CALCULUS II PRACTICE FINAL
Last Updated Fall 2000
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(1)
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The initial value problem y¢ =
Ö(1-y2),
y(p) = -1 has as its solution the function
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(a) y = sin(t)
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(b) y = sin(t-3p/2)
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(c) y = tan(t)
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(d) y =
-(1/2)Ö{4-(t-p)2}
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(2)
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The improper integral
ò01
x-1 dx
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(a) has value 1
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(b) has value 0
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(c) has value -1
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(d) diverges
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(3) Evaluating the indefinite integral
òcos2(x) dx
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(a) gives us the family x/2 + sin(2x)/4 + C
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(b) gives us the family x/2 - cos(2x)/4 + C
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(c) gives us the family cos3(x)/3 + C
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(d) gives us the family -2sin(x) + C
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(4) Evaluating the definite integral
ò01 xex
dx
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(a) gives us 1
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(b) gives us -1
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(c) gives us 0
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(d) gives us e
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(5) The general solution to the differential equation
y¢¢ = cos(t)
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(a) is the function y = -cos(t)
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(b) is the function family y = (1/3)cos3(t) + C, where C
is any real number
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(c) is the function family y = sin(t) + Bt, where B is any real number
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(d) is the function family y = A + Bt - cos(t), where A and B are any
real numbers
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(6) Which of the following is a Taylor series representation for f(t)
= [2/(1+(3t))]?
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(a)
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åj =
0+¥
2t^j,
| t | < 1
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(b)
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åj =
0+¥
2(t-1)^j, 0 <
t < 2
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(c)
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åj =
0+¥
2(-1)^j (3t)^j,
| t | < 1/3
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(d)
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åj =
0+¥ 2(-1)^j
(t-3)^j, 2 < t
< 4
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(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?
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(a) The volume is p
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(b) The volume is
p[(13-8Ö2)/6]
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(c) The volume is 5p/6
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(d) The volume is
p[(4-2Ö2)/3]
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(8) Consider the function f(x) = 1-x2 on the interval [0,1].
Any left-hand sum approximation for the value of
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ò01 f(x) dx
will be
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(a) an underestimate
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(b) an overestimate
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(c) exactly equal to the value
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(d) an over or underestimate, depending on the number of partitions
used
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(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
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(a)Q¢ = kQ
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(b)Q¢ = k(Q+Q0)
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(c)Q¢ = k/(Q-Q0)
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(d)Q¢ = k(Q-Q0)
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(10)
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The value of the series åj =
0+¥ (3 - 2
^j) / (4 ^2j) is
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(a) not applicable; the series diverges
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(b) 1/8
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(c) 2
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(d) 1/16
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(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
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(a) T2(x) = 2 + (x-1)/2 - (x-1)^2 / 3
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(b) T2(x) = 2 + (x-1) / 2 - (x-1)^2 / 6
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(c) T2(x) = 2 + x / 2 - x^2 / 3
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(d) T2(x) = 2 + (x-1) / 4] - (x-1)^2 /1 8
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(12)
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The indefinite integral ò(5x-7) /
((x-1)(x-2)) dx
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(a) gives us the function family
2ln| x-1 |+
3ln| x-2 |+ C
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(b) gives us the function family
2ln| x-2 |+
3ln| x-1 |+ C
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(c) gives us the function family
2ln| x-1 |+
3ln| x-2 |+ C
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(d) gives us the function family
7ln| x-2 |- 5(x-1)^
-1 + C
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(13)
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The indefinite integral ò
x^2cos(x) dx
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(a) gives us the function family C-x3cos(x)/3
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(b) gives us the function family 2xcos(x) - x2sin(x) + C
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(c) gives us the function family x2sin(x) + 2xcos(x) - sin(x)
+ C
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(d) gives us the function family x2cos(x) - 2xsin(x) - cos(x)
+ C
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(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
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(a)
ò0p
sin3(x) dx
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(b)
(p/4)ò0p sin2(x) dx
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(c)
(1/4)ò0p
sin3(x) dx
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(d)
(p/8)ò0p sin2(x) dx
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(15) The definite integral expressing the arc-length of the curve y
= x2 on the interval [1,2] is
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(a) ò12
x2 dx
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(b) pò12
x4 dx
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(c) ò12
Ö(1 + x4) dx
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(d) ò12
Ö(1+4x2) dx
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(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
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(a) pr/2 kilograms
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(b) 2pr/3 kilograms
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(c) 4pr/3 kilograms
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(d) pr kilograms
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(17)
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The radius of convergence for the series
åj =
0+¥ 3^j
(x-1)^j / j!
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(a) is infinite - the series converges for all real numbers
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(b) is zero - the series converges only for x = 1
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(c) is 1/2
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(d) is 2
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(18)
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The radius of convergence for the series
åj =
1+¥ 2^j
(x-3)^j / j
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(a) is infinite - the series converges for all real numbers
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(b) is zero - the series converges only for x = 3
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(c) is 1/2
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(d) is 2
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(19) The improper integral
ò1+¥
dx / (1+x2)
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(a) diverges
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(b) has value p/2
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(c) has value 1
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(d) has value p/4
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(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
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(a) 1.68035
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(b) 2.06565
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(c) 1.7119
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(d) 0.1353
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(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
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(a)
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åj =
1+¥ j (-1)^j
(x-1)^j-1, 0 < x
< 2
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(b)
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åj =
0+¥ (-1)^j
(x-1)^j+1 / (j+1) , 0 < x
< 2
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(c)
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åj =
0+¥ (-1)^j
(x-1)^j / (j+1) , 0 < x
< 2
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(d)
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åj =
0+¥ (-1)^j
(x-1)^j / j! , 0 < x
< 2
File translated from TEX by
TTH,
version 2.00.
On 27 Nov 2000, 15:44.