Review 2
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Review 2

Course Number: MAC 2233, Spring 2011

College/University: University of Florida

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MAC2233 Chapter 3 Review 1. Find the derivative of h(x) = (x 2)(2x + 3). 2. Find the derivative of f (t) = 3t4 2 + t5 . 2 t 2 4x 3. Find the derivative of f (x) = 3x x+2+2 . 2 2 t 4. Find the equation of the tangent line to y = t4 3+2+1 at t = 0. t2 5. The total worldwide box-oce receipts for a long-running movie are approximated by the function T (x) = 120x2 x2 + 4 where T (x) is measured in millions of...

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Chapter MAC2233 3 Review 1. Find the derivative of h(x) = (x 2)(2x + 3). 2. Find the derivative of f (t) = 3t4 2 + t5 . 2 t 2 4x 3. Find the derivative of f (x) = 3x x+2+2 . 2 2 t 4. Find the equation of the tangent line to y = t4 3+2+1 at t = 0. t2 5. The total worldwide box-oce receipts for a long-running movie are approximated by the function T (x) = 120x2 x2 + 4 where T (x) is measured in millions of dollars and x is the number of years since the moviess release. How fast are the total receipts changing two years after its release? 6. Find the derivative of g (u) = u5 3u2 + 5u 6. 7. Find the derivative of h(t) = (t2 6)2 (t2 + 5)3 . x2 +2 8. Find the derivative of f (x) = 2 . x +1 9. Find the rst, second, and third derivatives of f (x) = 3x2 + 7x 2. 10. Find the rst and second derivatives of f (x) = x2x . +1 11. The population of Americans age 55 yr and older as a percentage of the total population is approximated by the function f (t) = 10.72(0.9t + 10)0.3 (0 t 20) where t is measured in years, with t = 0 corresponding to 2000. Compute f (10) and interpret your result. 1 12. Use implicit dierentiation to nd y in the equation y 2 +2xy + x2 = 0. 1 1 13. Use implicit dierentiation to nd y in the equation x3 + y 3 = 5x. 14. Suppose the wholesale price of a certain brand of medium-sized eggs p (in dollars/carton) is related to the weekly supply x (in thousands of cartons) by the equation 625p2 x2 = 100. If 25,000 cartons of eggs are available at the beginning of a certain week and the price is falling at the rate of 2 cents/carton/week, at what rate is the supply falling? 15. Two cars start at the same point. One travels north at a constant speed of 45 mph and the other travels east at a constant speed of 50 mph. At what rate is the distance between the cars increasing three hours later? 16. Given y = f (x) = 2x + 1 determine the dierential of the function. For the values x = 4, dx = x = 0.1, calculate f (x + x), dy, y , and the value of the approximation to the function at x + x. 17. The relationship between Cunningham Realtys quarterly prots, P (x), and the amount of money x spent on advertising per quarter is described by the function P (x) = (1/8)x2 + 7x + 30 (0 x 50) where both P (x) and x are measured in thousands of dollars. Find the increase in prots when the advertising expenditure each quarter is increased from $24,000 to $26,000. Use dierentials to approximate this increase. 