Calculus Examples

$f (x) = 2 x$ , $(0, 6)$

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:

${x | x \in ℝ}$

$f (x)$ is continuous on $[0, 6]$ .

$f (x)$ 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.

$A (x) = \frac{1}{6 - 0} (\int_{0}^{6} 2 x d x)$

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

$A (x) = \frac{1}{6 - 0} (2 \int_{0}^{6} x d x)$

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

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

Simplify the answer.

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

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

Substitute and simplify.

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Evaluate $\frac{x^{2}}{2}$ at $6$ and at $0$ .

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

Simplify.

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Raise $6$ to the power of $2$ .

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

Cancel the common factor of $36$ and $2$ .

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Factor $2$ out of $36$ .

$A (x) = \frac{1}{6 - 0} (2 (\frac{2 \cdot 18}{2} - \frac{0^{2}}{2}))$

Cancel the common factors.

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Factor $2$ out of $2$ .

$A (x) = \frac{1}{6 - 0} (2 (\frac{2 \cdot 18}{2 (1)} - \frac{0^{2}}{2}))$

Cancel the common factor.

$A (x) = \frac{1}{6 - 0} (2 (\frac{2 \cdot 18}{2 \cdot 1} - \frac{0^{2}}{2}))$

Rewrite the expression.

$A (x) = \frac{1}{6 - 0} (2 (\frac{18}{1} - \frac{0^{2}}{2}))$

Divide $18$ by $1$ .

$A (x) = \frac{1}{6 - 0} (2 (18 - \frac{0^{2}}{2}))$

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

$A (x) = \frac{1}{6 - 0} (2 (18 - \frac{0}{2}))$

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

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Factor $2$ out of $0$ .

$A (x) = \frac{1}{6 - 0} (2 (18 - \frac{2 (0)}{2}))$

Cancel the common factors.

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Factor $2$ out of $2$ .

$A (x) = \frac{1}{6 - 0} (2 (18 - \frac{2 \cdot 0}{2 \cdot 1}))$

Cancel the common factor.

$A (x) = \frac{1}{6 - 0} (2 (18 - \frac{2 \cdot 0}{2 \cdot 1}))$

Rewrite the expression.

$A (x) = \frac{1}{6 - 0} (2 (18 - \frac{0}{1}))$

Divide $0$ by $1$ .

$A (x) = \frac{1}{6 - 0} (2 (18 - 0))$

Multiply $- 1$ by $0$ .

$A (x) = \frac{1}{6 - 0} (2 (18 + 0))$

Add $18$ and $0$ .

$A (x) = \frac{1}{6 - 0} (2 \cdot 18)$

Multiply $2$ by $18$ .

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

Simplify the denominator.

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Multiply $- 1$ by $0$ .

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

Add $6$ and $0$ .

$A (x) = \frac{1}{6} \cdot 36$

Cancel the common factor of $6$ .

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Factor $6$ out of $36$ .

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

Cancel the common factor.

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

Rewrite the expression.

$A (x) = 6$

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