Question

over which interval does ( f(t) ) have a negative average rate of change?
choose 1 answer:
(a) ( -9,-8 )
(b) ( 2,4 )
(c) ( -8,-2 )
(d) ( -5,-1 )

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \(y = f(t)\) over the interval \([a,b]\) is given by \(\frac{f(b)-f(a)}{b - a}\). For the average rate of change to be negative, \(f(b)-f(a)<0\) (since \(b - a>0\) when \(b>a\)). In terms of the graph, if \(b>a\), the function \(y = f(t)\) has a negative average rate of change over \([a,b]\) when \(f(b)<f(a)\) (the function is decreasing from \(t=a\) to \(t = b\)).

Step2: Analyze option A: \([-9,-8]\)

For the interval \([-9,-8]\), \(b=-8\), \(a = - 9\). The function is constant over \([-9,-8]\) (horizontal line). Using the formula \(\frac{f(-8)-f(-9)}{-8-(-9)}=\frac{f(-8)-f(-9)}{1}\). Since \(f(-8)=f(-9)\), the average rate of change is \(0\).

Step3: Analyze option B: \([2,4]\)

For the interval \([2,4]\), \(b = 4\), \(a=2\). From the graph, \(f(4)=0\) and \(f(2)=3\). Then \(\frac{f(4)-f(2)}{4 - 2}=\frac{0 - 3}{2}=-\frac{3}{2}<0\).

Step4: Analyze option C: \([-8,-2]\)

For the interval \([-8,-2]\), \(b=-2\), \(a=-8\). From the graph, \(f(-2)>f(-8)\). Using the formula \(\frac{f(-2)-f(-8)}{-2-(-8)}=\frac{f(-2)-f(-8)}{6}>0\) (since \(f(-2)-f(-8)>0\)).

Step5: Analyze option D: \([-5,-1]\)

For the interval \([-5,-1]\), \(b=-1\), \(a=-5\). From the graph, \(f(-1)>f(-5)\). Using the formula \(\frac{f(-1)-f(-5)}{-1-(-5)}=\frac{f(-1)-f(-5)}{4}>0\) (since \(f(-1)-f(-5)>0\)).

Answer:

B. \([2,4]\)