University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2015/2016
Assignment #2
This assignment is due at
September 24 September 30, 2015.
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
2014/2015
Solutions #7
f
f
= 2x + y and
= 32y 3 + x 6y 3y 2 so
x
y
2x + y
= 0
f = 0 if
. The
3
2
32y + x 6y 3y = 0
rst equation gives y = 2x and substituting
into th
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2015/2016
Assignment #5
The Midterm Test will be written on Saturday, October 24, 5:00 7:00 pm. See
the information sheet posted to the course website for details.
T
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2014/2015
Assignment #2
This assignment is due at
September 22 September 26, 2014.
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
Solutions #1
f (x) f (a)
2x2 + x 3 (2a2 + a 3)
= lim
=
x a
x a
xa
xa
1. (a) If a nite limit exists, f (a) = lim
2(x2 a2 ) + (x a)
= lim 2(x a) + 1 = 2(a +
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2015/2016
Assignment #3
This assignment is due
October 1 October 7, 2015.
at
the
start
of
your
tutorial
in
the
period
A. Suggested reading: Marsden & Tromba, Chapter
UNIVERSITY OF TORONTO SCARBOROUGH
Department of Computer and Mathematical Sciences
Final Exam, Winter  2015
STAB57H3: An Introduction to Statistics
Duration: Three hours (180 minutes)
LAST NAME:
FIRST NAME:
STUDENT NUMBER:
SIGNATURE:
TUTORIAL:
Aids Allow
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2015/2016
Assignment #4
The Term Test will take place on Saturday, October 24, 5:00 pm 7:00 pm.
This assignment is due at the start of your tutorial in the period
Oc
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2015/2016
Assignment #6
This assignment is due at
November 5 November 11, 2015.
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
2015/2016
Assignment #7
This assignment is due at
November 12 November 18, 2015.
the
start
of
your
tutorial
in
the
period
A. Suggested reading: Marsden & Tromba, Cha
Let a = (a1 , a2, , an) Rn be an extremum for f : Rn R,
subject to the constraint g(x1, x2, , xn) = c. To nd the coordinates a1, a2, , an of a we solve the system
f (a) = g(a)
g(a) c =
0
n + 1 equations in n + 1 unknowns:
, a1, a2, , an
is called a Lagr
Denition:
The average value or mean value of an integrable
function f (x, y) over the set D is the number
f = avD f =
1
area of D
f (x, y) dA .
D
Denition:
A set D in the plane is said to be connected if any two
points
D
in
can
be
joined
by
a
continuous
f
Example: Find the extreme values of f (x, y, z) = x+y z on the solid ellipsoid
x2 + 2 y 2 + 8 z 2 32.
We know that global extrema will exist since f is continuous and the
domain (the solid ellipsoid) is compact in R3 . Now f is dierentiable
everywhere so
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2014/2015
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
2014/2015
Solutions #9
1. (a)
D
2 y dA, where D is the region between y = x2 and y = x(1 x).
For this integral we will use vertical strips. To
obtain the limits of i
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2014/2015
Solutions #8
1. (a) f (x, y) = (x2 + 1)y subject to the constraint x2 + y 2 = 5. We dene h(x, y, ) =
(x2 + 1)y (x2 + y 2 5). The critical points of h will
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2014/2015
Solutions #4
1 x 0
2 1
(x) det
1. (a) det A = det x 2 1 = (1) det
3 y
z 3 y
x (xy z) = 2y 3 x2 y + xz = x2 y + xz + 2y 3.
x 1
z y
= 2y 3
(b) From part (a
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2014/2015
Solutions #3
1.
(i) f (x, y) = x2 + y 2 + 4. Domain is R2 . Now x2 + y 2 + 4 = c x2 + y 2 = c 4.
These are circles centered at (0, 0) with radius c 4. Plea
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2014/2015
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
2015/2016
Assignment #8
This assignment is due at
November 19 November 25, 2015.
A. Suggested reading: 1.
2.
the
start
of
your
tutorial
in
the
period
Marsden & Tromb
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2015/2016
Assignment #9
This assignment is due at
November 26 December 2, 2015.
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
2015/2016
Solutions #8
1. (a) f (x, y) = y subject to the constraint g(x, y) = 2x2 +y 2 = 4. We dene h(x, y, ) =
y (2x2 +y 2 4). The critical points of h will give t
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2014/2015
Solutions #6
f
f (0 + h, 0) f (0, 0)
00
(0, 0) = lim
= lim
= 0 and
h0
h0
x
h
h
f
f (0, 0 + h) f (0, 0)
00
(0, 0) = lim
= lim
= 0. Hence both partial deriva
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2014/2015
Solutions #5
1. Recall that a normal vector for the tangent plane to the graph of z = f (x, y) or
f f
f
f (x, y) z = 0 is
,
, 1 . Here we have f (x, y) = x
k n n o
7AF&o!
k
g
Ta hcfw_
%k g y ut U on&AaA&A&
m j j n h h n q no
cfw_ cfw_ ~cfw_ cfw_

cfw_ Scfw_
cfw_
t cfw_
cfw_
 t cfw_ t 3cfw_ t P
ut y t P
t
y ut t P
p& cfw_ cfw_ u cfw_
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2015/2016
Solutions #6
1. (a) (i) f (x, y) = x4 + x2 y 2 + xy 3 3y 4 . Now f (tx, ty) = t4 x4 + (t2 x2 )(t2 y 2 ) +
(tx)(t3 y 3 ) 3t4 y 4 = t4 x4 + t4 x2 y 2 + t4 xy
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2015/2016
Solutions #7
1. (a) f (x, y) = 2x2 + 3xy + y 3 . We rst compute the
f
f
partials:
= 4x+3y,
= 3x+3y 2 , so f = 0
x
y
4x + 3y = 0
if
. From the rst equation