University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
First Midterm Test
MATB41H3
Techniques of the Calculus of Several Variables I
Examiner: R. Grinnell Date: June 10, 2
MATB41H3 page 1
1 The temperature 1n C at points in the cry plane is given by the function
T(x y): ~22y where z and y are in centimetres.
(a) [5 points] Find the direction that a ladybug at the point
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
Second Midterm Test
MATB41H3
Techniques of the Calculus of Several Variables I
Examiner: R. Grinnell
Date: July 15,
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
First Midterm Test
MATB41H3
Techniques of the Calculus of Several Variables I
Examiner: R. Grinnell
Date: June 10, 2
_
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
Final Examination
MATB41H3
Techniques of the Calculus of Several Variables I
Examiner: R. Grinnell
Date: August 18
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2016/2017
Assignment #1
This assignment is due
January 16 January 20, 2017.
at
the
start
of
your
tutorial
in
t
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
Theorem
Let f : Rn R be differentiable and let a be an
interior point in the domain of f . If f has a local extremum at a,
then f (a) = 0.
Definition
Those points a, in the domain of f , at which f is
Theorem Change of Variables Theorem
Let B, B be regions in Rn and let g : B B be of class C 1 with
g
a bijection onto interior(B) and B = g(B ). Then for any
interior(B )
integrable function f : B R
Theorem: If B is bounded and f is continuous over B, then f is integrable
over B.
Definition The graph of a polar equation (equation in r and ) is the set
of points that corresponds to the polar coord
Definition: Let v = (v1, v2, v3), w = (w1, w2, w3) R3. We
define the cross product v w by
v w = (v2 w3 v3 w2, v3 w1 v1 w3, v1 w2 v2 w1) .
Properties:
1. v w v, w
2. v w = w v
3. v (u + w) = v u + v w
Definition.
A function, whose partial derivatives exist and are
continuous, is said to be of class C1.
If the partial derivatives of a C1 function f have, in
Definition.
turn, continuous partial deriv
To look at the rate of change in a function f at a point a along the
line p + t v, t R we have
Definition:
The directional derivative of f at a in direction
v, denoted by Dv f (a) , is given by
f (a +
Theorem 1. Let f, g : U Rn Rk be differentiable at a Rn
and let R. Then
(i) f is differentiable at a and
D f (a) = D f (a)
(ii) f + g is differentiable at a and
D f + g (a) = D f (a) + D g(a)
Theorem
We consider 2 practical approaches to computing the triple integral
ZZZ
Z
f dV
f dV =
B
B
1.
Z ZZ
f dA dz
Fix one variable, say (w.l.o.g.) z and let Rz be the planar region consisting of the points in
Taylor series around a = 0 that you should know.
X
xk
x
e
=
, |x| <
k!
k=0
cos x
sin x
=
X
=
k=0
X
k=0
x2k
(1)
(2k)!
=
ln(1 + x) =
k=0
X
arctan x =
, |x| <
x2k+1
, |x| <
(1)
(2k + 1)!
X
1
1+x
k
(1)
Z
D
(1 + xy) dA, where D = (x, y) R2 | 1 x2 + y 2 2, y 0 .
y=
y=
!
- 2
Z
"#2#
2-x
"#2#
1-x
-1
(1 + xy) dA =
Z
1
1
Z
2x2
Z
1
Z
!
2
2x2
(1 + xy) dy dx +
(1 + xy) dy dx +
2
1x
1
2x2
2x2
Z 1
Z 1
Z
1 2
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2005/2006
Assignment #2
This assignment is due
January 31 February 2, 2006.
at
the
start
of
your
tutorial
i
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2005/2006
Assignment #3
This assignment is due
February 7 February 9, 2006.
A. Suggested reading:
at
the
st
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2005/2006
Assignment #4
This assignment is due at
February 14 February 16, 2006.
the
start
of
your
tutorial
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2005/2006
Assignment #5
This assignment is due
February 28 March 2, 2006.
at
the
start
of
your
tutorial
in
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2005/2006
Assignment #6
This assignment will NOT be collected since the Midterm Test will take place on
Fri
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2005/2006
Assignment #7
This assignment is due
March 14 March 16, 2006.
at
the
start
of
your
tutorial
in
th
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT B42H
2005/2006
Assignment #8
This assignment is due
March 21 March 23, 2006.
A. Suggested reading:
at
the
start