Calculus Examples

$f (x) = 3 x - 6$ , $(0, 4)$

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.

$A (x) = \frac{1}{4 - 0} (\int_{0}^{4} 3 x - 6 d x)$

Step 5

Split the single integral into multiple integrals.

$A (x) = \frac{1}{4 - 0} (\int_{0}^{4} 3 x d x + \int_{0}^{4} - 6 d x)$

Step 6

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

$A (x) = \frac{1}{4 - 0} (3 \int_{0}^{4} x d x + \int_{0}^{4} - 6 d x)$

Step 7

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

$A (x) = \frac{1}{4 - 0} (3 (\frac{1}{2} x^{2}]_{0}^{4}) + \int_{0}^{4} - 6 d x)$

Step 8

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

$A (x) = \frac{1}{4 - 0} (3 (\frac{x^{2}}{2}]_{0}^{4}) + \int_{0}^{4} - 6 d x)$

Step 9

Apply the constant rule.

$A (x) = \frac{1}{4 - 0} (3 (\frac{x^{2}}{2}]_{0}^{4}) + - 6 x]_{0}^{4})$

Step 10

Substitute and simplify.

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Step 10.1

Evaluate $\frac{x^{2}}{2}$ at $4$ and at $0$ .

$A (x) = \frac{1}{4 - 0} (3 ((\frac{4^{2}}{2}) - \frac{0^{2}}{2}) + - 6 x]_{0}^{4})$

Step 10.2

Evaluate $- 6 x$ at $4$ and at $0$ .

$A (x) = \frac{1}{4 - 0} (3 (\frac{4^{2}}{2} - \frac{0^{2}}{2}) - 6 \cdot 4 + 6 \cdot 0)$

Step 10.3

Simplify.

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Step 10.3.1

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

$A (x) = \frac{1}{4 - 0} (3 (\frac{16}{2} - \frac{0^{2}}{2}) - 6 \cdot 4 + 6 \cdot 0)$

Step 10.3.2

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

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Step 10.3.2.1

Factor $2$ out of $16$ .

$A (x) = \frac{1}{4 - 0} (3 (\frac{2 \cdot 8}{2} - \frac{0^{2}}{2}) - 6 \cdot 4 + 6 \cdot 0)$

Step 10.3.2.2

Cancel the common factors.

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Step 10.3.2.2.1

Factor $2$ out of $2$ .

$A (x) = \frac{1}{4 - 0} (3 (\frac{2 \cdot 8}{2 (1)} - \frac{0^{2}}{2}) - 6 \cdot 4 + 6 \cdot 0)$

Step 10.3.2.2.2

Cancel the common factor.

$A (x) = \frac{1}{4 - 0} (3 (\frac{2 \cdot 8}{2 \cdot 1} - \frac{0^{2}}{2}) - 6 \cdot 4 + 6 \cdot 0)$

Step 10.3.2.2.3

Rewrite the expression.

$A (x) = \frac{1}{4 - 0} (3 (\frac{8}{1} - \frac{0^{2}}{2}) - 6 \cdot 4 + 6 \cdot 0)$

Step 10.3.2.2.4

Divide $8$ by $1$ .

$A (x) = \frac{1}{4 - 0} (3 (8 - \frac{0^{2}}{2}) - 6 \cdot 4 + 6 \cdot 0)$

Step 10.3.3

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

$A (x) = \frac{1}{4 - 0} (3 (8 - \frac{0}{2}) - 6 \cdot 4 + 6 \cdot 0)$

Step 10.3.4

Cancel the common factor of $0$ and $2$ .

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Step 10.3.4.1

Factor $2$ out of $0$ .

$A (x) = \frac{1}{4 - 0} (3 (8 - \frac{2 (0)}{2}) - 6 \cdot 4 + 6 \cdot 0)$

Step 10.3.4.2

Cancel the common factors.

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Step 10.3.4.2.1

Factor $2$ out of $2$ .

$A (x) = \frac{1}{4 - 0} (3 (8 - \frac{2 \cdot 0}{2 \cdot 1}) - 6 \cdot 4 + 6 \cdot 0)$

Step 10.3.4.2.2

Cancel the common factor.

$A (x) = \frac{1}{4 - 0} (3 (8 - \frac{2 \cdot 0}{2 \cdot 1}) - 6 \cdot 4 + 6 \cdot 0)$

Step 10.3.4.2.3

Rewrite the expression.

$A (x) = \frac{1}{4 - 0} (3 (8 - \frac{0}{1}) - 6 \cdot 4 + 6 \cdot 0)$

Step 10.3.4.2.4

Divide $0$ by $1$ .

$A (x) = \frac{1}{4 - 0} (3 (8 - 0) - 6 \cdot 4 + 6 \cdot 0)$

Step 10.3.5

Multiply $- 1$ by $0$ .

$A (x) = \frac{1}{4 - 0} (3 (8 + 0) - 6 \cdot 4 + 6 \cdot 0)$

Step 10.3.6

Add $8$ and $0$ .

$A (x) = \frac{1}{4 - 0} (3 \cdot 8 - 6 \cdot 4 + 6 \cdot 0)$

Step 10.3.7

Multiply $3$ by $8$ .

$A (x) = \frac{1}{4 - 0} (24 - 6 \cdot 4 + 6 \cdot 0)$

Step 10.3.8

Multiply $- 6$ by $4$ .

$A (x) = \frac{1}{4 - 0} (24 - 24 + 6 \cdot 0)$

Step 10.3.9

Multiply $6$ by $0$ .

$A (x) = \frac{1}{4 - 0} (24 - 24 + 0)$

Step 10.3.10

Add $- 24$ and $0$ .

$A (x) = \frac{1}{4 - 0} (24 - 24)$

Step 10.3.11

Subtract $24$ from $24$ .

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

Step 11

Simplify the denominator.

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Step 11.1

Multiply $- 1$ by $0$ .

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

Step 11.2

Add $4$ and $0$ .

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

Step 12

Multiply $\frac{1}{4}$ by $0$ .

$A (x) = 0$

Step 13

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