Question

you began membership in a new health and fitness club which has access to a dietician and personal trainer. they help you develop a special eight - week diet and exercise program. the data in the following table represents your weight, w, as a function of time, t, over an eight - week period.

time (weeks)	0	1	2	3	4	5	6	7	8

during which time period is the average rate of change zero?
a. from week 7 to week 8
b. from week 3 to week 4
c. from week 0 to week 1
d. from week 5 to week 6
please select the best answer from the choices provided

Explanation:

The average rate of change of a function \( w(t) \) over the interval \([a, b]\) is given by the formula:
\[
\text{Average Rate of Change} = \frac{w(b) - w(a)}{b - a}
\]
We need to find the interval where \( w(b) - w(a) = 0 \) (since \( b - a
eq 0 \) for a time period), which means \( w(a) = w(b) \).

Step 1: Analyze Option a (Week 7 to Week 8)

\( w(7) = 132 \), \( w(8) = 129 \)
\[
\frac{129 - 132}{8 - 7} = \frac{-3}{1} = -3
eq 0
\]

Step 2: Analyze Option b (Week 3 to Week 4)

\( w(3) = 141 \), \( w(4) = 140 \)
\[
\frac{140 - 141}{4 - 3} = \frac{-1}{1} = -1
eq 0
\]

Step 3: Analyze Option c (Week 0 to Week 1)

\( w(0) = 150 \), \( w(1) = 144 \)
\[
\frac{144 - 150}{1 - 0} = \frac{-6}{1} = -6
eq 0
\]

Step 4: Analyze Option d (Week 5 to Week 6)

\( w(5) = 137 \), \( w(6) = 137 \)
\[
\frac{137 - 137}{6 - 5} = \frac{0}{1} = 0
\]

Answer:

d. From week 5 to week 6