(i) Compute the average rate of change over the shortest interval to the left of the point of interest (for instance, for the interval from 6 to 9).
(ii) Compute the average rate of change over the shortest interval to the right of the point of interest (for instance, for the interval from 9 to 12).
(iii) Now average the results from steps (i) and (ii), i.e., compute the average of the average rates of change on the left and right sides.
Round your answers to one decimal place.
t 0 2 4 6 9 12
P 6 11 21 32 44 48
b) f′(9) ≈
c) f′(4) ≈
To compute the average rate of change over the shortest interval to the left or right of a point of interest, you need to use the formula:
Average Rate of Change = (f(x2) - f(x1))/(x2 - x1)
Where x1 and x2 are the endpoints of the interval, and f(x1) and f(x2) are the corresponding function values.
(i) Compute the average rate of change over the shortest interval to the left of the point of interest (for the interval from 6 to 9):
Average Rate of Change = (f(6) - f(4))/(6 - 4) = (32 - 21)/(6 - 4) = 11/2 = 5.5
(ii) Compute the average rate of change over the shortest interval to the right of the point of interest (for the interval from 9 to 12):
Average Rate of Change = (f(12) - f(9))/(12 - 9) = (48 - 44)/(12 - 9) = 4/3 ≈ 1.3
(iii) Now average the results from steps (i) and (ii), i.e., compute the average of the average rates of change on the left and right sides:
Average of Average Rate of Change = (5.5 + 1.3)/2 = 6.8/2 = 3.4
Round your answers to one decimal place:
(b) f'(9) ≈ 1.3 (rounded to one decimal place).
(c) f'(4) ≈ 5.5 (rounded to one decimal place).