Precalculus Examples

$f (x) = 3^{x} + 2$ ; over the interval $1 \leq x \leq 4$

Step 1

Write $f (x) = 3^{x} + 2$ as an equation.

$y = 3^{x} + 2$

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.

$\frac{f (4) - f (1)}{(4) - (1)}$

Step 2.2

Substitute the equation $y = 3^{x} + 2$ for $f (4)$ and $f (1)$ , replacing $x$ in the function with the corresponding $x$ value.

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

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $3$ to the power of $4$ .

$\frac{81 + 2 - (3^{1} + 2)}{4 - (1)}$

Step 3.1.2

Evaluate the exponent.

$\frac{81 + 2 - (3 + 2)}{4 - (1)}$

Step 3.1.3

Add $3$ and $2$ .

$\frac{81 + 2 - 1 \cdot 5}{4 - (1)}$

Step 3.1.4

Multiply $- 1$ by $5$ .

$\frac{81 + 2 - 5}{4 - (1)}$

Step 3.1.5

Add $81$ and $2$ .

$\frac{83 - 5}{4 - (1)}$

Step 3.1.6

Subtract $5$ from $83$ .

$\frac{78}{4 - (1)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $1$ .

$\frac{78}{4 - 1}$

Step 3.2.2

Subtract $1$ from $4$ .

$\frac{78}{3}$

Step 3.3

Divide $78$ by $3$ .

$26$