Answer:
(a, b) -2
(c) -1, -1.5, -2, -2.5
Explanation:
You want the average and instantaneous rates of change at various times in year 3, given the money in the bank after t years is 180+3t-t².
(a) Change
The amount at the beginning of year 3 is ...
180 +3t -t² = 180 +t(3 -t)
180 +2(3 -2) = 182 . . . . . . t=2
The amount at the end of year 3 is ...
180 +3(3 -3) = 180 . . . . . . t=3
The amount increased by 180 -182 = -2 thousand dollars.
(b) Rate of change
This change occurred in one year, so the average rate of change is ...
change/years = -2/1 = -2 thousand dollars per year
(c) Instantaneous rate
The derivative of the amount function will give its instantaneous rate of change:
da/dt = 3 -2t
The values of t at the beginning of the quarters in year 3 are ...
t = 2: da/dt = 3 -2·2 = -1 thousand per year at start of 1st quarter
t = 2.25: da/dt = 3 -2(2.25) = -1.5 thousand per year at start of Q2
t = 2.50: da/dt = 3 -2(2.50) = -2 thousand per year at start of Q3
t = 2.75: da/dt = 3 -2(2.75) = -2.5 thousand per year at start of Q4