Calculus Examples

$f (x) = 8 x - 6$ , $[0, 3]$

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.

$(- \infty, \infty)$

${x | x \in ℝ}$

$f (x)$ is continuous on $[0, 3]$ .

$f (x)$ is continuous

The average value of function $f$ over the interval $[a, b]$ is defined as $A (x) = \frac{1}{b - a} \int_{a}^{b} f (x) d x$ .

$A (x) = \frac{1}{b - a} \int_{a}^{b} f (x) d x$

Substitute the actual values into the formula for the average value of a function.

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

Since integration is linear, the integral of $8 x - 6$ with respect to $x$ is $\int_{0}^{3} 8 x d x + \int_{0}^{3} - 6 d x$ .

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

Since $8$ is constant with respect to $x$ , the integral of $8 x$ with respect to $x$ is $8 \int_{0}^{3} x d x$ .

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

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

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

Combine fractions.

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Write $x^{2}$ as a fraction with denominator $1$ .

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

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

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

Since $- 6$ is constant with respect to $x$ , the integral of $- 6$ with respect to $x$ is $- 6 x$ .

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

Simplify the answer.

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Evaluate $\frac{x^{2}}{2}$ at $3$ and at $0$ .

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

Evaluate $- 6 x$ at $3$ and at $0$ .

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

Raise $3$ to the power of $2$ .

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

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

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

Reduce the expression by cancelling the common factors.

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Factor $2$ out of $0$ .

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

Cancel the common factors.

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Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $0$ by $1$ .

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

Multiply $- 1$ by $0$ .

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

Add $\frac{9}{2}$ and $0$ .

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

Write $8$ as a fraction with denominator $1$ .

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

Multiply $\frac{8}{1}$ and $\frac{9}{2}$ .

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

Multiply $8$ by $9$ .

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

Reduce the expression by cancelling the common factors.

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Factor $2$ out of $72$ .

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

Cancel the common factors.

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Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $36$ by $1$ .

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

Multiply $- 6$ by $3$ .

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

Multiply $6$ by $0$ .

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

Add $- 18$ and $0$ .

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

Subtract $18$ from $36$ .

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

Simplify the denominator.

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Multiply $- 1$ by $0$ .

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

Add $3$ and $0$ .

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

Cancel the common factor of $3$ .

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Write $18$ as a fraction with denominator $1$ .

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

Factor out the greatest common factor $3$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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Multiply $\frac{1}{1}$ and $\frac{6}{1}$ .

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

Divide $6$ by $1$ .

$A (x) = 6$

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