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,
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2015/2016
Solutions #4
kT
V 1 = 2 .
V
V
P
Since P and V are always positive so is T . Hence
< 0 which means that
V
pressure is a decreasing function of volume at a c
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 definition.
f
f (h, 0) f (0, 0)
00
(0, 0) = lim
= lim
=0
h0
h0
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 #2
1. (a) The slope of the line ` is m(x) =
line is vertical).
y2 y1
ex 0
ex
=
= , x 6= 0 (when x = 0 the
x2 x1
x0
x
(b) We know the graph of y =
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MATB24H3F
2014/2015
Assignment #2
You are expected to work on this assignment prior to your tutorial in the week of September 15, 2014. You may ask questions about this assig
Equation of Tangent Plane at point (a, b)
z=f ( a ,b )+ f x ( a , b )( xa ) +f y ( a , b ) ( yb )
f (a , b , c )( ( x , y , z )(a , b , c)
Angle of 2 Lines
cos1
(
ab
ab
)
Chain Rule Example
Let f: R4 R4 be given by f(x, y, z, w) = (xw, yz, xy, zw) and
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
MATB44 November 29
Jordan Bell
Department of Mathematics, University of Toronto
December 5, 2016
An ODE
x0 (t) = P (t)x(t) + g(t)
is called nonhomogeneous when g = 0. To solve this we use variation of
parameters.
To solve a nonhomogeneous ODE we first sol
MATB44 November 15 and 17
Jordan Bell
Department of Mathematics, University of Toronto
November 28, 2016
System of linear first order ODE
x01 (t) = p1,1 (t)x1 (t) + + p1,n (t)xn (t) + g1 (t)
.
.
x0n (t) = pn,1 (t)x1 (t) + + pn,n (t)xn (t) + gn (t).
This c
MATB44 November 8 and 10
Jordan Bell
Department of Mathematics, University of Toronto
November 15, 2016
Regular singular points For y 00 + py 0 + qy = 0, x0 is a regular singular
point if p(x) and q(x) are defined at x = x0 but (x x0 )p(x) and (x x0 )2 q(
MATB44 November 22 and 24
Jordan Bell
Department of Mathematics, University of Toronto
December 3, 2016
Let A be a square matrix. Define
X
An
.
exp(A) =
n!
n=0
The term n = 0 is defined by
A0
0!
= I, the identity matrix. Let (t) = exp(At).
d X (At)n
(t)
28.12.2011
THERMODYNAMICS
(TER 201 E , 2011-2012 Fall, CRN: 13888)
2nd Midterm Exam
1) A 0.2 m3 rigid tank equipped with a pressure regulator contains steam(superheated water
vapour) at 2 MPa and 300C. The steam in the tank is heated. The regulator keeps
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2016/2017
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)
(2 x + 1)( x + 2)
(5)(4)
lim
= 4.
=
x4
5
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
1. (a)
(b)
(c)
2. (a)
2016/2017
Solutions #4
0 x 2
z
x
y
+
det A = det x 1 y = x det
y 2
y z 2
y
x 1
2
= x (2x y ) + 2(xz y) =
2 det
y z
2x2 + xy 2 + 2xz 2y.
x
From
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2016/2017
Solutions #3
1.
x
f (x, y) = p
.
Domain is cfw_(x, y) R2 | (x, y) 6= (0, 0). Putting
x2 + y 2
x
= c.
f (x, y) = c we have p
2
x + y2
For c = 0, the level c
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2016/2017
Solutions #2
1. (a) We will compare the square of each side.
kx yk2 = (x y) (x y) = kxk2 2 x y + kyk2 .
x
x
y
y
2
2
2
2
2
k
kxk kyk k
x y
= kxk kyk
=
kxk2
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2016/2017
Assignment #5
This assignment is due at the start of your tutorial in the period October 28 November 3,
2016.
A. Suggested reading: 1.
Marsden & Tromba, Ch
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2016/2017
Assignment #4
The Term Test will take place on Wednesday, October 19, 5:00 pm 7:00 pm.
This assignment is due at the start of your tutorial in the period
O
University of Toronto at Sarborough
Department of Computer & Mathematial Sienes
MAT B42H
2003/2004
Assignment #10
This assignment is due at the start of your tutorial in the week of Marh 29, 2004.
A. Suggested reading: 1. Marsden & Tromba, Chapter 8, seti
University of Toronto at Sarborough
Department of Computer & Mathematial Sienes
MAT B42H
2003/2004
Assignment #9
This assignment is due at the start of your tutorial in the week of Marh 22, 2004.
Marsden & Tromba, Chapter 8, setions 8.4 and 8.6
A. Suggest