Finite Math Examples

$f (x) = x - e^{- x}$ , $[0, 1]$

Step 1

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

$y = x - e^{- x}$

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.

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

Step 2.2

Substitute the equation $y = x - e^{- x}$ for $f (1)$ and $f (0)$ , replacing $x$ in the function with the corresponding $x$ value.

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

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 $1$ .

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

Step 3.1.2

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

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

Step 3.1.3

Simplify each term.

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Step 3.1.3.1

Multiply $- 1$ by $0$ .

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

Step 3.1.3.2

Anything raised to $0$ is $1$ .

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

Step 3.1.3.3

Multiply $- 1$ by $1$ .

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

Step 3.1.4

Subtract $1$ from $0$ .

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

Step 3.1.5

Multiply $- 1$ by $- 1$ .

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

Step 3.1.6

Add $1$ and $1$ .

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

Step 3.1.7

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

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

Step 3.1.8

Combine the numerators over the common denominator.

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

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $0$ .

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

Step 3.2.2

Add $1$ and $0$ .

$\frac{\frac{2 e - 1}{e}}{1}$

Step 3.3

Divide $\frac{2 e - 1}{e}$ by $1$ .

$\frac{2 e - 1}{e}$