Precalculus Examples

$f (x) = 6 e^{x}$ , $[- 3, 3]$

Step 1

Write $f (x) = 6 e^{x}$ as an equation.

$y = 6 e^{x}$

Step 2

Substitute using the average rate of change formula.

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Step 2.1

The average rate of change of a function can be found by calculating the change in $y$ values of the two points divided by the change in $x$ values of the two points.

$\frac{f (3) - f (- 3)}{(3) - (- 3)}$

Step 2.2

Substitute the equation $y = 6 e^{x}$ for $f (3)$ and $f (- 3)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(6 e^{3}) - (6 e^{- 3})}{(3) - (- 3)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Multiply $- 1$ by $6$ .

$\frac{6 e^{3} - 6 e^{- 3}}{3 - (- 3)}$

Step 3.1.2

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

$\frac{6 e^{3} - 6 \frac{1}{e^{3}}}{3 - (- 3)}$

Step 3.1.3

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

$\frac{6 e^{3} + \frac{- 6}{e^{3}}}{3 - (- 3)}$

Step 3.1.4

Move the negative in front of the fraction.

$\frac{6 e^{3} - \frac{6}{e^{3}}}{3 - (- 3)}$

Step 3.1.5

To write $6 e^{3}$ as a fraction with a common denominator, multiply by $\frac{e^{3}}{e^{3}}$ .

$\frac{\frac{6 e^{3} e^{3}}{e^{3}} - \frac{6}{e^{3}}}{3 - (- 3)}$

Step 3.1.6

Combine the numerators over the common denominator.

$\frac{\frac{6 e^{3} e^{3} - 6}{e^{3}}}{3 - (- 3)}$

Step 3.1.7

Multiply $e^{3}$ by $e^{3}$ by adding the exponents.

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Step 3.1.7.1

Move $e^{3}$ .

$\frac{\frac{6 (e^{3} e^{3}) - 6}{e^{3}}}{3 - (- 3)}$

Step 3.1.7.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{6 e^{3 + 3} - 6}{e^{3}}}{3 - (- 3)}$

Step 3.1.7.3

Add $3$ and $3$ .

$\frac{\frac{6 e^{6} - 6}{e^{3}}}{3 - (- 3)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $- 3$ .

$\frac{\frac{6 e^{6} - 6}{e^{3}}}{3 + 3}$

Step 3.2.2

Add $3$ and $3$ .

$\frac{\frac{6 e^{6} - 6}{e^{3}}}{6}$

Step 3.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{6 e^{6} - 6}{e^{3}} \cdot \frac{1}{6}$

Step 3.4

Simplify terms.

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Step 3.4.1

Multiply $\frac{6 e^{6} - 6}{e^{3}}$ by $\frac{1}{6}$ .

$\frac{6 e^{6} - 6}{e^{3} \cdot 6}$

Step 3.4.2

Cancel the common factor of $6 e^{6} - 6$ and $6$ .

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Step 3.4.2.1

Factor $6$ out of $6 e^{6}$ .

$\frac{6 (e^{6}) - 6}{e^{3} \cdot 6}$

Step 3.4.2.2

Factor $6$ out of $- 6$ .

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

Step 3.4.2.3

Factor $6$ out of $6 (e^{6}) + 6 (- 1)$ .

$\frac{6 (e^{6} - 1)}{e^{3} \cdot 6}$

Step 3.4.2.4

Cancel the common factors.

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Step 3.4.2.4.1

Factor $6$ out of $e^{3} \cdot 6$ .

$\frac{6 (e^{6} - 1)}{6 \cdot e^{3}}$

Step 3.4.2.4.2

Cancel the common factor.

$\frac{6 (e^{6} - 1)}{6 \cdot e^{3}}$

Step 3.4.2.4.3

Rewrite the expression.

$\frac{e^{6} - 1}{e^{3}}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Rewrite $e^{6}$ as ${(e^{2})}^{3}$ .

$\frac{{(e^{2})}^{3} - 1}{e^{3}}$

Step 3.5.2

Rewrite $1$ as $1^{3}$ .

$\frac{{(e^{2})}^{3} - 1^{3}}{e^{3}}$

Step 3.5.3

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = e^{2}$ and $b = 1$ .

$\frac{(e^{2} - 1) ({(e^{2})}^{2} + e^{2} \cdot 1 + 1^{2})}{e^{3}}$

Step 3.5.4

Simplify.

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Step 3.5.4.1

Rewrite $1$ as $1^{2}$ .

$\frac{(e^{2} - 1^{2}) ({(e^{2})}^{2} + e^{2} \cdot 1 + 1^{2})}{e^{3}}$

Step 3.5.4.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = e$ and $b = 1$ .

$\frac{(e + 1) (e - 1) ({(e^{2})}^{2} + e^{2} \cdot 1 + 1^{2})}{e^{3}}$

Step 3.5.4.3

Multiply $e^{2}$ by $1$ .

$\frac{(e + 1) (e - 1) ({(e^{2})}^{2} + e^{2} + 1^{2})}{e^{3}}$

Step 3.5.5

Simplify each term.

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Step 3.5.5.1

Multiply the exponents in ${(e^{2})}^{2}$ .

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Step 3.5.5.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{(e + 1) (e - 1) (e^{2 \cdot 2} + e^{2} + 1^{2})}{e^{3}}$

Step 3.5.5.1.2

Multiply $2$ by $2$ .

$\frac{(e + 1) (e - 1) (e^{4} + e^{2} + 1^{2})}{e^{3}}$

Step 3.5.5.2

One to any power is one.

$\frac{(e + 1) (e - 1) (e^{4} + e^{2} + 1)}{e^{3}}$