CALCULUS II PRACTICE FINAL

Last Updated Fall 2000

(1)
The initial value problem y¢ = Ö(1-y2),    y(p) = -1 has as its solution the function
(a) y = sin(t)
(b) y = sin(t-3p/2)
(c) y = tan(t)
(d) y = -(1/2)Ö{4-(t-p)2}
(2)
The improper integral   ò01  x-1  dx
(a) has value 1
(b) has value 0
(c) has value -1
(d) diverges
(3)  Evaluating the indefinite integral   òcos2(x) dx
(a)  gives us the family x/2 + sin(2x)/4 + C
(b)  gives us the family x/2 - cos(2x)/4 + C
(c)  gives us the family cos3(x)/3 + C
(d)  gives us the family -2sin(x) + C
(4)  Evaluating the definite integral   ò01  xex  dx
(a)  gives us 1
(b)  gives us -1
(c)  gives us 0
(d)  gives us e
(5)  The general solution to the differential equation y¢¢ = cos(t)
(a)  is the function y = -cos(t)
(b)  is the function family y = (1/3)cos3(t) + C, where C is any real number
(c)  is the function family y = sin(t) + Bt, where B is any real number
(d)  is the function family y = A + Bt - cos(t), where A and B are any real numbers
(6)  Which of the following is a Taylor series representation for f(t) = [2/(1+(3t))]?
(a)
åj = 0+¥ 2t^j, | t | < 1
(b)
åj = 0+¥ 2(t-1)^j,     0 < t < 2
(c)
åj = 0+¥ 2(-1)^j (3t)^j,     | t | < 1/3
(d)
åj = 0+¥ 2(-1)^j (t-3)^j,   2 < t < 4
(7)  The region enclosed between the curves y = x and y = Öx is revolved about the x-axis on the interval [1,2]. What is the volume of the resulting solid?
(a)  The volume is p
(b)  The volume is p[(13-8Ö2)/6]
(c)  The volume is 5p/6
(d)  The volume is p[(4-2Ö2)/3]
(8)  Consider the function f(x) = 1-x2 on the interval [0,1]. Any left-hand sum approximation for the value of
ò01 f(x) dx will be
(a)  an underestimate
(b)  an overestimate
(c)  exactly equal to the value
(d)  an over or underestimate, depending on the number of partitions used
(9)  The rate of change for a given quantity is directly proportional to the difference between the quantity present at any given time and the initial quantity Q0. If we let Q denote the quantity as a function of time, and if we assume Q0 ¹ 0, then a differential equation describing Q is
(a)Q¢ = kQ
(b)Q¢ = k(Q+Q0)
(c)Q¢ = k/(Q-Q0)
(d)Q¢ = k(Q-Q0)
(10)
The value of the series åj = 0+¥ (3 - 2 ^j) / (4 ^2j) is
(a)  not applicable; the series diverges
(b)  1/8
(c)  2
(d)  1/16
(11)  A function f is known to have the following properties: f(1) = 2, f ¢(1) = 1/2, and f ¢¢(1) = -1/3. The second-degree Taylor polynomial for f at x = 1 is therefore
(a)  T2(x) = 2 + (x-1)/2 - (x-1)^2 / 3
(b)  T2(x) = 2 + (x-1) / 2  - (x-1)^2 / 6
(c)  T2(x) = 2 + x / 2 - x^2 / 3
(d)  T2(x) = 2 + (x-1) / 4] - (x-1)^2 /1 8
(12)
The indefinite integral   ò(5x-7)  /  ((x-1)(x-2)) dx
(a)  gives us the function family 2ln| x-1 |+ 3ln| x-2 |+ C
(b)  gives us the function family 2ln| x-2 |+ 3ln| x-1 |+ C
(c)  gives us the function family 2ln| x-1 |+ 3ln| x-2 |+ C
(d)  gives us the function family 7ln| x-2 |- 5(x-1)^ -1 + C
(13)
The indefinite integral ò x^2cos(x) dx
(a)  gives us the function family C-x3cos(x)/3
(b)  gives us the function family 2xcos(x) - x2sin(x) + C
(c)  gives us the function family x2sin(x) + 2xcos(x) - sin(x) + C
(d)  gives us the function family x2cos(x) - 2xsin(x) - cos(x) + C
(14)  A solid has as its base the region enclosed between the x-axis and the curve y = sin(x) on the interval [0,p]. Vertical cross sections of this solid taken perpendicular to the x-axis are semi-circles. The definite integral expressing the volume of this solid is
(a)  ò0p sin3(x) dx
(b)  (p/4)ò0p sin2(x) dx
(c)  (1/4)ò0p sin3(x) dx
(d)  (p/8)ò0p sin2(x) dx
(15)  The definite integral expressing the arc-length of the curve y = x2 on the interval [1,2] is
(a)  ò12 x2 dx
(b)  pò12 x4 dx
(c)  ò12 Ö(1 + x4) dx
(d)  ò12 Ö(1+4x2) dx
(16)  A tank in the form of a right circular cone is filled with liquid having density r kilograms per cubic meter. If the tank is two meters tall and two meters wide at the top, then the mass of the liquid is
(a)    pr/2 kilograms
(b)  2pr/3 kilograms
(c)  4pr/3 kilograms
(d)    pr kilograms
(17)
The radius of convergence for the series åj = 0+¥ 3^j  (x-1)^j / j!
(a)  is infinite - the series converges for all real numbers
(b)  is zero - the series converges only for x = 1
(c)  is 1/2
(d)  is 2
(18)
The radius of convergence for the series åj = 1+¥ 2^j (x-3)^j / j
(a)  is infinite - the series converges for all real numbers
(b)  is zero - the series converges only for x = 3
(c)  is 1/2
(d)  is 2
(19)  The improper integral ò1+¥ dx / (1+x2)
(a)  diverges
(b)  has value p/2
(c)  has value 1
(d)  has value p/4
(20)  Consider the differential equation y¢ = Ö{1+cos(px)}. Starting at the point (0,1), and using Euler's method with four steps, we can estimate the value of y at x = 1 to be
(a)  1.68035
(b)  2.06565
(c)  1.7119
(d)  0.1353
(21)  The Taylor series representation for f(x) = x-1 at x = 1 is åj = 0+¥ (-1)^j (x-1)^j   for 0 < x < 2. Using this fact, the Taylor series representation for g(x) = ln(x) at x = 1 is
(a)
åj = 1+¥ j (-1)^j (x-1)^j-1,   0 < x < 2
(b)
åj = 0+¥ (-1)^j (x-1)^j+1 / (j+1) , 0 < x < 2
(c)
åj = 0+¥ (-1)^j (x-1)^j / (j+1) , 0 < x < 2
(d)
åj = 0+¥ (-1)^j (x-1)^j / j! , 0 < x < 2

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On 27 Nov 2000, 15:44.