Economics 3364A/B Lecture Notes - Lecture 5: Product Rule, Quotient Rule

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carminewalrus808

22 Aug 2015

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Do not simplify. (a) f (x) = x tan x, (b) f (x) = tan 1 x x2+1 , (c) f (x) = sec2( x2 + x),(d) f (x) = 3arccos x. Solution to 1-(a): by the product rule of di erentiaion, we have f (x) = ( x) tan x + x(tan x) = Solution to 1-(b): by the quotient rule of di erentiation, we have f (x) = (tan 1 x) (x2 + 1) (tan 1 x)(x2 + 1) (x2 + 1)2. 1 x2+1 (x2 + 1) (tan 1 x)(2x) (x2 + 1)2. 1 2x tan 1 x (x2 + 1)2. Solution to 1-(c): by the chain rule of di erentiation, we have f (x) = 2 sec(px2 + x) [sec(px2 + x)] . = 2 sec(px2 + x)[sec(px2 + x) tan(px2 + x)] (px2 + x) . = 2 sec2(px2 + x) tan(px2 + x) Solution to 1-(d): f (x) = 3arccos x.

Related Questions

1. Suppose marginal cost of abating X units of pollution is given by: MC(X) = 2X and marginal benefits are MB(X) = 16-2x when X<= 3 and 10 when X > 3. Current pollution = 6.

a.) What is the optimal level of pollution abatement, X*? How much pollution is there when X = X*?

COMMENT: For this part I set 2x = 16-2x and got 4 but 4>3 so I don't think that's a valid solution. So I set 2x = 10 when X >3 and got 5, what's the correct approach?

b.) What are total costs, marginal costs, and total net benefits at X = X*?

c.) What is the total net benefit of making (and encofring) a rule that bans pollution completely? What is the marginal benefit from removing the last little bit of pollution (i.e. the marginal benefit around the point X = Xmax)

d.) Suppose air pollution abatement in the U.S. is currently X* (that is, we currently have the ideal amount of air pollution given the costs and benefits of abatement). Now suppose that an effective vaccine against asthma is invented and given to everyone. Would this increase or decrease X*? Come up with your own example of a new technology that would move X* in the opposite direction and explain why it would have this effect.

Poster below me asked for irrelevant information that is not needed to answer the question.

Suppose there are two firms (firms 1 and 2) producing good X. Firm 1 has a marginal cost of $10 per unit
of X produced, while firm 2 has a marginal cost of $20 per unit of X produced. Suppose the two firms
decide to form a cartel and have the following collusive agreement: each firm will produce q = 30. In this
case, Q = q1 + q2 and that firms always receive the same price P at any given time. (10 marks)
a) What profit will each of the firms receive at the cartel agreement? (2)
b) Does either firm have an incentive to cheat (by increasing output to q = 40)? (2)
c) What is the Nash Equilibrium in this case? (3)
d) Is there anything Firm 2 can do to keep both firms producing according to the cartel agreement? (3)
Hint: Compare the profits Firms 1 and 2 earn in the cartel agreement and the Nash. Think back to the
Coase Theorem we looked at a few weeks ago…

maroonarmadillo294

Starting with the data fro Problem 6 and the data on the price of a related commodity for the years 1986 to 2005 given below, we estimated the regression for the quantity demanded a commodity (which we now reliable (Q x), on the price of the commodity (which we now reliable (Qx), on the price of the commodity (which we now live Px), consumer income (which we now label Y), and price of the related commodity (Pz), and we obtained the following results.

Qx=121.86-9.50Px+ 0.04Y - 2.21P

(-5.12) (2.18) (-0.68)

R2=0.9633 F=167.33 D-W=2.38

The year 1986 1987 1988 1989 1990
Pz($) 14 15 15 16 17
The year 1991 1992 1993 1994 1995
Pz($) 18 17 18 19 20
The year 1996 1997 1998 1999 2000
Pz($) 20 19 21 21 22
The year 2001 2002 2003 2004 2005
Pz($) 23 23 24 25 25

(b) Evaluate the regression results. 1.P15(b) is to evaluate the above regression results in terms of the signs of the coefficients, the statistical significance of the coefficients, and the explanatory power of the regression (R²). The number in parentheses below the estimated slope coefficients refer to the estimated t values. The rule of thumb for testing the significance of the coefficients is if the absolute t value is greater than 2, the coefficient is significant, which means the coefficient is significantly different from 0. For example, the absolute t value for Px is 5.12, which is greater than 2; therefore, the coefficient of Px, (-9.50) is significant. In order words, Px does affect Qx. If the price of the commodity X increases by $1, the quantity demanded (Qx) will decrease by 9.50 units.

apricotpig911

Economics 3364A/B Lecture Notes - Lecture 5: Product Rule, Quotient Rule

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