Calculus Examples

f (x) = 3 x - 6

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

(0, 4)

$(0, 4)$

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)

$(- \infty, \infty)$

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