Calculus Examples

$f (x) = 4 e^{x}$ , $[- 2, 2]$

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 $[- 2, 2]$ .

$f (x)$ 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.

$A (x) = \frac{1}{2 + 2} (\int_{- 2}^{2} 4 e^{x} d x)$

Step 5

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$A (x) = \frac{1}{2 + 2} (4 \int_{- 2}^{2} e^{x} d x)$

Step 6

The integral of $e^{x}$ with respect to $x$ is $e^{x}$ .

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

Step 7

Evaluate $e^{x}$ at $2$ and at $- 2$ .

$A (x) = \frac{1}{2 + 2} (4 (e^{2} - e^{- 2}))$

Step 8

Simplify.

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Step 8.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$A (x) = \frac{1}{2 + 2} (4 (e^{2} - \frac{1}{e^{2}}))$

Step 8.2

Apply the distributive property.

$A (x) = \frac{1}{2 + 2} (4 e^{2} + 4 (- \frac{1}{e^{2}}))$

Step 8.3

Multiply $4 (- \frac{1}{e^{2}})$ .

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Step 8.3.1

Multiply $- 1$ by $4$ .

$A (x) = \frac{1}{2 + 2} (4 e^{2} - 4 \frac{1}{e^{2}})$

Step 8.3.2

Combine $- 4$ and $\frac{1}{e^{2}}$ .

$A (x) = \frac{1}{2 + 2} (4 e^{2} + \frac{- 4}{e^{2}})$

Step 8.4

Move the negative in front of the fraction.

$A (x) = \frac{1}{2 + 2} (4 e^{2} - \frac{4}{e^{2}})$

Step 9

Add $2$ and $2$ .

$A (x) = \frac{1}{4} \cdot (4 e^{2} - \frac{4}{e^{2}})$

Step 10

Apply the distributive property.

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

Step 11

Cancel the common factor of $4$ .

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Step 11.1

Factor $4$ out of $4 e^{2}$ .

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

Step 11.2

Cancel the common factor.

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

Step 11.3

Rewrite the expression.

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

Step 12

Cancel the common factor of $4$ .

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Step 12.1

Move the leading negative in $- \frac{4}{e^{2}}$ into the numerator.

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

Step 12.2

Factor $4$ out of $- 4$ .

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

Step 12.3

Cancel the common factor.

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

Step 12.4

Rewrite the expression.

$A (x) = e^{2} + \frac{- 1}{e^{2}}$

Step 13

Move the negative in front of the fraction.

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

Step 14