Question

a function is shown in the table below. on which interval of x is the average rate of change of the function the smallest?
answer
x = 0 to x = 1 x = 1 to x = 6
x = 6 to x = 19 x = 19 to x = 28

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \(y = f(x)\) over the interval \([a,b]\) is given by \(\frac{f(b)-f(a)}{b - a}\).

Step2: Calculate average rate of change for \(x = 0\) to \(x = 1\)

Here \(a = 0\), \(b = 1\), \(f(0)=0\), \(f(1) = 2\).
\(\frac{f(1)-f(0)}{1-0}=\frac{2 - 0}{1}=2\).

Step3: Calculate average rate of change for \(x = 1\) to \(x = 6\)

Here \(a = 1\), \(b = 6\), \(f(1)=2\), \(f(6)=10\).
\(\frac{f(6)-f(1)}{6 - 1}=\frac{10 - 2}{5}=\frac{8}{5}=1.6\).

Step4: Calculate average rate of change for \(x = 6\) to \(x = 19\)

Here \(a = 6\), \(b = 19\), \(f(6)=10\), \(f(19)=14\).
\(\frac{f(19)-f(6)}{19 - 6}=\frac{14 - 10}{13}=\frac{4}{13}\approx0.31\).

Step5: Calculate average rate of change for \(x = 19\) to \(x = 28\)

Here \(a = 19\), \(b = 28\), \(f(19)=14\), \(f(28)=19\).
\(\frac{f(28)-f(19)}{28 - 19}=\frac{19 - 14}{9}=\frac{5}{9}\approx0.56\).

Answer:

\(x = 6\) to \(x = 19\)