1. Let f(p) represent the daily demand for San Francisco \'49ers T-shirts when t
ID: 2895213 • Letter: 1
Question
1. Let f(p) represent the daily demand for San Francisco '49ers T-shirts when the price for a shirt is p dollars. In other words, f(p) gives the number of shirts purchased daily if the selling price is p dollars. (a) Is f increasing or decreasing? (b) What are the units of p. f(p), and f'(p)? (c) Explain, in terms of shirts and dollars, the practical meaning of the following: i. f(20) 150 ii. f'(20)--5 iii. f(30) (d) Let d represent demand. Then d = f(p), so the function f takes as an input and gives as an output. On the other hand, the inverse function f-l takes as an input and gives as an output, so f-1 (e) Give practical interpretations of f(25) and f-1(25).Explanation / Answer
Basically, f(p) is the dependent variable
and p is the independent variable
Clearly, when p the price increases,
we'd expect demand to drop
So, a) f is DECREASING
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b)
p --> dollars
f(p) --> number of shirts sold, which is just a number
f'(p) --> rate of change of shirts sold per dollar
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c)
f(20) = 150
Tis means when p = 20, f(p) = 150
As in, when the price is 20 dollars, the number of shirts purchaes is 150
f'(20) = -5 :
At this instant, the rate of change of shirts purchased per dollar is
decreasing at the rate of 5 shirts sold per dollar of increased selling price
f(30) :
This tells us the number of shirts sold when the selling price
is 30 dollars per shirt
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f takes p as an input and gives d as the output.
f^-1 takes d as input and returns p as output
So, f^-1(d) = p
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f(25) ---> number of shirts sold when each shirt is sold at 25 dollars
f^-1(25) ---> Cost of each shirt given that the number of shirts sold was 25 in number
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