You are working as assistant to a major movie producer from Hollywood, but mainly you look after pre-release film publicity. You are an old hand at publicity, having worked in several production companies before. Based on previous experience you have developed a model for the most effective expenditure of money in radio and television advertising in the last ten days prior to the release of the film. The advertising campaign starts with massive media blitz ten days before the release of the film. Money spent, (in 100,000 dollars) as a function of the number of days after the initial media blitz, , is given by:

M(n)=(n^3-48n+1100)/200

The producer, your boss, comes in and asks you to prepare a report on the funding needs for Radio and Television advertising in the days between the media blitz and the release, highlighting the rate at which spending goes up or down each day. In particular he seems interested in the rate at which spending will change in the 'th day. To estimate this,using your formula, calculate the average rate of change for each of the following intervals (in 100,000 dollars per day):

Average rate of change on [0,1]:
Average rate of change on [1,2]:
Average rate of change on [2,3]:
Average rate of change on [3,4]:
Average rate of change on [4,5]:
Average rate of change on [5,6]:
Average rate of change on [6,7]:
Average rate of change on [7,8]:
Average rate of change on [8.9]:
Average rate of change on [9,10]:
Why was the manager interested in the 'th day?
a)NO reason
b)Spending starts increasing day after day
c)the day is peak of expenditures
d)Spending starts decreasing day after day

Your best estimate for the instantaneous rate of change at the 4'th day:

please help with this questions...i really don't get how to slve!
mary

Well they say estimate so I would not calculate the cubic equation for each day and subtract. I would use the derivative.

Since they want the average rate of change between 0 and 1 for example, I think it appropriate to use the rate of change at the derivative at 0 and the derivative at 1 and average them for 0 - 1
Of course we must get that derivative

M(n)=(n^3-48n+1100)/200
200 M(n) = n^3 - 48 n + 1100
200 dM/dt = 3 n^2 -48
so
rate n = dM/dt = (3 n^2 -48)/200
calculate that at
n = 0
n = 1
n = 2
n = 3 etc
then
rate 0 - 1 = (1/2)( rate 0 + rate 1)

by the way the maximum spending will be when dM/dt = 0
3 n^2 - 48 = 0
n^2 = 16
n = 4
That is why they want you to calculate M there and that is why the boss wants to know these rates.

By the way, I would personally program a spreadsheet to do those numbers, but a fancy programable calculator would do it pretty fast.

To calculate the average rate of change for each interval, we need to find the difference in the spending between the two days and divide it by the number of days in the interval.

For example, to calculate the average rate of change on [0,1], we need to find the difference in spending between Day 0 and Day 1. The formula for spending on Day n is given by M(n)=(n^3-48n+1100)/200.

So, substituting n=0 into the formula, we have M(0)=(0^3-48(0)+1100)/200 = 1100/200 = 5.5.

Similarly, substituting n=1 into the formula, we have M(1)=(1^3-48(1)+1100)/200 = 3.6.

The difference in spending between Day 0 and Day 1 is 3.6 - 5.5 = -1.9.

Finally, we divide this difference by the number of days in the interval (which is 1) to get the average rate of change: -1.9/1 = -1.9.

Similarly, you can calculate the average rate of change for each interval using the same method.

To estimate the average rate of change on the 'th day, you can use the same approach but specify the interval as [n, n+1]. For example, to estimate the average rate of change on the 4th day, you can calculate the difference in spending between Day 4 and Day 5 and divide it by 1.

To find the instantaneous rate of change at the 4th day, you need to calculate the derivative of the spending function with respect to time and evaluate it at Day 4. However, since the formula provided only gives the spending as a function of the number of days after the initial media blitz, we cannot directly calculate the instantaneous rate of change without more information.