2 Solutions: 1. h (x) = d dx d dx (x 2)(2x + 3) = (2x2 + 3x 4x 6) = d dx (2x2 x 6) = 4x 1. Or using the product rule: h (x) = d dx (x 2)(2x + 3) = (x 2) (2x + 3) + (x 2) (2x + 3) = (1)(2x + 3) + (x 2)(2) = 4x 1. 2. f (t) = d dt (3t4 2 + t5 ) = 2 t d dt (3t4 2t1 + 5t2 ) = 12t3 + 2t2 10t3 . 2 (3x2 4x+2) (x2 +2)(3x2 4x+2)(x2 +2) d 4x 3. f (x) = dx 3x x+2+2 = 2 (x2 +2)2 = (6x4)(x2 +2)(3x2 4x+2)(2x) (x2 +2)2 = 4x2 +8x8 (x2 +2)2 = = (6x3 4x2 +12x8)(6x3 8x2 +4x) (x2 +2)2 4(x2 +2x2) (x2 +2)2 . 2 t d 4. y = dt t4 3+2+1 = t2 (t2 +2) (t4 3t2 +1)(t2 +2)(t4 3t2 +1) (t4 3t2 +1)2 = (2t)(t4 3t2 +1)(t2 +2)(4t3 6t) (t4 3t2 +1)2 = 2t5 8t3 +14t (t4 3t2 +1)2 = = 2t5 6t3 +2t4t5 +6t3 8t3 +12t (t4 3t2 +1)2 2t(t4 +4t2 7) (t4 3t2 +1)2 . 2(0)((0)4 +4(0)2 7) = 0 and the equation of the tangent Thus y (0) = ((0)4 3(0)2 +1)2 line at this point is given by y = y (0)t + b = 0 t + b = b. To nd the value (0)2 +2 of b, we note that y (0) = (0)4 3(0)2 +1 = 2 so that the point on the curve corresponding to t = 0 is (0, y (0)) = (0, 2); inserting this into the equation of the tangent line yields 2 = b. Thus the tangent line is the horizontal line y = 2. 5. T (x) = = (120x2 ) (x2 +4)(120x2 )(x2 +4) (x2 +4)2 240x3 +960x240x3 (x2 +4)2 = = (240x)(x2 +4)(120x2 )(2x) (x2 +4)2 960x (x2 +4)2 . 3 960(2) T (2) = ((2)2 +4)2 = 30 million dollars/year. 6. g (u) = d du (u5 3u2 + 5u 6)(1/2) = (1/2)(u5 3u2 + 5u 6)(1/2) (u5 3u2 + 5u 6) 4 = (1/2)(u5 3u2 + 5u 6)(1/2) (5u4 6u + 5) = 55u 62u+5 2 u 3u +5u6 . 7. h (t) = [(t2 6)2 ] (t2 + 5)3 + (t2 6)2 [(t2 + 5)3 ] = [2(t2 6)(t2 6) ](t2 + 5)3 + (t2 6)2 [3(t2 + 5)2 (t2 + 5) ] = [2(t2 6)(2t)](t2 + 5)3 + (t2 6)2 [3(t2 + 5)2 (2t)] = 4t(t2 6)(t2 + 5)3 + 6t(t2 6)2 (t2 + 5)2 = (t2 6)(t2 + 5)2 [4t(t2 + 5) + 6t(t2 6)] = (t2 6)(t2 + 5)2 [4t3 + 20t 6t3 + 36t] = (t2 6)(t2 + 5)2 [10t3 16t] = 2t(t2 6)(t2 + 5)2 (5t2 8). 2 [x2 +2] (x2 +1)(1/2) (x2 +2)[(x2 +1)(1/2) ] d 8. f (x) = dx (x2x +2 /2) = (1 +1) [(x2 +1)(1/2) ]2 = [2x](x2 +1)(1/2) (x2 +2)[(1/2)(x2 +1)(1/2) (2x)] x2 +1 = 2x(x2 +1)(1/2) x(x2 +2)(x2 +1)(1/2) x2 +1 = 2x(x2 +1)(1/2) x(x2 +2)(x2 +1)(1/2) x2 +1 = (x2 +1)(3/2) x3 9. f (x) = d dx (x2 +1)(1/2) (x2 +1)(1/2) = 2x(x2 +1)x(x2 +2) (x2 +1)(3/2) . (3x2 + 7x 2) = 6x + 7. f (x) = d dx f (x) = f (x) = d dx f (x) = 10. f (x) = d (6x dx d (6) dx + 7) = 6. = 0. (x) (x2 +1)(x)(x2 +1) (x2 +1)2 = (1)(x2 +1)(x)(2x) (x2 +1)2 4 = x2 +12x2 (x2 +1)2 = x2 +1 (x2 +1)2 . f (x) = = (x2 +1) (x2 +1)2 (x2 +1)[(x2 +1)2 ] (x2 +1)4 2x(x2 +1)(x2 +12x2 +2) (x2 +1)4 = 2x(x2 +3) (x2 +1)3 = (2x)(x2 +1)2 (x2 +1)(2)(x2 +1)(2x) (x2 +1)4 = 2x(x2 3) (x2 +1)3 . 11. f (t) = 10.72(0.3)(0.9t + 10)0.7 (0.9t + 10) = 10.72(0.3)(0.9t + 10)0.7 (0.9) = 2.894(0.9t + 10)0.7 . f (t) = 2.894(0.7)(0.9t + 10)1.7 (0.9t + 10) = 2.894(0.7)(0.9t + 10)1.7 (0.9) = 1.823(0.9t + 10)1.7 . f (10) = 0.01222 percentage/year2 ; Since f (t) > 0, the second derivative (because it is negative) tells us that the rate of increase in the percentage of older Americans is decreasing by 0.01222 percentage/year per year in 2010. 