Calculus Examples

$y = 3 e^{- x}$ , $[0, 6]$

Step 1

Write $y = 3 e^{- x}$ as a function.

$f (x) = 3 e^{- x}$

Step 2

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 3

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

$f (x)$ is continuous

Step 4

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 5

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

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

Step 6

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

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

Step 7

Let $u = - x$ . Then $d u = - d x$ , so $- d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 7.1

Let $u = - x$ . Find $\frac{d u}{d x}$ .

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Step 7.1.1

Differentiate $- x$ .

$\frac{d}{d x} [- x]$

Step 7.1.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$- \frac{d}{d x} [x]$

Step 7.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$- 1 \cdot 1$

Step 7.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 7.2

Substitute the lower limit in for $x$ in $u = - x$ .

$u_{lower} = - 0$

Step 7.3

Multiply $- 1$ by $0$ .

$u_{lower} = 0$

Step 7.4

Substitute the upper limit in for $x$ in $u = - x$ .

$u_{upper} = - 1 \cdot 6$

Step 7.5

Multiply $- 1$ by $6$ .

$u_{upper} = - 6$

Step 7.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 0$

$u_{upper} = - 6$

Step 7.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

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

Step 8

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

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

Step 9

Multiply $- 1$ by $3$ .

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

Step 10

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

$A (x) = \frac{1}{6 - 0} (- 3 (e^{u}]_{0}^{- 6}))$

Step 11

Substitute and simplify.

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Step 11.1

Evaluate $e^{u}$ at $- 6$ and at $0$ .

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

Step 11.2

Simplify.

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Step 11.2.1

Anything raised to $0$ is $1$ .

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

Step 11.2.2

Multiply $- 1$ by $1$ .

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

Step 12

Simplify.

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Step 12.1

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

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

Step 12.2

Apply the distributive property.

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

Step 12.3

Combine $- 3$ and $\frac{1}{e^{6}}$ .

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

Step 12.4

Multiply $- 3$ by $- 1$ .

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

Step 12.5

Move the negative in front of the fraction.

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

Step 13

Simplify the denominator.

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Step 13.1

Multiply $- 1$ by $0$ .

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

Step 13.2

Add $6$ and $0$ .

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

Step 14

Simplify terms.

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Step 14.1

Apply the distributive property.

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

Step 14.2

Cancel the common factor of $3$ .

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Step 14.2.1

Move the leading negative in $- \frac{3}{e^{6}}$ into the numerator.

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

Step 14.2.2

Factor $3$ out of $6$ .

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

Step 14.2.3

Factor $3$ out of $- 3$ .

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

Step 14.2.4

Cancel the common factor.

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

Step 14.2.5

Rewrite the expression.

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

Step 14.3

Multiply $\frac{1}{2}$ by $\frac{- 1}{e^{6}}$ .

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

Step 14.4

Cancel the common factor of $3$ .

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Step 14.4.1

Factor $3$ out of $6$ .

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

Step 14.4.2

Cancel the common factor.

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

Step 14.4.3

Rewrite the expression.

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

Step 14.5

Move the negative in front of the fraction.

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

Step 15