Precalculus Examples

$f (x) = e^{x}$ on $0$ , $2$

Step 1

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

$y = e^{x}$

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.

$\frac{f (2) - f (0)}{(2) - (0)}$

Step 2.2

Substitute the equation $y = e^{x}$ for $f (2)$ and $f (0)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(e^{2}) - (e^{0})}{(2) - (0)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Anything raised to $0$ is $1$ .

$\frac{e^{2} - 1 \cdot 1}{2 - (0)}$

Step 3.1.2

Multiply $- 1$ by $1$ .

$\frac{e^{2} - 1}{2 - (0)}$

Step 3.1.3

Rewrite $e^{2} - 1$ in a factored form.

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Step 3.1.3.1

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

$\frac{e^{2} - 1^{2}}{2 - (0)}$

Step 3.1.3.2

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 $b = 1$ .

$\frac{(e + 1) (e - 1)}{2 - (0)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $0$ .

$\frac{(e + 1) (e - 1)}{2 + 0}$

Step 3.2.2

Add $2$ and $0$ .

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