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A y 3 x b y x 2 20 x 100 c y x 2 20 x 100 d y x 2 20

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(a) y = 3 x . (b) y = x 2 + 20 x + 100 (c) y = x 2 – 20 x + 100 (d) y = x 2 – 20 x + 106 (e) y = x 3 – 27 x (f) y = 1 4 + x x 2. The Production Possibility Frontier (PPF) for Leutonia, which produces only 2 goods, X and Y , has the form Y = 100 – 0.25 X 2 , where 0 ≤ X ≤ 20. (a) When Leutonia is producing 16 units of X , what is its maximum possible pro- duction of Y (that is, the value of Y given by the PPF equation)? At this point, what is the slope of the PPF? (b) When the slope of the PPF is –2, what are the values of X and Y? 3. You are asked to find the maximum value of the equation y = f ( x ) = 32 x – .5 x 2 . Then, you are given a second function, g ( x ) = 10 + 8 x + .25 x 2 , and told to maximize the function h ( x ) = f ( x ) – g ( x ). What are your results? 4. The demand function for Super Frosteeos, the breakfast cereal made from 100%- pure refined sugar (“Part of a healthy breakfast!”) is given by the equation Q = 6 P 4 2 = 64 P –2 , where Q is measured in boxes demanded per period and P is in $/box. The Total Revenue (TR) function for Super Frosteeos is given by the equation TR = PQ , with MATH MODULE 10: CALCULUS RESULTS FOR THE NON-CALCULUS SPEAKER M10-7
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TR measured in $/period. The Total Cost (TC) function is initially TC = 0.5 Q , with Total Costs also measured in $/period. Profits (in $/period) = TR – TC. (a) Give the inverse demand function for Super Frosteeos, with P as a function of Q . (b) Express the TR function (i) as a function of P and (ii) as a function of Q . (c) Express Profits ( ) as a function of Q , and by setting the derivative of Profits with respect to Q = 0, find the level of Q at which profits are maximized, and the value of P and at this point. (d) Repeat (c) if the Total Cost function becomes TC = 0.25 Q . 5. Max Yu derives utility from consuming 2 goods, imaginatively known as X and Y . His utility function U ( X , Y ), which measures his satisfaction from consuming X and Y , has the form U = X 2 Y . Max wants to maximize his utility subject to his budget constraint, which has the form M = 120 – 4 X – 2 Y = 0. [This means that Max has $120 to spend this period; the price of X is $4/unit; the price of Y is $2/unit; and Max spends all of his budget on X and Y .] (a) Solve the budget constraint for Y as a function of X , substitute this expression into Max’s utility function, and show him how to maximize his utility by setting dU / dX = 0 to get the utility-maximizing quantities of X and Y . (b) Repeat (a) if the price of X falls to $2/unit, with M = $120 and the price of Y = $2/unit. M10-8 MATH MODULE 10: CALCULUS RESULTS FOR THE NON-CALCULUS SPEAKER
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