Calculus II Midterm 2 June, 2011 2 1. (3 marks each; total 9 marks) Answer each question in the space provided. You MUST show your work.a)Show that x x y cos 2 1 is a solution of the differential equation x y y sin . b)Given the parametric equations ttx 2 sin cos 2 and tty 2 cos sin 2 , find dx dy when 0 t.c)Kirchhoff’s Law (has to do with electrical circuits) gives us the differential equation Q dtdQ 4 12 . If0 ) 0 ( Q , use Euler’s method with step size 1 . 0 h to estimate ) 3 . 0 ( Q .
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Calculus II Midterm 2 June, 2011 2
1. (3 marks each; total 9 marks) Answer each question in the space provided. You
MUST show your work.
a) Show that x x y cos2
1 is a solution of the differential equation x y y sin .
b) Given the parametric equations t t x 2sincos2 and t t y 2cossin2 ,
finddx
dywhen 0t .
c) Kirchhoff’s Law (has to do with electrical circuits) gives us the differential
equation Qdt
dQ412 . If 0)0( Q , use Euler’s method with step size
1.0h to estimate )3.0(Q .
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Calculus II Midterm 2 June, 2011 3
2. (5 marks) Find the slope of the tangent line to the polar curve lnr at the
point e .
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Calculus II Midterm 2 June, 2011 4
3. (9 marks) Suppose that a corpse was discovered in a motel room at midnight and its
temperature was F
80 . The temperature of the room is kept constant at F
60 . Two
hours later the temperature of the corpse dropped to F 75 . Find the time of death. Note:
The temperature of a corpse at time of death is F 6.98
Hint: Newton’s Law of Cooling is given by )( sT T k dt
dT
where T s is the
temperature of the surroundings.
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Calculus II Midterm 2 June, 2011 5
4. (6 marks) Find the area of the region that lies inside the curve sin3r and
outside sin1r .
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Calculus II Midterm 2 June, 2011 6
5. (Total 15 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 some cases if the final answer is wrong).