Precalculus Examples

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

Step 1

Write $f (x) = 4 e^{x}$ as an equation.

$y = 4 e^{x}$

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.

$\frac{f (2) - f (- 2)}{(2) - (- 2)}$

Step 2.2

Substitute the equation $y = 4 e^{x}$ for $f (2)$ and $f (- 2)$ , replacing $x$ in the function with the corresponding $x$ value.

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

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Rewrite $4 e^{2}$ as ${(2 e)}^{2}$ .

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

Step 3.1.2

Rewrite $4 e^{- 2}$ as ${(2 e^{- 1})}^{2}$ .

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

Step 3.1.3

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

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

Step 3.1.4

Simplify.

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Step 3.1.4.1

Factor $2$ out of $2 e + 2 e^{- 1}$ .

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

Step 3.1.4.2

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

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

Step 3.1.4.3

Multiply $2$ by $- 1$ .

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

Step 3.1.5

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

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

Step 3.1.6

Combine the numerators over the common denominator.

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

Step 3.1.7

Multiply $e e$ .

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Step 3.1.7.1

Raise $e$ to the power of $1$ .

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

Step 3.1.7.2

Raise $e$ to the power of $1$ .

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

Step 3.1.7.3

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

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

Step 3.1.7.4

Add $1$ and $1$ .

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

Step 3.1.8

Simplify each term.

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Step 3.1.8.1

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

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

Step 3.1.8.2

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

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

Step 3.1.8.3

Move the negative in front of the fraction.

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

Step 3.1.9

To write $2 e$ as a fraction with a common denominator, multiply by $\frac{e}{e}$ .

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

Step 3.1.10

Combine the numerators over the common denominator.

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

Step 3.1.11

Multiply $2 e e$ .

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Step 3.1.11.1

Raise $e$ to the power of $1$ .

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

Step 3.1.11.2

Raise $e$ to the power of $1$ .

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

Step 3.1.11.3

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

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

Step 3.1.11.4

Add $1$ and $1$ .

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

Step 3.1.12

Combine exponents.

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Step 3.1.12.1

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

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

Step 3.1.12.2

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

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

Step 3.1.12.3

Raise $e$ to the power of $1$ .

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

Step 3.1.12.4

Raise $e$ to the power of $1$ .

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

Step 3.1.12.5

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

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

Step 3.1.12.6

Add $1$ and $1$ .

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

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $- 2$ .

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

Step 3.2.2

Add $2$ and $2$ .

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

Step 3.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.4

Simplify terms.

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Step 3.4.1

Combine.

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

Step 3.4.2

Cancel the common factor of $2$ and $4$ .

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Step 3.4.2.1

Factor $2$ out of $2 (e^{2} + 1) (2 e^{2} - 2) \cdot 1$ .

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

Step 3.4.2.2

Cancel the common factors.

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Step 3.4.2.2.1

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

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

Step 3.4.2.2.2

Cancel the common factor.

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

Step 3.4.2.2.3

Rewrite the expression.

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

Step 3.4.3

Cancel the common factor of $2 e^{2} - 2$ and $2$ .

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Step 3.4.3.1

Factor $2$ out of $((e^{2} + 1) (2 e^{2} - 2)) \cdot 1$ .

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

Step 3.4.3.2

Cancel the common factors.

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Step 3.4.3.2.1

Factor $2$ out of $e^{2} \cdot 2$ .

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

Step 3.4.3.2.2

Cancel the common factor.

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

Step 3.4.3.2.3

Rewrite the expression.

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

Step 3.4.4

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

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

Step 3.5

Simplify the numerator.

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Step 3.5.1

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

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

Step 3.5.2

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 $b = 1$ .

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