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
2015/2016
Solutions #2
1. (a) The slope of the line
line is vertical).
is m(x) =
y2 y1
ex 0
ex
=
= , x = 0 (when x = 0 the
x2 x1
x0
x
(b) We know the graph of y = ex
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 #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
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
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
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
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
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2015/2016
Solutions #5
1. (a) We have f (0, 0) = 0, so we can compute the partial derivatives from the denition.
f
f (h, 0) f (0, 0)
00
(0, 0) = lim
= lim
=0
h0
h0
x
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
2014/2015
Solutions #1
 t2 2 
1. (a) lim
= lim
t 2
t 2
t2 + 2 2t + 2
t 2
= .
= lim
t 2 t +
2
t
2t + 2
2
t+ 2
x2 + 3x 1 (a2 + 3a 1)
f (x) f (a)
= lim
xa
xa
xa
Spherical coordinates
, ,
z
= distance from P to the origin.
=
P
plane makes with the positive xaxes.
angle projection of line into xy
y
= angle line makes with the positive
zaxes.
x
x = sin cos
0<
y = sin sin
0 < 2
z = cos
0
Notes:
> is picked up
We consider 2 practical approaches to computing the triple integral
f dV =
B
1.
f dA
f dV
B
dz
Fix one variable, say (w.l.o.g.) z and let Rz be the planar region
consisting of the points in B with z xed (the cross section).
f dV =
Then
f dA
dz.
Rz
B
We no
1?
1?
Denition:
The average value or mean value of an integrable
function f (x, y) over the set D is the number
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
func
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 & 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
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
Midterm Test
MATB41H Techniques of the Calculus of Several Variables I
Examiner: E. Moore
1. [8 points]
represents.
Date: October 28, 2013
Duration: 110 minutes
In this quest
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 define h(x, y, ) =
y (2x2 +y 2 4). The critical points of h will give
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2015/2016
Solutions #9
1. (a)
(b)
Z
(x2 + y 2 ) dA, where D is the region bounded by the positive x and y axes and
D
the line 3x + 4y = 10.
Z
(x2
+
y2)
dA
=
ZD10/3 Z
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 first compute the
f
f
y 3
partials:
= 4x+3y,
= 3x+3y 2 , so f = 0
4
x
y
0
4x + 3y = 0
if
. From the fir
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2015/2016
Solutions #3
1. (a) Since f (x, y) = x2 y 2 = (x y)(x + y) =
0, the level curves of f (x, y) = 0 are the
lines y = x and y = x. Between these
curves f (x,