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
2y +
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 3x + 1 (a2 3a + 1)
= lim
=
xa
xa
xa
xa
(b) If a nite li
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
in
the
period
A. Suggested reading: Marsden & Tromba, C
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 say that f is dierentiable at
a U if the partial derivati
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
( x + 3)( x 2)
(5)(4)
(2 x + 1)( x + 2)
lim
=
= 4.
x4
(
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
period
A. Suggested reading: Marsden & Tromba, Chapter
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)
x2 3x + 1 (a2 3a + 1)
(b) If a nite limit exists, f (a)
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
week
of
A. Suggested reading: Marsden & Tromba, Chapte
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
1
1
1
lim
= lim
=
=
t0 t 1 + t (1 +
t0
(1)(1 + 1)
2
1
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 #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
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 th col
vj1
vj2
t
umn of V is . . Now the i j th entry
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
in
the
period
A. Suggested reading: Marsden & Tromba, C
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 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
Solutions #8
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 the
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
MAT B41H
2010/2011
Solutions #10
1. Fixing x and y, we have 4x2 + y 2 z 2 y 2 (see the picture). The projection into
the xyplane is given by cfw_ (x, y)  4x2 + y 2 2 y 2 =
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
2011/2012
Term Test Solutions
1. (a) From the lecture notes we have
Let f : U Rn Rk be a given function. We say that f is dierentiable at
a U if the partial derivati
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 22, 2011
Duration: 110 minutes
1. [8 points]
(a) Carefully complete th
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
FINAL EXAMINATION
MATB41H Techniques of the Calculus of Several Variables I
Examiner: E. Moore
1. [4 points] Evaluate
Date: December 10, 2010
Duration: 3 hours
x2 y
or show t
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2009/2010
Assignment #3
This assignment is due
October 8 October 9, 2009.
at
the
start
of
your
tutorial
in
the
period
A. Suggested reading: Marsden & Tromba, Chapter
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2009/2010
Assignment #6
This assignment is due at
November 5 November 6, 2009.
the
start
of
your
tutorial
in
the
period
A. Suggested reading: Marsden & Tromba, Chapt
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2009/2010
Assignment #4
The Term Test will take place on Wednesday, October 28, 5:00 pm 7:00 pm.
This assignment is due
October 15 October 16, 2009.
A. Suggested rea
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2009/2010
Assignment #5
The Midterm Test will take place on Wednesday, October 28, 5:00 pm 7:00 pm.
Term Test Room Assignments
Surname
go to room
A to J
K to Z
HW 21
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2009/2010
Assignment #7
This assignment is due at
November 12 November 13, 2009.
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
2009/2010
Assignment #8
This assignment is due at
November 19 November 20, 2009.
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
2016/2017
Solutions #9
1. (a)
Z
Z
1
0
0
2
3
Z
y
y2
1Z y
Z
x dx dy.
x dx dy
y2
1
y
0
3/2
y
1
=
Z
1
0
5/2
2 2y
y dy =
3
5
3
y
2 x3/2
dy =
3
y2
1
1
y4
= .
4 0 10
x = y
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2016/2017
Solutions #6
f
f
2
= 2 (x 1) e(x1) cos y,
(1, 0) = 0;
x
x
f
2f
2f
f
2
2
(x1)2
= e(x1) sin y,
(1, 0) = 0;
=
2
(2x
4x
+
3)
e
cos
y,
(1, 0) = 2;
y
y
x2
x2
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2016/2017
Solutions #7
f (x, y) = sin x cos y. For critical points
1. (a)
f
= cos x cos y = 0
y
x
0
. To solve we

f = sin x sin y = 0
y
1
first note that cos t a
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2016/2017
Solutions #5
y (x4 + 4 x2 y 2 y 4 )
f
(x, y) =
and
x
(x2 + y 2 )2
f
x (x4 4 x2 y 2 y 4 )
(x, y) =
.
y
(x2 + y 2 )2
1. (a) For (x, y) 6= (0, 0),
f
y (y 4 )
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2016/2017
Solutions #8
1. (a) f (x, y) = x + y subject to the constraint x2 + y 2 = 4. We define h(x, y, ) =
x + y (x2 + y 2 4). The critical points of h will give t