Questions: Calculate the average rate of change between (x=4) and (x=6) for the function [ f(x)=7+frac18 x ] Average rate of change (=)

$Calculate the average rate of change between (x=4) and (x=6) for the function [ f(x)=7+frac18 x ] Average rate of change (=)$

Transcript text: Calculate the average rate of change between $x=4$ and $x=6$ for the function \[ f(x)=7+\frac{1}{8} x \] Average rate of change $=$ $\square$

Solution

Solution Steps

To find the average rate of change of a function between two points, we use the formula: $\frac{f(b) - f(a)}{b - a}$, where $a$ and $b$ are the given points. Here, $a = 4$ and $b = 6$. We need to evaluate the function $f(x) = 7 + \frac{1}{8}x$ at these points and then apply the formula.

Step 1: Evaluate the Function at Given Points

To find the average rate of change, we first evaluate the function $ f(x) = 7 + \frac{1}{8}x $ at the given points $ x = 4 $ and $ x = 6 $.

\[ f(4) = 7 + \frac{1}{8} \times 4 = 7.5 \]

\[ f(6) = 7 + \frac{1}{8} \times 6 = 7.75 \]

Step 2: Calculate the Average Rate of Change

The average rate of change of the function between $ x = 4 $ and $ x = 6 $ is given by the formula:

\[ \frac{f(b) - f(a)}{b - a} \]

Substituting the values we found:

\[ \frac{7.75 - 7.5}{6 - 4} = \frac{0.25}{2} = 0.125 \]