12. d d2 (y + 2xy + x2 ) = (0) dx dx d d d2 (y ) + (2xy ) + (x2 ) = 0 dx dx dx 2yy + (2x) (y ) + (2x)(y ) + 2x = 0 2yy + 2y + 2xy + 2x = 0 2yy + 2xy = 2y 2x y (2y + 2x) = 2y 2x y = (2y 2x)/(2y + 2x) = 1. 13. d d 3 (x + y 3 ) = (5x) dx dx d 3 d 3 d x+ y= (5x) dx dx dx 3x4 3y 4 y = 5 3y 4 y = 5 + 3x4 5 y = (y 4 /3)(5 + 3x4 ). 14. In this problem we assume that both p and x are functions of time, t, where time is given in weeks. We proceed by dierentiating both sides of the given equation with respect to t and obtain d d [625p2 x2 ] = 100 dt dt d d2 d 625p2 x= 100 dt dt dt 1250pp 2xx = 0. The question is asking for the rate at which the supply of eggs is falling; in the equation above this would be x ; we are given that x = 25 (recall that x has units of thousands of cartons) and that p = 0.02 (recall that p has units of dollars/carton and the fact that the price is falling implies the negative sign for p ). To solve for x we need to know the value of p; this can be obtained by substituting x = 25 into the original equation; in so doing we nd 625p2 (25)2 = 100 625p2 (25)2 = 100 625p2 = 725 p = 29/5; hence we take p = 29/5 and placing all of the above values into the dierentiated equation obtain 1250( 29/5)(0.02) 2(25)x = 0 5 29 50x = 0 x = 29/10. Thus, the supply of eggs is falling at a rate of 29/10 thousands of cartons per week. 15. We can think of the paths of the cars as the sides of a right triangle with the distance between the two cars being given by the length of the hypotenuse. If we consider time to be given in hours, then after t hours the rst car has traveled a distance of 45t miles (the length of the rst side of the triangle) while the second car has traveled a distance of 50t miles (the length of the second side of the triangle). Thus using the Pythagorean Theorem, the distance, D(t), (in miles) between the two cars at time t is given by 6 D(t) = (45t)2 + (50t)2 = 2025t2 + 2500t2 = 4525 t = 5 181 t. The rate of increase of the distance between the two cars is given by the derivative with respect to time of D(t); therefore d 5 181 t = 5 181 dt where the units are given in mph; since the derivative is a constant, D (3) = 5 181 mph. D (t) = dy 1 16. Using the chain rule we obtain dx = 2x+1 so that the dierential dx is given by dy = df (x) = 2x+1 . f (4 + 0.1) = f (4.1) = dy = 0.1 2(4)+1 = 0.1 3 2(4.1) + 1 = = y = f (4.1) f (4) = 1 30 9.2 3.0332. 0.0333. 9.2 9= 9.2 3 0.0332. Using the formula f (x + x) f (x) + dy we nd f (4.1) f (4) + dy 3 + 0.0333 = 3.0333 which is very close to the actual value of the function as calculated above. 17. The actual quarterly increase in prots is given by P = P (26) P (24) = 127.5 126 = 1.5 thousand dollars. The approximation to the quarterly increase is given by dP with x = 24 and x = 26 24 = 2. Since dP = (1/4)x + 7 we obtain the dierential dx dP = [(1/4)x + 7]dx; inserting the above values gives dP = [(1/4)(24) + 7](2) = 2 thousand dollars. 7

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(2a)-P.1Growth Theory: Broad Outline1. Foundation: The Solow Model. Romer ch.1.- Basic version: Mechanics of production, savings, and capital accumulation.- Take technological progress for granted. Take population growth as given.- Extended versions:
