Precalculus Examples

$f (x) = e^{x - 1} + 3$ , $[1, 5]$

Step 1

Write $f (x) = e^{x - 1} + 3$ as an equation.

$y = e^{x - 1} + 3$

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.

$\frac{f (5) - f (1)}{(5) - (1)}$

Step 2.2

Substitute the equation $y = e^{x - 1} + 3$ for $f (5)$ and $f (1)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(e^{(5) - 1} + 3) - (e^{(1) - 1} + 3)}{(5) - (1)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Subtract $1$ from $5$ .

$\frac{e^{4} + 3 - (e^{1 - 1} + 3)}{5 - (1)}$

Step 3.1.2

Simplify each term.

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Step 3.1.2.1

Subtract $1$ from $1$ .

$\frac{e^{4} + 3 - (e^{0} + 3)}{5 - (1)}$

Step 3.1.2.2

Anything raised to $0$ is $1$ .

$\frac{e^{4} + 3 - (1 + 3)}{5 - (1)}$

Step 3.1.3

Add $1$ and $3$ .

$\frac{e^{4} + 3 - 1 \cdot 4}{5 - (1)}$

Step 3.1.4

Multiply $- 1$ by $4$ .

$\frac{e^{4} + 3 - 4}{5 - (1)}$

Step 3.1.5

Subtract $4$ from $3$ .

$\frac{e^{4} - 1}{5 - (1)}$

Step 3.1.6

Rewrite $e^{4} - 1$ in a factored form.

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Step 3.1.6.1

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

$\frac{{(e^{2})}^{2} - 1}{5 - (1)}$

Step 3.1.6.2

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

$\frac{{(e^{2})}^{2} - 1^{2}}{5 - (1)}$

Step 3.1.6.3

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

$\frac{(e^{2} + 1) (e^{2} - 1)}{5 - (1)}$

Step 3.1.6.4

Simplify.

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Step 3.1.6.4.1

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

$\frac{(e^{2} + 1) (e^{2} - 1^{2})}{5 - (1)}$

Step 3.1.6.4.2

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 $b = 1$ .

$\frac{(e^{2} + 1) (e + 1) (e - 1)}{5 - (1)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $1$ .

$\frac{(e^{2} + 1) (e + 1) (e - 1)}{5 - 1}$

Step 3.2.2

Subtract $1$ from $5$ .

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