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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).
a) (1 mark) Plot the polar point
4,2
.
Answer: _____________________________________________________
b) (2 marks) Consider the parametric equations t t yt x )sin(4,ln . What is the
corresponding Cartesian equation.
Answer: _____________________________________________________
c) (2 marks) Set up (but do NOT evaluate) the integral for the volume of the solid
obtained by rotating the area between the curves2
x y and x y 2 about the x-
axis.
Answer: _____________________________________________________
d) (2 marks) Set up, but do not evaluate, an integral that represents the arc length
of the curve sin5 r for3
5
6
.
Answer: _____________________________________________________
e) (1 mark) Convert the Cartesian equation 3 y x to a polar equation
Answer: _____________________________________________________
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Calculus II Midterm 2 June, 2011 7
f) (2 marks) The direction field shown below corresponds to the differential
equation2
)2(
y
x x
dx
dy. (Answer true (T) or false (F) and JUSTIFY your answer)
Answer: _________ Justification:
g) (1 mark) State the order of the following differential equation: xe y y 5)(2 .
Answer: __________________________________________________________
h) (2 marks) The following graph corresponds to the following set of parametric
curves t x cos10 and t y sin . (Answer true (T) or false (F) and JUSTIFY
your answer)
Answer: _________ Justification:
i) (2 marks) The ode 03ln y x y y is separable. (Answer true (T) or
false (F) and JUSTIFY your answer)
Answer: ________ Justification:
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Calculus II Midterm 2 June, 2011 8
6. (6 marks) Let
elsewhere
x xkx x f
0
10)1()(
2
(a) For what value of k is f a probability density function? (3 marks)
(b) Find the mean. [Note: if you did not complete part a), just choose an arbitrary k value.] (3 marks)
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