University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2011/2012
Assignment #9
This assignment is due at
November 24 November 30, 2011.
the
start
of
your
tutorial
in
the
period
A. Suggested reading: Marsden & Tromba, Cha
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2007/2008
Solutions #5
1. Marsden & Tromba, page 140, #10.
f(x, y) = x2 + y 2 and g(x, y) = x2 y 2 + xy 3. At (x, y) = (0, 0), we have fx (0, 0) =
= 0, fy (0, 0)
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2010/2011
Solutions #5
1. We will follow the hint. Away from (0, 0), a little long division gives
xy 2 x2 y + 3x3 y 3
2x3
= x y + 2
.
Hence we can rewrite f (x, y) a
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2005/2006
Solutions #8
2
2
2
1. (a) f (x, y ) = x (y 1) 3 = x 3 (y 1) 3 . f is dened
for all (x, y ) R2 . Computing the partials we
2
f = 2(y 1) 3 = 0
x
1
3x 3
h
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2005/2006
Solutions #9
y
1. We write both equations as functions of y giving x = 1 and x = y 2. We will
2
minimize the square of the distance between a point on t
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2008/2009
Assignment #5
The Term Test will take place on Friday, October 31, 7:00 pm 9:00 pm.
This assignment is due at
October 21 October 24, 2008.
the
start
of
you
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2010/2011
Assignment #4
The Term Test will take place on Friday, October 29, 7:00 pm 9:00 pm.
This assignment is due
October 19 October 21, 2010.
A. Suggested readin
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 27, 2012
Duration: 110 minutes
1. [12 points]
(a) Dene the following t
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2010/2011
Assignment #6
This assignment is due at
November 9 November 11, 2010.
the
start
of
your
tutorial
in
the
period
A. Suggested reading: Marsden & Tromba, Chap
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2010/2011
Assignment #8
This assignment is due at
November 23 November 25, 2010.
the
start
of
your
tutorial
in
the
period
A. Suggested reading: 1. Marsden & Tromba,
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2010/2011
Assignment #7
This assignment is due at
November 16 November 18, 2010.
the
start
of
your
tutorial
in
the
period
A. Suggested reading: Marsden & Tromba, Cha
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
x2 + 2x x2 2x
|y
x
= lim
x
= lim
x
x2
+ 2x
x2
x
2|y +
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
3
4
2 + 3 13 = 3 32 + 17 + 16.
To solve we need Newt
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2010/2011
Solutions #4
1. (a) det A = x det
xy yz + xz 2 .
y z
z 2
0 + det
y
y
x z
= x (2y + z 2 ) + (yz xy) =
(b) From part (a) we have f (x, y, z) = xyyz+xz 2 , s
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 have x = 0, the yaxis. If c = 0 we have a family of parab
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2010/2011
Solutions #6
1. f (x, y, z, w) = exyz sin(xw)
f
x
=
yz exyz sin(xw) + w exyz cos(xw),
2f
z x
=
(y + xy 2 z) exyz sin(xw)
3f
= (xy + x2 y 2 z) exyz cos(xw
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2010/2011
Solutions #7
1. (a) f (x, y) = x3 x2 + y 3 y 2.
f
f
= 3x2 2x and
= 3y 2 2y so
x
y
x(3x 2) = 0
. Solving gives critical
y(3y 2) = 0
points (0, 0), (0, 2 ),
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2010/2011
Assignment #3
This assignment is due
October 12 October 14, 2010.
at
the
start
of
your
tutorial
in
the
period
A. Suggested reading: Marsden & Tromba, Chapt
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2005/2006
Solutions #10
| x|
1
1. (a)
1
1
1
2|x|
| x|
ex+y dy dx.
e
dy dx
=
2|x|
1
=
1
1
1
x+y
1
ex+|x| ex2|x| dx
| x|
ex+y 2|x|
y
dx
0
(e 0
=
1
-1
-1
1
1
3x 0
e
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2005/2006
Solutions #11
1. Fixing x and y , we have 4x2 + y 2 z 2 y 2 (see the picture). The projection into
the xy plane is given by cfw_ (x, y ) | 4x2 + y 2 2 y
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2005/2006
Assignment #4
This assignment is due at
October 18 October 20, 2005.
the
start
of
your
tutorial
in
the
week
of
A. Suggested reading: Marsden & Tromba, C
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2005/2006
Assignment #5
This assignment is due at
October 25 October 27, 2005.
the
start
of
your
tutorial
in
the
week
of
A. Suggested reading: 1. Marsden & Tromba
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2005/2006
Assignment #3
This assignment is due at
October 11 October 13, 2005.
the
start
of
your
tutorial
in
the
week
of
A. Suggested reading: Marsden & Tromba, C
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2005/2006
Assignment #2
This assignment is due
October 4 October 6, 2005.
at
the
start
of
your
tutorial
in
the
week
of
A. Suggested reading: Marsden & Tromba, Cha
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2005/2006
Assignment #6
The Midterm Test will take place on Friday, October 28, 7:00 pm 9:00 pm.
Term Test Room Assignments
Surname
go to room
A to H
I to Z
HW215
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2005/2006
Assignment #7
This assignment is due at
November 8 November 10, 2005.
the
start
of
your
tutorial
in
the
week
of
A. Suggested reading: Marsden & Tromba,
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2005/2006
Assignment #8
This assignment is due at
November 15 November 17, 2005.
the
start
of
your
tutorial
in
the
week
of
A. Suggested reading: Marsden & Tromba,
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2005/2006
Assignment #9
This assignment is due at
November 22 November 24, 2005.
the
start
of
your
tutorial
in
the
week
of
A. Suggested reading: 1. Marsden & Trom