MATH 1451, EXAM 3b (solution), 14 NOVEMBER, 2011
(Each of the following ve problems is worth 12 points, 6 points for each part. When in
doubt, draw a picture. BE SURE TO SHOW ALL YOUR WORK.)
(1) Be sure to use correct units in the following work problems.
Math 1451 Exam 1 Spring 2016
Solui’ims
Name: (4 points)
Math 1451 Section 102
Quiz Section Time:
Instructions:
1. Please ﬁll in your name and quiz section time.
2. You will have 50 minutes to complete the exam.
3. The use of electronic devices is prohibit
Math 1451 Calculus 2 Namagluhm—
May 3, 2016 Quiz Section Time: 8 AM 9 AM 10 AM
Group Activity 11
d
1. Consider the initial value problem i = 1 — m — y, y(0) = 0.
(a) Use Euler’s method with n = 2 intervals to estimate y(0.8).
AX: 056730: OJ, 5» 513:0] z.-
MATH 1451, QUIZ 3a (solution), 15 SEP., 2011
d
(1) Compute the derivative:
dx
x3
cos(t2 ) dt.
2
3
x
d
d
cos(t2 ) dt =
(F (x3 )
Suppose F (t) = cos(t ). Then
dx 2
dx
F (2) = 3x2 F (x3 ) = 3x2 cos(x3 )2 ) = 3x2 cos(x6 ).
2
(2) On the moon, the acceleration
MATH 1451, QUIZ 7a (solution), 27 OCT., 2011
(1) A probability density function is given by
p(x) =
0
ce2x
if x < 0
if x 0,
where c is a positive constant. What must c be?
1=
ce
dx =
lim
ce
2x
dx = lim
T
0
T
T
2x
c 2T c
c
e
+
= . Hence c = 2.
2
2
2
ce2x
MATH 1451, QUIZ 5a (solution), 06 OCT., 2011
(1) Does
1
3
9
x2
1
dx converge? And if so, what does it converge to?
B
2
2
2
= .
B 9 x
B x
B 3
3
9
B
x
9
1
(2) Use an appropriate comparison benchmark to show that
dx
2+x+1
x
1
converges.
3
2
dx = lim
1
3
2
dx
MATH 1451, QUIZ 6a (solution), 13 OCT., 2011
Set upbut do not solveintegrals that represent the volumes of the solids described
below.
(1) The area between the x-axis and the curve y = 4 x2 is rotated about the
y-axis.
After rotation, the solid has a at c
MATH 1451, QUIZ 10a (solution), 01 DEC., 2011
(1) Given that (x) = tan(x + C) is the general solution to the ODE
nd the specic solution that goes through the point 0, 1.
1 = tan(0 + C), so C = /4. Hence (x) = tan(x +
dy
= 1 + y2,
dx
).
4
dy
= 3x2 , y(0) =
MATH 1451, QUIZ 1a (solution), 01 SEP., 2011
(1) A meter on a water pipe reads ow ratesin gallons per hourof 50 at 6am and
of 100 at 9am. Assuming the ow rate to be increasing steadily (i.e., linearly)
during that time interval, how much water ows through
MATH 1451, QUIZ 11a (solution), 08 DEC., 2011
(1) If
dy
y
= and y(0) = 10, what is y(3)?
dx
3
1
t
dy
=
dx, or ln |y| = + C.
Separation of variables gives
y
3
3
t
3
Thus we get y = Ae for the general solution. When t = 0, y = 10;
t
hence A = 10 and y(t) =
MATH 1451, QUIZ 9a (solution), 10 NOV., 2011
(1) Find the radius and interval of convergence for the series
n=1
1
(n+1)2
lim
1
n
n2
(x 2)n
.
n2
1
= 1. Thus the radius of conver1
1+ n
gence is = 1/L = 1. The end points of the interval of convergence
(1)n
a
MATH 1451, QUIZ 2a (solution), 08 SEP., 2011
1
dt.
t2
1
t3 t1
t3 1
2
2
2
t 2 dt = (t t ) dt =
+ C = + + C.
t
3
1
3
t
dq
(2) Find the solution of the initial value problem
= 2 + sin(z), q(0) = 5.
dz
t2
(1) Evaluate the indenite integral
First antidierent
MATH 1451, QUIZ 4a (solution), 29 SEP., 2011
6
x2 dx.
(1) Compute MID(3) for the integral
0
Were subdividing an interval of length 6 into three equal pieces, so
x = 2. The endpoints of the subintervals are 0, 2, 4, and 6, so the
respective midpoints are 1
MATH 1451, EXAM 1b (solution), 22 SEPTEMBER, 2011
(Each of the following ve problems is worth 12 points, 6 points for each part. Be sure to
show all your work.)
(1) (a) Find the unique solution to the initial value problem
dy
1
1
= + , y(1) = 6.
dt
t t
Fi
MATH 1451, EXAM 2bd, 14,17 OCTOBER, 2011
(Each of the following ve problems is worth 12 points, 6 points for each part. Be sure to
show all your work. And, when in doubt, DRAW A PICTURE.)
3
(1) (a) Exactly one of the integrals
0
and why?
dx
and
x2 + 4
3
0
Convergence tests
Integral test: Suppose an = f (n), where f is both positive and decreasing. Then
Z
f (x) dx converges.
converges if and only if
X
an
1
1
Comparison test: Suppose 0 an bn for all n.
If
X
bn converges, then
n=1
If
X
X
an also converges.