Precalculus Examples

$f (x) = \ln (x)$ , $[1, e]$

Step 1

Write $f (x) = \ln (x)$ as an equation.

$y = \ln (x)$

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.

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

Step 2.2

Substitute the equation $y = \ln (x)$ for $f (e)$ and $f (1)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(\ln (e)) - (\ln (1))}{(e) - (1)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

The natural logarithm of $e$ is $1$ .

$\frac{1 - \ln (1)}{e - (1)}$

Step 3.1.2

The natural logarithm of $1$ is $0$ .

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

Step 3.1.3

Multiply $- 1$ by $0$ .

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

Step 3.1.4

Add $1$ and $0$ .

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

Step 3.2

Multiply $- 1$ by $1$ .

$\frac{1}{e - 1}$