Finite Math Examples

$f (x) = e^{- \frac{x}{2}}$ , $[- 1, 16]$

Step 1

Write $f (x) = e^{- \frac{x}{2}}$ as an equation.

$y = e^{- \frac{x}{2}}$

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.

$\frac{f (16) - f (- 1)}{(16) - (- 1)}$

Step 2.2

Substitute the equation $y = e^{- \frac{x}{2}}$ for $f (16)$ and $f (- 1)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(e^{- \frac{16}{2}}) - (e^{- \frac{- 1}{2}})}{(16) - (- 1)}$

Step 3

Simplify the expression.

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Step 3.1

Cancel the common factor of $16$ and $2$ .

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Step 3.1.1

Factor $2$ out of $16$ .

$\frac{e^{- \frac{2 \cdot 8}{2}} - (e^{- \frac{- 1}{2}})}{(16) - (- 1)}$

Step 3.1.2

Cancel the common factors.

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Step 3.1.2.1

Factor $2$ out of $2$ .

$\frac{e^{- \frac{2 \cdot 8}{2 (1)}} - (e^{- \frac{- 1}{2}})}{(16) - (- 1)}$

Step 3.1.2.2

Cancel the common factor.

$\frac{e^{- \frac{2 \cdot 8}{2 \cdot 1}} - (e^{- \frac{- 1}{2}})}{(16) - (- 1)}$

Step 3.1.2.3

Rewrite the expression.

$\frac{e^{- \frac{8}{1}} - (e^{- \frac{- 1}{2}})}{(16) - (- 1)}$

Step 3.1.2.4

Divide $8$ by $1$ .

$\frac{e^{- 1 \cdot 8} - (e^{- \frac{- 1}{2}})}{(16) - (- 1)}$

Step 3.2

Simplify the numerator.

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Step 3.2.1

Multiply $- 1$ by $8$ .

$\frac{e^{- 8} - e^{- \frac{- 1}{2}}}{16 - (- 1)}$

Step 3.2.2

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

$\frac{\frac{1}{e^{8}} - e^{- \frac{- 1}{2}}}{16 - (- 1)}$

Step 3.2.3

Move the negative in front of the fraction.

$\frac{\frac{1}{e^{8}} - e^{- - \frac{1}{2}}}{16 - (- 1)}$

Step 3.2.4

Multiply $- - \frac{1}{2}$ .

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Step 3.2.4.1

Multiply $- 1$ by $- 1$ .

$\frac{\frac{1}{e^{8}} - e^{1 (\frac{1}{2})}}{16 - (- 1)}$

Step 3.2.4.2

Multiply $\frac{1}{2}$ by $1$ .

$\frac{\frac{1}{e^{8}} - e^{\frac{1}{2}}}{16 - (- 1)}$

Step 3.2.5

To write $- e^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{e^{8}}{e^{8}}$ .

$\frac{\frac{1}{e^{8}} - e^{\frac{1}{2}} \cdot \frac{e^{8}}{e^{8}}}{16 - (- 1)}$

Step 3.2.6

Combine $- e^{\frac{1}{2}}$ and $\frac{e^{8}}{e^{8}}$ .

$\frac{\frac{1}{e^{8}} + \frac{- e^{\frac{1}{2}} e^{8}}{e^{8}}}{16 - (- 1)}$

Step 3.2.7

Combine the numerators over the common denominator.

$\frac{\frac{1 - e^{\frac{1}{2}} e^{8}}{e^{8}}}{16 - (- 1)}$

Step 3.2.8

Multiply $e^{\frac{1}{2}}$ by $e^{8}$ by adding the exponents.

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Step 3.2.8.1

Move $e^{8}$ .

$\frac{\frac{1 - (e^{8} e^{\frac{1}{2}})}{e^{8}}}{16 - (- 1)}$

Step 3.2.8.2

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

$\frac{\frac{1 - e^{8 + \frac{1}{2}}}{e^{8}}}{16 - (- 1)}$

Step 3.2.8.3

To write $8$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\frac{1 - e^{8 \cdot \frac{2}{2} + \frac{1}{2}}}{e^{8}}}{16 - (- 1)}$

Step 3.2.8.4

Combine $8$ and $\frac{2}{2}$ .

$\frac{\frac{1 - e^{\frac{8 \cdot 2}{2} + \frac{1}{2}}}{e^{8}}}{16 - (- 1)}$

Step 3.2.8.5

Combine the numerators over the common denominator.

$\frac{\frac{1 - e^{\frac{8 \cdot 2 + 1}{2}}}{e^{8}}}{16 - (- 1)}$

Step 3.2.8.6

Simplify the numerator.

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Step 3.2.8.6.1

Multiply $8$ by $2$ .

$\frac{\frac{1 - e^{\frac{16 + 1}{2}}}{e^{8}}}{16 - (- 1)}$

Step 3.2.8.6.2

Add $16$ and $1$ .

$\frac{\frac{1 - e^{\frac{17}{2}}}{e^{8}}}{16 - (- 1)}$

Step 3.3

Simplify the denominator.

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Step 3.3.1

Multiply $- 1$ by $- 1$ .

$\frac{\frac{1 - e^{\frac{17}{2}}}{e^{8}}}{16 + 1}$

Step 3.3.2

Add $16$ and $1$ .

$\frac{\frac{1 - e^{\frac{17}{2}}}{e^{8}}}{17}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

$\frac{1 - e^{\frac{17}{2}}}{e^{8}} \cdot \frac{1}{17}$

Step 3.5

Multiply $\frac{1 - e^{\frac{17}{2}}}{e^{8}}$ by $\frac{1}{17}$ .

$\frac{1 - e^{\frac{17}{2}}}{e^{8} \cdot 17}$

Step 3.6

Move $17$ to the left of $e^{8}$ .

$\frac{1 - e^{\frac{17}{2}}}{17 e^{8}}$