Precalculus Examples

$f (x) = e^{\frac{x}{2}}$ ; $[0, 14]$

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.

$\frac{f (14) - f (0)}{(14) - (0)}$

Step 2.2

Substitute the equation $y = e^{\frac{x}{2}}$ for $f (14)$ and $f (0)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(e^{\frac{14}{2}}) - (e^{\frac{0}{2}})}{(14) - (0)}$

Step 3

Simplify the expression.

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Step 3.1

Reduce the expression by cancelling the common factors.

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Step 3.1.1

Cancel the common factor of $14$ and $2$ .

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Step 3.1.1.1

Factor $2$ out of $14$ .

$\frac{e^{\frac{2 \cdot 7}{2}} - (e^{\frac{0}{2}})}{(14) - (0)}$

Step 3.1.1.2

Cancel the common factors.

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Step 3.1.1.2.1

Factor $2$ out of $2$ .

$\frac{e^{\frac{2 \cdot 7}{2 (1)}} - (e^{\frac{0}{2}})}{(14) - (0)}$

Step 3.1.1.2.2

Cancel the common factor.

$\frac{e^{\frac{2 \cdot 7}{2 \cdot 1}} - (e^{\frac{0}{2}})}{(14) - (0)}$

Step 3.1.1.2.3

Rewrite the expression.

$\frac{e^{\frac{7}{1}} - (e^{\frac{0}{2}})}{(14) - (0)}$

Step 3.1.1.2.4

Divide $7$ by $1$ .

$\frac{e^{7} - (e^{\frac{0}{2}})}{(14) - (0)}$

Step 3.1.2

Cancel the common factor of $0$ and $2$ .

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Step 3.1.2.1

Factor $2$ out of $0$ .

$\frac{e^{7} - e^{\frac{2 (0)}{2}}}{(14) - (0)}$

Step 3.1.2.2

Cancel the common factors.

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Step 3.1.2.2.1

Factor $2$ out of $2$ .

$\frac{e^{7} - e^{\frac{2 \cdot 0}{2 \cdot 1}}}{(14) - (0)}$

Step 3.1.2.2.2

Cancel the common factor.

$\frac{e^{7} - e^{\frac{2 \cdot 0}{2 \cdot 1}}}{(14) - (0)}$

Step 3.1.2.2.3

Rewrite the expression.

$\frac{e^{7} - e^{\frac{0}{1}}}{(14) - (0)}$

Step 3.1.2.2.4

Divide $0$ by $1$ .

$\frac{e^{7} - e^{0}}{(14) - (0)}$

Step 3.2

Simplify the numerator.

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Step 3.2.1

Anything raised to $0$ is $1$ .

$\frac{e^{7} - 1 \cdot 1}{14 - (0)}$

Step 3.2.2

Multiply $- 1$ by $1$ .

$\frac{e^{7} - 1}{14 - (0)}$

Step 3.3

Simplify the denominator.

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Step 3.3.1

Multiply $- 1$ by $0$ .

$\frac{e^{7} - 1}{14 + 0}$

Step 3.3.2

Add $14$ and $0$ .

$\frac{e^{7} - 1}{14}$