UCSB - ECON - 240A
(2b)-P.1Applications of Growth Theory I:Growth Accounting Objective: Use empirical data on output, capital stocks, and labor supply, to interpret history(accounting), to compare across countries, or to make projections. Data sets: observations (Yt, K
UCSB - ECON - 240A
(2c)-P.1New Growth: The Economics of Ideas(Main reference: Jones ch.4-5.) Neoclassical growth modeling: Focus on capital accumulation New growth theory: Focus on technology, ideas, explaining economic growth "endogenously" rather than assuming a growth
UCSB - ECON - 240A
Optimal Growth in Continuous Time Key assumption: Households maximize utility over consumption- They choose an optimal path of consumption and asset accumulation.- They discount future utility at a fixed rate, called the rate of time preference; symbol
UCSB - ECON - 240A
Standard Optimal Control: The Hamiltonian Approach(3c)-P.1 General approach to control problems (Barro/Sala-i-Martin, Appendix A3).- Presented with key example: Problem of representative household (or social planner):Maximize U = cfw_e0 t(u[C(t )]
UCSB - ECON - 240A
Dynamic Properties of the Optimal Growth ModelI. Graphical Analysis Restate the key differential equations (in effective units for convenience): ! c = 1 (r % n % g % $ ) = 1 ( f ' (k ) % # % " % !g ) 1. Euler equation: c ! ! ! k = f (k ) " c " (n + g + !
UCSB - ECON - 240A
Digression: Discrete-Time Optimization[For now: As motivation for continuous time. For later: Preview of discrete-time macro.] Consider optimal consumption and capital accumulation problem over T periods:TU = t 1u(ct ) = u(c1 ) + u(c2 ) + . + T 1u(cT
UCSB - ECON - 240A
(204A -3e)-P.1Fiscal Policy:I. Government Spending Assumption: Public spending G per efficiency unit of labor.- Assume spending is tax-financed: T=G, lump-sum. - Here abstract from productivity and population growth (could be added) - Best interpret a
UCSB - ECON - 240A
(3f)-P.1Introduction to Money How does money fit into modern macro models? - Money M = = nominal units issued by the government; p = price level. - Consider discrete periods: Household hold money and interest-bearing assets:ct + at +1 + M t +1 / pt = w
UCSB - ECON - 240A
(4a)-P.1Part 4: Overlapping Generations Models Basic Version: Each birth-cohort lives for two periods, young and old age.- Individuals are identical except for their date of birth => Each generation has a representative agent.- Young individuals earn
UCSB - ECON - 240A
(4b)-P.1Overlapping Generations & Fiscal Policy Focus on Intergenerational Redistribution and other issues excluded in representative agent models. Assume lump-sum taxes:T1t= per-capita net taxes on the youngT2t+1 = per-capita net taxes on the old.
UCSB - ECON - 240A
(4c)-P.1Overlapping Generations & Dynamic InefficiencyMotivating example: Why inefficiency may be empirically relevant and deserves analysis. Assume log-utility, Cobb-Douglas production, depreciation >0 Steady state:=>rt +1 = f ' (k t +1 ) = k t +1