Question

identifying decreasing average rates of change
choose the intervals where the graph has a
decreasing average rate of change
( x = 0 ) to ( x = 1.3 )
( x = 3 ) to ( x = 6 )
( x = 4 ) to ( x = 6 )
( x = 8 ) to ( x = 10 )

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 \(\frac{f(b)-f(a)}{b - a}\). A decreasing average rate of change means \(\frac{f(b)-f(a)}{b - a}<0\), or \(f(b)<f(a)\) (since \(b>a\) for an interval \([a,b]\)).

Step2: Analyze each interval

• Interval \(x = 0\) to \(x=13\):

Let \(a = 0\), \(b = 13\). From the graph, \(f(0)=0\) and \(f(13)>0\). Then \(\frac{f(13)-f(0)}{13 - 0}=\frac{f(13)}{13}>0\).

• Interval \(x = 3\) to \(x = 6\):

Let \(a = 3\), \(b = 6\). From the graph, \(f(3)>f(6)\). Then \(\frac{f(6)-f(3)}{6 - 3}=\frac{\text{(negative value)}}{3}<0\).

• Interval \(x = 4\) to \(x = 8\):

Let \(a = 4\), \(b = 8\). From the graph, \(f(4)>f(8)\). Then \(\frac{f(8)-f(4)}{8 - 4}=\frac{\text{(negative value)}}{4}<0\).

• Interval \(x = 8\) to \(x = 10\):

Let \(a = 8\), \(b = 10\). From the graph, \(f(8)<f(10)\). Then \(\frac{f(10)-f(8)}{10 - 8}=\frac{\text{(positive value)}}{2}>0\).

Answer:

\(x = 3\) to \(x = 6\), \(x = 4\) to \(x = 8\)