calculus

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

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  1. 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.

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

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