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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)