MATH 2720
Winter 2012
Assignment 5
Questions 3 and 4 were marked.
1. Using an appropriate change of variables, evaluate
ex
2 y 2
dA
R2
Solution. Of course, we switch to polar coordinates. We have
ex
2 y 2
2
dA =
R2
0
0
= 2
2
er r dr d
2
er r dr
0
0
= 2
1
MATH 2720
F12
Solutions 5
(1)(a)
(x2 +y 2 )
e
R
2
er r ddr
dA =
D
0
0
R
=
re
2
dr = er
2
r2
0
R
0
1 eR .
=
2
(1) (b) The volume is given by
coordinates.
D
(x2 +y 2 ) dA, which is easier to compute in polar
1
2
/2
2
r2 rddr
(x + y ) dA =
D
0
0
1
=
0
3
r
DEPARTMENT OF MATHEMATICS
MATH 2720 Multivariable Calculus
Winter 2016
Instructor:
Dr. A. Prymak
Office:
423 Machray Hall
Phone:
480-1246
Email:
[email protected] (you must use UofM email and include student
number; I normally respond to emails o
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MATH 2720
Winter 2012
Assignment 2
Comments
Questions 3, 5, 7 and 9 were marked. Each question is worth 5 marks, with 1 mark for legibility and
presentation.
1. Sketch the level curves of the function f : R2 R dened by f (x, y) = x3 y.
Solution. In the xy
MATH 2720
Winter 2012
Assignment 1 Solutions
Comments
Questions 1, 2, 6 and 9 were marked. Each question is worth 5 marks, with 1 mark for legibility and
presentation.
1. Describe, i.e., write inequalities for, the region in 3-space bounded by the sphere
MATH 2720
F12
Solutions 3
1. (a) We compute that
f (x, y) =
(b) At (2, 0) we have that
u = ( 3/2, 1/2)
x2
2y
2x
2
2
+ 2xyex y , 2
+ x2 e x y .
2
2
+y
x +y
f (2, 0) = (1, 4). Thus the directional derivative is (letting
=
f (2, 0) u = (1, 4)
Du f (2, 0) =
MATH 2720
Winter 2012
Assignment 4
Questions 1, 2, 6 and 8 were marked.
1. Show that the curve c(t) = (t2 , 2t1, t), t > 0, is a ow line of the vector eld F (x, y, z) = (y+1, 2, 1/2z).
Solution. The curve is a ow line if
c (t) = F (c(t).
We have
c (t) = (
MATH 2720
Winter 2012
Assignment 6
Exercises 1, 5 and 7 were marked.
1. Find a parametrization of the surface x2 y 2 = 1, where x > 0, 1 y 1 and 0 z 1. Use this to
express the area of the surface as an integral.
Solution. Let S be the surface under consid
MATH 2720 Review sheet. Questions were taken from Term Test 1 in the Winter 2009
term. There is no guarantee the term test this term will in any way resemble this or any
other past test.
1. Given r(t) = t, sin 3t, cos 3t .
(a) Sketch the curve
(b) Find th
MATH 2720 Review sheet. Questions were taken from the Winter 2008 and 2009 term.
There is no guarantee the term test this term will in any way resemble this or any other past
test.
1. Find the unit tangent vector T(t), the unit normal vector N(t) and the
UNIVERSITY OF MANITOBA
DATE: February 8, 2012
MIDTERM EXAMINATION 1
TITLE PAGE
COURSE: MATH 2720
TIME: 60 minutes
EXAMINATION: Multivariable Calculus
EXAMINER: J. Arino
INSTRUCTIONS TO STUDENTS:
This is a 60 minute exam.
Please show your work clearly. C
MATH 2720
Winter 2012
Assignment 3
Questions 2-5 were marked.
1. State the (, ) denition of the limit for a function f : Rn Rm .
Solution.
lim f (x) = L
xx0
( > 0, > 0, 0 < x x0 < f (x) L < ) .
2. Use polar coordinates and the (, ) denition of a limit on
MATH 2720
F12
Solutions 2
1. (a) Solving the initial value problem we have that the position as a function of time is
r(t) = (8t, 10 + 5t 5t2 ). This intersects the ground when y(t) = 10 + 5t 5t2 = 0.
This has two solutions t = 1 and t = 2. Since the ball
UNIVERSITY OF MANITOBA
DATE: March 7, 2012
MIDTERM EXAMINATION 2
TITLE PAGE
COURSE: MATH 2720
TIME: 120 minutes
EXAMINATION: Multivariable Calculus
EXAMINER: J. Arino
GENERAL COMMENTS:
Most of you did well on the course questions. These are easy question
MATH 2720 2013F Midterm Exam (1 hour)
1. Evaluate the limit, or Show that the limit does not exist.
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