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Unformatted text preview: Math 250A (Kennedy)  Exam 3  Fall ’07
SHOW YOUR WORK. Correct answers with no work will get no credit. 1. (20 points) Find the solution of the differential equation and initial con—
dition: .7; xx
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buﬂ/ 9! ’9 Ji’C’ 4 2. (18 points) The graph of f is shown below. 0.7
0.4
0.3
0.2 0.1 i 0 0 0.125 0.25 0.375 0.5 0.625 0.75 0,875 1 a Compute the trapezoid and midpoint approximations for 2 :5 dx with
1 "Tm: '.2¥( i 7%) + %(.2r/*ﬂf/ + ﬂﬂh Wu) / _Zf(é(‘f().e'2f +‘o(+‘0,+é(:06)/ .4—
.2 024,02/ r: (b) Which of the two approximations in part (a) is an overestimate and which
is an underestimate? Funt‘h'oa \.‘ W‘W‘G'e “4/”? 4/" Map (‘1 an over erf'é‘h ' ‘ b m‘IApomi ;( an uncle €I‘IL\ q 3. (12 points )The growth of the population P (t) of Freedonia is shown in the
table. We want to model the growth with the diﬁerential equation 0171; : kP.
Use the data to estimate k. Explain your work. (There are several valid ways to do this.) 4. (12 points) A calculus professor jumps out of an airplane but forgets his
parachute. A good model for his descent is
d
m—U = mg — kv2 dt where t is time, 12 is velocity, 9 = 9.8m/sec2, and his mass m is 70kg. His
velocity increases as he falls, but it has become almost constant at 95m/5
when he hits the ground. What is k? Term—‘14:? Utlocr‘l'7/ Vol I? €?U\l'/ILP«D\W Jo(u74m~ : mj ” IQ ‘g’l ; y agec
m3 ?¢5)°7m0 07; Lil—.— m 5. (18 points) Consider the differential equation and initial condition —$ = go), 21(0) = 0.5 Some values of g are given in the table. Use Euler’s method with a step size
of h = 0.25 to estimate y(1). 5’0 : f
a 9 5"? .7“; _
Is: ‘f’f‘ ‘7”? (14‘)? '2973
7lgv’+\2r &(,37J') 1;, (Lng * (IQ? 6. (20 points) For the differential equation Ivy’  y1n(y/fv) — y = 0 (a) Show this is a differential equation with homogeneous coefﬁcients. 42 _ 2 i J; a, an A.) 4
(«ﬁlm i} g, c <0) 00 4), c 9: a“ C‘r C 1 gaj— (Me
re 5; f «x we / (b) Use the substitution u = y/x to solve the equation. Assume LE > 0 and
y > 0. /
Mfr/K“ mu ‘6’: a L‘— 0 v/ a 0 ...
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 Fall '11
 Kennedy
 Calculus

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