Calculus Examples

f (x) = 8 x - 6

$f (x) = 8 x - 6$ ,

[0, 3]

$[0, 3]$

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

$[0, 3]$ .

$f (x)$ 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.

$A (x) = \frac{1}{3 - 0} (\int_{0}^{3} 8 x - 6 d x)$

Step 5

Split the single integral into multiple integrals.

$A (x) = \frac{1}{3 - 0} (\int_{0}^{3} 8 x d x + \int_{0}^{3} - 6 d x)$

Step 6

Since

$8$ is constant with respect to

$x$ , move

$8$ out of the integral.

$A (x) = \frac{1}{3 - 0} (8 \int_{0}^{3} x d x + \int_{0}^{3} - 6 d x)$

Step 7

By the Power Rule, the integral of

$x$ with respect to

$x$ is

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

$A (x) = \frac{1}{3 - 0} (8 (\frac{1}{2} x^{2}]_{0}^{3}) + \int_{0}^{3} - 6 d x)$

Step 8

Combine

$\frac{1}{2}$ and

$x^{2}$ .

$A (x) = \frac{1}{3 - 0} (8 (\frac{x^{2}}{2}]_{0}^{3}) + \int_{0}^{3} - 6 d x)$

Step 9

Apply the constant rule.

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

Step 10

Substitute and simplify.

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Step 10.1

Evaluate

$\frac{x^{2}}{2}$ at

$3$ and at

$0$ .

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

Step 10.2

Evaluate

$- 6 x$ at

$3$ and at

$0$ .

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

Step 10.3

Simplify.

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Step 10.3.1

Raise

$3$ to the power of

$2$ .

$A (x) = \frac{1}{3 - 0} (8 (\frac{9}{2} - \frac{0^{2}}{2}) - 6 \cdot 3 + 6 \cdot 0)$

Step 10.3.2

Raising

$0$ to any positive power yields

$0$ .

$A (x) = \frac{1}{3 - 0} (8 (\frac{9}{2} - \frac{0}{2}) - 6 \cdot 3 + 6 \cdot 0)$

Step 10.3.3

Cancel the common factor of

$0$ and

$2$ .

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Step 10.3.3.1

Factor

$2$ out of

$0$ .

$A (x) = \frac{1}{3 - 0} (8 (\frac{9}{2} - \frac{2 (0)}{2}) - 6 \cdot 3 + 6 \cdot 0)$

Step 10.3.3.2

Cancel the common factors.

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Step 10.3.3.2.1

Factor

$2$ out of

$2$ .

$A (x) = \frac{1}{3 - 0} (8 (\frac{9}{2} - \frac{2 \cdot 0}{2 \cdot 1}) - 6 \cdot 3 + 6 \cdot 0)$

Step 10.3.3.2.2

Cancel the common factor.

$A (x) = \frac{1}{3 - 0} (8 (\frac{9}{2} - \frac{2 \cdot 0}{2 \cdot 1}) - 6 \cdot 3 + 6 \cdot 0)$

Step 10.3.3.2.3

Rewrite the expression.

$A (x) = \frac{1}{3 - 0} (8 (\frac{9}{2} - \frac{0}{1}) - 6 \cdot 3 + 6 \cdot 0)$

Step 10.3.3.2.4

Divide

$0$ by

$1$ .

$A (x) = \frac{1}{3 - 0} (8 (\frac{9}{2} - 0) - 6 \cdot 3 + 6 \cdot 0)$

Step 10.3.4

Multiply

$- 1$ by

$0$ .

$A (x) = \frac{1}{3 - 0} (8 (\frac{9}{2} + 0) - 6 \cdot 3 + 6 \cdot 0)$

Step 10.3.5

Add

$\frac{9}{2}$ and

$0$ .

$A (x) = \frac{1}{3 - 0} (8 (\frac{9}{2}) - 6 \cdot 3 + 6 \cdot 0)$

Step 10.3.6

Combine

$8$ and

$\frac{9}{2}$ .

$A (x) = \frac{1}{3 - 0} (\frac{8 \cdot 9}{2} - 6 \cdot 3 + 6 \cdot 0)$

Step 10.3.7

Multiply

$8$ by

$9$ .

$A (x) = \frac{1}{3 - 0} (\frac{72}{2} - 6 \cdot 3 + 6 \cdot 0)$

Step 10.3.8

Cancel the common factor of

$72$ and

$2$ .

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Step 10.3.8.1

Factor

$2$ out of

$72$ .

$A (x) = \frac{1}{3 - 0} (\frac{2 \cdot 36}{2} - 6 \cdot 3 + 6 \cdot 0)$

Step 10.3.8.2

Cancel the common factors.

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Step 10.3.8.2.1

Factor

$2$ out of

$2$ .

$A (x) = \frac{1}{3 - 0} (\frac{2 \cdot 36}{2 (1)} - 6 \cdot 3 + 6 \cdot 0)$

Step 10.3.8.2.2

Cancel the common factor.

$A (x) = \frac{1}{3 - 0} (\frac{2 \cdot 36}{2 \cdot 1} - 6 \cdot 3 + 6 \cdot 0)$

Step 10.3.8.2.3

Rewrite the expression.

$A (x) = \frac{1}{3 - 0} (\frac{36}{1} - 6 \cdot 3 + 6 \cdot 0)$

Step 10.3.8.2.4

Divide

$36$ by

$1$ .

$A (x) = \frac{1}{3 - 0} (36 - 6 \cdot 3 + 6 \cdot 0)$

Step 10.3.9

Multiply

$- 6$ by

$3$ .

$A (x) = \frac{1}{3 - 0} (36 - 18 + 6 \cdot 0)$

Step 10.3.10

Multiply

$6$ by

$0$ .

$A (x) = \frac{1}{3 - 0} (36 - 18 + 0)$

Step 10.3.11

Add

$- 18$ and

$0$ .

$A (x) = \frac{1}{3 - 0} (36 - 18)$

Step 10.3.12

Subtract

$18$ from

$36$ .

$A (x) = \frac{1}{3 - 0} (18)$

Step 11

Simplify the denominator.

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Step 11.1

Multiply

$- 1$ by

$0$ .

$A (x) = \frac{1}{3 + 0} \cdot 18$

Step 11.2

Add

$3$ and

$0$ .

$A (x) = \frac{1}{3} \cdot 18$

Step 12

Cancel the common factor of

$3$ .

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Step 12.1

Factor

$3$ out of

$18$ .

$A (x) = \frac{1}{3} \cdot (3 (6))$

Step 12.2

Cancel the common factor.

$A (x) = \frac{1}{3} \cdot (3 \cdot 6)$

Step 12.3

Rewrite the expression.

$A (x) = 6$

Step 13

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