cdot 0)$

Step 13.4.15

Cancel the common factor of $768$ and $3$ .

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Step 13.4.15.1

Factor $3$ out of $768$ .

$A (x) = \frac{1}{4 - 0} (128 + \frac{3 \cdot 256}{3} - 7 \cdot 4 + 7 \cdot 0)$

Step 13.4.15.2

Cancel the common factors.

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Step 13.4.15.2.1

Factor $3$ out of $3$ .

$A (x) = \frac{1}{4 - 0} (128 + \frac{3 \cdot 256}{3 (1)} - 7 \cdot 4 + 7 \cdot 0)$

Step 13.4.15.2.2

Cancel the common factor.

$A (x) = \frac{1}{4 - 0} (128 + \frac{3 \cdot 256}{3 \cdot 1} - 7 \cdot 4 + 7 \cdot 0)$

Step 13.4.15.2.3

Rewrite the expression.

$A (x) = \frac{1}{4 - 0} (128 + \frac{256}{1} - 7 \cdot 4 + 7 \cdot 0)$

Step 13.4.15.2.4

Divide $256$ by $1$ .

$A (x) = \frac{1}{4 - 0} (128 + 256 - 7 \cdot 4 + 7 \cdot 0)$

Step 13.4.16

Add $128$ and $256$ .

$A (x) = \frac{1}{4 - 0} (384 - 7 \cdot 4 + 7 \cdot 0)$

Step 13.4.17

Multiply $- 7$ by $4$ .

$A (x) = \frac{1}{4 - 0} (384 - 28 + 7 \cdot 0)$

Step 13.4.18

Multiply $7$ by $0$ .

$A (x) = \frac{1}{4 - 0} (384 - 28 + 0)$

Step 13.4.19

Add $- 28$ and $0$ .

$A (x) = \frac{1}{4 - 0} (384 - 28)$

Step 13.4.20

Subtract $28$ from $384$ .

$A (x) = \frac{1}{4 - 0} (356)$

Step 14

Simplify the denominator.

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Step 14.1

Multiply $- 1$ by $0$ .

$A (x) = \frac{1}{4 + 0} \cdot 356$

Step 14.2

Add $4$ and $0$ .

$A (x) = \frac{1}{4} \cdot 356$

Step 15

Cancel the common factor of $4$ .

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Step 15.1

Factor $4$ out of $356$ .

$A (x) = \frac{1}{4} \cdot (4 (89))$

Step 15.2

Cancel the common factor.

$A (x) = \frac{1}{4} \cdot (4 \cdot 89)$

Step 15.3

Rewrite the expression.

$A (x) = 89$

Step 16