Question

use the graph of $f(x)$ to evaluate the following:
average rate of change of $f(x)$ over the interval $1,4=$
question help: video

Explanation:

Step1: Recall the formula for average rate of change

The formula for the average rate of change of a function \(y = f(x)\) over the interval \([a,b]\) is \(\frac{f(b)-f(a)}{b - a}\). Here, \(a = 1\) and \(b=4\).

Step2: Find \(f(1)\) and \(f(4)\) from the graph

From the graph, when \(x = 1\), \(f(1)=5\); when \(x = 4\), \(f(4)=1\).

Step3: Substitute values into the formula

Substitute \(a = 1\), \(b = 4\), \(f(1)=5\), and \(f(4)=1\) into \(\frac{f(b)-f(a)}{b - a}\). We get \(\frac{1 - 5}{4-1}\).

Step4: Simplify the expression

\(\frac{1 - 5}{4 - 1}=\frac{-4}{3}=-\frac{4}{3}\)

Answer:

\(-\frac{4}{3}\)