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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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UCSB - ECON - 240A

Econ 204A - Midterm ExamFall 2010This exam is closed book. Most points are given for the correct set-up of a problem and foreconomically insightful interpretations.Problem 1 (50p)Consider a Solow model with general production function Y = F (K , AL)

UCSB - ECON - 240A

Note on Linear Differential EquationsEcon 204A - Prof. Bohn1 We will have to work with differential equations throughout this course. Differential equations and their discrete-time analogs: difference equations are economically interesting because they l

UCSB - ECON - 240A

Note on Linearized Solutions to the Optimal Growth ModelEcon 204A - Prof. Bohn This note reviews the linearized dynamics of the optimal growth model and derives log-linearized solutions. General Problem: Linearization Linearization is a common approach i

UCSB - ECON - 240A

Supplementary Note on the OG ModelEcon 204A - Prof. BohnHere are some comments on Romers exposition and on general OG dynamics. Read Romers Section2.9 carefully as it lays out the individual problem. In Romers Section 2.10, the key equation for thedyn

UCSB - ECON - 240A

Collection of Practice ProblemsEcon 204AHenning Bohn*In previous years, students have often asked me about practice problems in addition to the problemsets. Here is a collection. Some will be assigned for the weekly problem sets. I hope the others are

UCSB - ECON - 240A

(1.Intro)-P.1Econ 204A: Organization Class Page: www.econ.ucsb.edu/~bohn/204A/204Aindex.html- Information is updated throughout the quarter.- Check for announcements. Class page announcements are assumed known. Open door policy for graduate students.

UCSB - ECON - 240A

(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