University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2016/2017
Term Test Solutions
1. (a) (i) Let A Rn . A point a A is called an interior point of A if Br (a) A,
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2011/2012
Solutions #1
sin 1
sin 1
1
= lim 1 x = 1lim 1 x = 1.
x x x
0+
x
x
1. (a) lim x sin
x
f (x) f (a)
x2
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2013/2014
Term Test Solutions
1. (a) From the lecture notes we have
Let f : U Rn Rk be a given function. We sa
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
Midterm Test
MATB41H Techniques of the Calculus of Several Variables I
Examiner: E. Moore
Date: October 31, 2014
Durati
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2010/2011
Solutions #1
| y2 2 |
= lim
1. (a) (i) lim
y 2
y 2
y 2 + 2 2y + 2
y 2
= .
= lim
y 2 y +
2
(ii) lim
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2009/2010
Assignment #1
This assignment is due at
September 24 September 25, 2009.
the
start
of
your
tutorial
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2009/2010
Solutions #1
2x x + x 8 x 4 f actorize
(2 x + 1)( x + 2)( x 2)
1. (a) (i) lim
lim
=
=
x4
x4
x + x 6
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2009/2010
Assignment #2
This assignment is due
October 1 October 2, 2009.
at
the
start
of
your
tutorial
in
the
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2005/2006
Solutions #1
1. (a) lim x sin
x
sin 1
sin 1
1
= lim 1 x = 1lim 1 x = 1.
x x x
0+
x
x
f (x) f (a)
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2010/2011
Assignment #2
This assignment is due
October 5 October 7, 2010.
at
the
start
of
your
tutorial
in
the
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2009/2010
Solutions #9
1
y
1
y2
y
1. (a)
0
x dx dy.
1
x dx dy
y2
1
0
2
3
y
3/2
0
y
1
=
0
5/2
2 2y
y dy =
3
5
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2009/2010
Solutions #2
1. (a) Let v i = (vi1 , vi2 , , vin ) so the ith row of V is (vi1 vi2 vin ) and the j t
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2012/2013
Solutions #1
1
1 1+t
1+ 1+t
1 1+t
1
= lim
= lim
1. (a) lim
t0
t0 t 1 + t
t0
t 1+t t
t 1+t
1+ 1+t
t
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2010/2011
Solutions #3
1.
x
x
(i) f (x, y) = . Domain is (x, y) R2 | y > 0 . = c x2 = c2 y.
y
y
If c = 0 we ha
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2010/2011
Solutions #2
1 2
3
expand on (2) det
0
=
1. det(A I) = det 2
column 2
3
0 4
1
3
() det
= 2 8 + 2
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2011/2012
Assignment #1
This assignment is due at
September 22 September 28, 2011.
the
start
of
your
tutorial
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2012/2013
Solutions #8
1. (a) f (x, y ) = x2 + y 2 subject to the constraint g (x, y ) = 2x + 3y 6 = 0. We den
Let a = (a1, a2, , an) Rn, x = (x1, x2, , xn) Rn and
f : Rn R. L R is called the limit of f as x approaches a
lim f (x) = L if f (x) can be made arbitrarily close to L by
xa
taking x sufficiently clos
2
V 4 *
Vector valued functions
V 1
Acceleration and Newtons law
Earth
b
m
r
F
object
b
M
a> 4.1: Gravitational force is represented as vectors
Example 1.1. The force acting on a particle of mass m.
Definition:
Let A Mn(R). If a nonzero vector v Rn and
R have the property A v = v then is called an eigenvalue
of A and v is said to be an eigenvector of A associated to the
eigenvalue .
Theorem: If
2
V 7 *
Integrals over Paths and
Surfaces
V 1
Path Integral
Path integral
R2 or R3 . A parameterized curve c can be written as c(t) = (x(t), y(t), z(t). If
x(t), y(t), z(t) are continuous then we say
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2016/2017
Assignment #9
This assignment is due at
November 25 December 1, 2016.
the
start
of
your
tutorial
in
Definition:
The average value or mean value of an integrable
function f (x, y) over the set D is the number
ZZ
1
f = avD f =
f (x, y) dA .
area of D
D
Definition:
A set D in the plane is said to be co
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2016/2017
Assignment #8
This assignment is due at
November 18 November 24, 2016.
A. Suggested reading: 1.
2.
t
sh is
ar stu
ed d
y
v i re
aC s
o
ou urc
rs e
eH w
er as
o.
co
m
Th
https:/www.coursehero.com/file/12581323/MATB41-Week-08/
sh is
ar stu
ed d
y
v i re
aC s
o
ou urc
rs e
eH w
er as
o.
co
m
Th
https:/w
V 5 *
Double and Triple integrals
Double integral, triple integral,
V 1
Double integral as volumes
For the convenience of presentation we assume the domain of a function on a
rectangle R given by
R =
300
V 6 *
Change of variables
V 1
Geometry of Maps from R2 to R2
If x : [a, b] R is C 1 function and f : R R is integrable, then
b
Z
f (x(t)x (t)dt =
a
Z
x(b)
f (x)dx
(6.1)
x(a)
Maps of a region to a
Vector Calculus
2008
9Z4 4cfw_9
ii
1
Much of the material presented here and some figures are copied from Vector Calculus of J. Marsden and A. Tromba for teaching purpose only.
Please do not distribu
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2017/2018
Assignment #6
This assignment is due at
November 6 November 10, 2017.
the
start
of
your
tutorial
in
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2017/2018
Assignment #7
This assignment is due at
November 13 November 17, 2017.
the
start
of
your
tutorial
in
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2017/2018
Assignment #8
This assignment is due at
November 20 November 24, 2017.
A. Suggested reading: 1.
2.
t
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2017/2018
Assignment #9
This assignment is due at
November 27 December 1, 2017.
the
start
of
your
tutorial
in