Precalculus Examples

$f (x) = 10^{2 x - 2}$ at $[1, 3]$

Step 1

Write $f (x) = 10^{2 x - 2}$ as an equation.

$y = 10^{2 x - 2}$

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.

$\frac{f (3) - f (1)}{(3) - (1)}$

Step 2.2

Substitute the equation $y = 10^{2 x - 2}$ for $f (3)$ and $f (1)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(10^{2 (3) - 2}) - (10^{2 (1) - 2})}{(3) - (1)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Multiply $2$ by $3$ .

$\frac{10^{6 - 2} - 10^{2 (1) - 2}}{3 - (1)}$

Step 3.1.2

Subtract $2$ from $6$ .

$\frac{10^{4} - 10^{2 (1) - 2}}{3 - (1)}$

Step 3.1.3

Raise $10$ to the power of $4$ .

$\frac{10000 - 10^{2 (1) - 2}}{3 - (1)}$

Step 3.1.4

Multiply $2$ by $1$ .

$\frac{10000 - 10^{2 - 2}}{3 - (1)}$

Step 3.1.5

Subtract $2$ from $2$ .

$\frac{10000 - 10^{0}}{3 - (1)}$

Step 3.1.6

Anything raised to $0$ is $1$ .

$\frac{10000 - 1 \cdot 1}{3 - (1)}$

Step 3.1.7

Multiply $- 1$ by $1$ .

$\frac{10000 - 1}{3 - (1)}$

Step 3.1.8

Subtract $1$ from $10000$ .

$\frac{9999}{3 - (1)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $1$ .

$\frac{9999}{3 - 1}$

Step 3.2.2

Subtract $1$ from $3$ .

$\frac{9999}{2}$