University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2012/2013
Assignment #2
This assignment is due
January 28 February 1, 2013.
at
the
start
of
your
tutorial
in
the
period
B. Problems:
1. (a) Find the Fourier series o
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2012/2013
Assignment #1
This assignment is due
January 21 January 25, 2013.
at
the
start
of
your
tutorial
in
the
period
B. Problems:
1. For each of the following fun
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2012/2013
Assignment #5
The Term Test will take place on Wednesday, March 6, 5:00 7:00 pm .
This assignment is due
February 25 March 1, 2013.
at
the
start
of
your
tu
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2012/2013
Assignment #4
This assignment is due at
February 11 February 15, 2013.
the
start
of
your
tutorial
in
the
period
A. Suggested Reading: Marsden & Tromba, Cha
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2012/2013
Assignment #3
This assignment is due
February 4 February 8, 2013.
A. Suggested reading:
at
the
start
of
your
tutorial
in
the
period
1. Marsden & Tromba, Ch
UNIVERSITY OF TORONTO SCARBOROUGH DEPARTMENT OF COMPUTER AND MATHEMATICAL SCIENCES MATB42H Examiner: P. Selick Date: August 16, 2007 Duration: 3 hours Question 1. a) [ 7 points] Find the volume of the gure bounded by y = x2 , y + z = 4, and z = 0. b) [6 p
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
FINAL EXAMINATION
MATB42H Techniques of the Calculus of Several Variables II
Examiner: E. Moore
Date: April 26, 2012
Duration: 3 hours
1. [12 points] Carefully state the foll
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2012/2013
Term Test Solutions
0 , x <
2
1 , x < , we will rst nd the Fourier coecients.
1. Given f (x) =
2
2
0 ,
x<
2
2
2
1
1
even 2
f (x) dx =
a0 =
f (x) dx =
1
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
FINAL EXAMINATION
MATB42H Techniques of the Calculus of Several Variables II
Examiner: E. Moore
Date: April 18, 2011
Duration: 3 hours
1. [10 points] Let f (x) be dened by
1
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2011/2012
Solutions #10
f
f
f
dx +
dy +
dz .
x
y
z
f
f f f
f
f
f =
f=
=
,
,
,
,
e1 +
e2 +
e3 .
x y z
x y z
x
y
z
Both have the same component coecients one in terms
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2011/2012
Assignment #1
This assignment is due
January 26 January 27, 2012.
at
the
start
of
your
tutorial
in
the
period
B. Problems:
1. Show that
0
, k = n
.
, k =
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2011/2012
Assignment #3
This assignment is due
February 9 February 10, 2012.
A. Suggested reading:
at
the
start
of
your
tutorial
in
the
1.
Marsden & Tromba, Chapter
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2011/2012
Solutions #11
1. We can parametrize by (t) = t2 cos t +
, t2 sin t +
2
2
2
(t) = 2t cos t +
t sin t +
, 2t sin t +
+
2
2
2
t2 cos t +
and (t) =
2
(4t2 +
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2011/2012
Assignment #2
This assignment is due
February 2 February 3, 2012.
at
the
start
of
your
tutorial
in
the
period
B. Problems:
1. Find the Fourier series of th
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2011/2012
Solutions #7
1. (a) The surface S is given by (u, v ) = (u2 +
v, v, u + v 2 ). Now u = (2u, 0, 1), v =
(1, 1, 2v ) and u v = (1, 1 4uv, 2u). To
nd the valu
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2011/2012
Solutions #8
1. (a) In the xz plane the line segment from (3, 0) to (2, 1) can be given by (3 t, t),
0 t 1 so, in R3 , a point P on the line is given by (3
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2011/2012
Solutions #9
1. We can parametrize S by (u, v )
=
(u cos v, u sin v, u), 0
v
2 ,
1 u 2. Now u = (cos v, sin v, 1),
v = (u sin v, u cos v, 0) and u u =
(u c
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2011/2012
Solutions #2
1. f (x) = x (x 2 ) = x2 2x on [0, 2 ].
Since the periodic extension is even bk =
1 2 2
k = 1, 2, . a0 =
(x 2x) dx
0
2
1 x3
4 2
2
=
x
. For k
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2011/2012
Solutions #6
1. (a) (i) We are given that = 1 so the curve lies on
=
8 , so we can parametrize
(using spherical polars) by (t)
cos 8t sin t, sin 8t sin t,
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2011/2012
Solutions #1
1. Recalling that sin A sin B =
1
2
cos(A B ) cos(A + B )
we have
1
sin(k x) sin(n x) dx =
cos (k n) x cos (k + n) x dx =
2
1
1
cos (k n) x d
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2011/2012
Solutions #4
1. (a) Since T (t) = 1, we have T (t) 2 = T (t) T (t) = 1. Dierentiating w.r.t.
d
t gives
(T (t) T (t) = T (t) T (t) + T (t) T (t) = 2 T (t) T
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2011/2012
Solutions #3
1. We are given that the path followed is (t) =
(t) =
et ,
1
,
sin
t2 6
t
6
t
1
et , , cos
t
6
. The particle leaves the curve at (1) and f
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2011/2012
Solutions #5
1. (a) = y exy dx + x exy dy . Put F1 (x, y ) = y exy and F2 (x, y ) = x exy . Since
F1
F2
= (1 + xy ) exy =
, is closed. Since F1 and F2 are