University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2005/2006
Solutions #5
1. (a) To nd the equation of the tangent plane to the surface given by x3 z + x2 y 2
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2007/2008
Assignment #2
This assignment is due
October 2 October 4, 2007.
at
the
start
of
your
tutorial
in
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2015/2016
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
MAT B41H
2011/2012
Assignment #9
This assignment is due at
November 24 November 30, 2011.
the
start
of
your
tutorial
in
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 . C
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2007/2008
Solutions #1
2x2 + x 3 (2a2 + a 3)
f (x) f (a)
= lim
=
xa
xa
xa
xa
1. (a) If a nite limit exists,
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 +
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
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
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.
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
m
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2005/2006
Solutions #6
1. (a) f (x, y ) = x cos(xy 2). We rst compute the (rst order) partial derivatives.
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2011/2012
Assignment #7
This assignment is due at
November 10 November 16, 2011.
the
start
of
your
tutorial
in
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2005/2006
Solutions #7
x
1
=x
2 y2
1x
1 x2 + y 3
1
1
have
=
=
(1)n (t)n =
1t
1 + (t)
n=0
1. (a) f (x, y ) =
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 int
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
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
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
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
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2005/2006
Solutions #3
1.
x
1
x
. Domain is (x, y ) R2 | y = 0 .
= c x = cy or y = x,
y
y
c
c = 0. If c = 0
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2005/2006
Solutions #4
1. (a) A direction vector for the line is (2, 3, 1) (1, 1, 3) = (3, 4, 2), so a para
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2011/2012
Assignment #3
The Term Test will take place on Saturday, October 22, 9:00 am 11:00 am.
This assignme
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2011/2012
Assignment #4
The Term Test will take place on Saturday, October 22, 9:00 am 11:00 am.
This assignme
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2011/2012
Assignment #2
This assignment is due
September 29 October 5, 2011
at
the
start
of
your
tutorial
in
t
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2011/2012
Assignment #8
This assignment is due at
November 17 November 23, 2011.
the
start
of
your
tutorial
in
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2011/2012
Assignment #10
The Final Examination will take place on December 19, from 7 pm 10 pm in IC 130
This
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
i
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
i
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
i
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
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2017/2018
Assignment #2
This assignment is due at
September 25 September 29, 2017.
the
start
of
your
tutorial
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2017/2018
Solutions #3
1.
(i) f (x, y) = 3 2x2 3y 2 . Domain is R2 . Putting 3 2x2 3y 2 = c gives
2x2 + 3y 2 =
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2017/2018
Solutions #2
1. A = A2 = O = A2 A = A(AI). Hence 0 = det(A(AI) = det A det(AI) =
det A = 0 or det(A
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2017/2018
Assignment #5
The Midterm Test will be written on Monday, October 23, 5:00 7:00 pm. See the
informat
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2017/2018
Solutions #4
p
1. (a) If r = (x, y, z), r = r(x, y, z) = kr(x, y, z)k = x2 + y 2 + z 2 . Taking the