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Unformatted text preview: 03%“ E Calculus II Midterm 2 March, 2009 2 1. (3 marks each; total 9 marks) Answer each question in the space provided. You
MUST show your work. a) A tank contains 100 L of brine, with a concentration of 0.3 kg of salt per liter.
Pure water enters the tank at a rate of 5 L/min. The solution is kept thoroughly
mixed and drains from the tank at the same rate. What is the amount of salt in the
tank, A(Z), as a function of time I? OQA us our A
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following initial value problem: y'=x—y, y(1)=0 wPCNgl=§jl= X—‘g
i3(\):0 :2 x0: A, 130:0 x‘: x°+ln= [+O.\= l.\ 3‘: $04, (A Raw.) —. 04040—0): ot\ XL: X\\l’\ = \.\+o.\=\\i (6° ‘31?) 30.2)) ‘31: \Aerbx Pm,qu o.\+o.\ (IA0.0: 0.1 so $0.1): L31=01 l _ _ ' dy _ ﬁ/
c) If x — cost and y — 2s1n(3l), ﬁnd Ewhen l— f6. Calculus II Midterm 2 March, 2009 3 2. (5 marks) Suppose the average time waiting in line to purchase movie tickets at some
CinepleX is 8 minutes. What is the probability of waiting more than 10 minutes? ~¥A HWL ﬂ
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2', {M5 Calculus II Midterm 2 March, 2009 4 3. (9 marks) Suppose that a corpse was discovered in a motel room at midnight and its
temperature was 80°F . The temperature of the room is kept constant at 60°F . Two
hours later the temperature of the corpse dropped to 75°F . Find the time of death. Note:
The temperature of a corpse at time of death is 98.6°F dT
Hint: Newton’s Law of Cooling is given by E : k(T _ where T S is the temperature of the surroundings.
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LN 3c, ’\‘.Ob 97:240M. Calculus II Midterm 2 March, 2009 5 4. (9 marks) Find the area of one loop of the polar curve 7" Z 2 005(36) .
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3‘76 Calculus II Midterm 2 March, 2009 6 5. (2 marks each; total 8 marks) Indicate whether each of the following statements is
true (T) or false (F) and JUSTIFY your answer. You will receive credit only if your
answer is correct AND properly justiﬁed. a) All solutions of the differential equation y' : y2 + 5 are increasing ﬁinctions. Answer: T Q“ g /
Justiﬁcation: % = 31413 7 O b) y : sin x + cosx is a solution of the differential equation y ' — y : 0. Answer: (PALSg Justiﬁcation: \al/tk : CCOOX ‘MXB — (Mxk—me) = »,Lm}««>c 7‘ O. c) The parametric curve shown below represents the parametric equations
x:cost, yzlz. Answer: F ALSg Justiﬁcation: on (:(LMH , xeL—‘Lrﬂ , bur xr—cooireLMQ .
(ALSO wlmlmo, ya: x: 4,310 , bUT C\,o) hat on THE crews d) The two parametric curves y : 3 sin I and x : cost can be converted to the
2 Cartesian equation x2 + y— : 1? 9 Answer: T QUE  — 1   1 1
Justiﬁcation: Mk: A; 7 30 M1}: Jrcm k_ A Q; + X Calculus II Midterm 2 March, 2009 7 6. (Total 10 marks) Answer each of the following in the space provided. You do NOT
have to show your work for this question, but you may do so if you wish (e. g. for part
marks in case the ﬁnal answer is wrong). a) (2 marks) If f(X) = 800x is the force necessary to extend a spring X meters
beyond its natural length of 0.1 meters, calculate the work needed to stretch the spring to 0.12 meters. 0,0; 0‘01
OAQ ( W :& %Qox;l>< = LKQQXL] s Lm«L0.01)L—__Q_\C
O O b) (2 marks) Set up an integral that calculates the volume of the solid obtained by
rotating the curve y : of}, where O S x S 1, about the xaXis. S‘TTX (ix A(><)=‘IT(\V?<)L d
c) (2 marks) Find the length ofthe curve X = 008249, y = sin26, O S 6 S 7! . 1,.
) 1&912W— %%=—Lﬁﬂe, ﬁgcamﬁze
O l C935: )1’rf‘ﬁaf31 #134. d) (2 marks) Consider the direction ﬁeld shown below. fiffff “Hahn ##M/fff
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y e) (2 marks) Find a Cartesian equation for the curve represented by I” = 2 . _
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 Spring '08
 Mihai
 Calculus

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