Question 1. For the iterated double integral
y=3 xzx/Qey2
/ / 2y dcc dy
y=0 $=/9~y2
sketch the region of integration. Then evaluate the integral by interchanging the order of
integration. [6 IEL
1. Z .
a. 5 1: 7 2:3 CAV 59 raA'u5 3
Ind-6
8 e [0,3] 6?
3M ~
MATH 2X3 (Fall 2014) Assignment 2
Due Date: Before Noon, October 3, 2013
1. Let a, b, c denote the lengths of the sides of a triangle and be the angle
opposite the side with length c. The Law of Cosines states that
c2 = a2 + b2 2ab cos .
Suppose a = 6 = b
MATH 2X3 (Fall 2014) Assignment 4
Due Date: Before 12:30 pm, October 24, 2014
1. Find the points of the paraboloid z = x2 + y 2 1 at which the normal line
to the surface coincides with the line joining the origin to the point. What is
the acute angle betw
Sample Questions
MATH 2X3
Disclaimer: The appearance or non-appearance of any topic in the following has no implication regarding its inclusion or exclusion in the actual test. The questions only indicate the
approximate complexity of the actual test ques
QUADRATIC FORMS AND THE SECOND DERIVATIVE TEST
M. WANG, MATH 2X3 FALL 2014
1. Preliminaries on Matrices
Let A be an n n real matrix and I be the identity matrix of the same size in this section.
Denition 1.1. (a) A real number is called a real eigenvalue
@[
Question 1.
Find the equation of the tangent plane to the graph of f(3:,y) : $2 + 33y + y 1 When
(27,31) = (4, 5). [10 marks]
§'(#,5):1é* amt-41
:- 80
E} : I;
w Um) aw- 33 (m) H 8H
_
35" [955) = 3H! J = IS
(45)
c: gag/two n 01; [PLKHG M
L2: 80+ 2§(y
Question 1.
Find all the critical points of the function f (3,9) = 459;, _ 5L.4 _ y4. For each critical point,
decide if it is a local maximum, a local minimum, or a saddle. [13 marks]
5 39 :LL3~ 4X3
7 - ax
24 g: q-)(_435 3
,W'cao FM (:> 3: x3 , x: 3
:2 a
Question 1. For the iterated double integral
f: (
Sketch the region of integration. Then evaluate the integral by interchanging the order of
integration. [6 marks]
firm/4i?
6w dy dm
y=-W
continued on p. 3 NAME:
Question 2. A hemispherical dome over the
Username: Gary MongioviBook: A First Course in Probability, Ninth Edition. No part of any book may be reproduced or transmitted
in any form by any means without the publisher's prior written permission. Use (other than pursuant to the qualified fair use p
Stats2D03 - Fall 2016
Midterm Test 2 Information
Our second test will be the evening of MONDAY, NOVEMBER 21
The test will last for 60 minutes. To find the room and time to which you have been
assigned, please look on the course web site on Avenue. The roo
MATH 2X3 (Fall 2016) Review Problems
Question 1.
(a) For the surface x2 + y 2 z 2 = 0, find the equation of the tangent plane
at the point (3, 4, 5) as well as the equation of the line perpendicular to
the surface at (3, 4, 5).
(b) Find the surface
area o
McMASTER UNIVERSITY
MATH 2XX3 TEST 1
Dr. M. Wang
DATE: October 9, 2015.
NAME: _ STUDENT NUMBER:
Time allowed: 45 minutes
This paper contains 3 questions (with parts) on a total of 6 pages. You are
responsible for ensuring that your copy of the paper is co
Question 1. Let F = mi yj + zk. Compute the work done by F in moving a unit mass
along the path r(t) = (cos 1%, sin 73, i), 0 S t S 27F. [5 marks]
A 7%,): (Wt/cw/ 1+2)
17:
@ g Elsif1 _ S (CheatjustlTi:>($mf/wsf TTLCB M
Y3 0
'_ SLTICQGOSJCQLL + cfw_$1 3
MATH 2X3 (Fall 2014) Assignment 5
Due Date: Before 12:30 pm, November 28, 2014
1. Consider the system of equations
x5 v 2 + 2y 3 u = 3
3yu xuv 3 = 2.
Show that near the point (x, y, u, v) = (1, 1, 1, 1), this system denes u and v
implicitly as functions o
MATH 2X3 (Fall 2014) Assignment 4
Due Date: Before 12:30 pm, November 7, 2014
1. Let f (x, y) = 1 + xy + x 2y and D be the closed triangular region with
vertices (1, 2), (5, 2), (1, 2). Find the maximum and minimum points of f
over D. (This is problem 44,
MATH 2X3 (Fall 2014) Assignment 1
Due Date: Before Noon, September 19, 2014
1. Let f (x, y) =
x
y 2 4x
for this question.
a. Find the domain of f , i.e., the largest subset of R2 on which f is dened.
b. Describe the level curves of f .
c. Determine the ra
MATH 2X3 (Fall 2015) Assignment 4
Due Date: Before 12:30 pm, November 20, 2014
1
1. Let F(x, y) = x2 +y2 (x3 + xy 2 y, y 3 + x2 y + x). Evaluate the work done
by F in moving a unit mass in the anti-clockwise direction along the ellipse
whose equation is 9
Math 2X3 Assignment 5
Date due: December 4, 2015
1. Let S denote the part of the cone x =
whose x-component is positive. Evaluate
y 2 + z 2 with 0 x 2 oriented with the normal
( F ) dS
S
where F = (tan1 (x2 yz 2 ), x2 y, x2 z 2 ). [tan1 denotes the inve
Math 2X03 Important Topics & Recommended
Problems Weeek 1
May 2nd
12.1: Coordinate systems, equations of planes and surfaces in R3 , spheres
Problems: 12.1.8, 12.1.12 (a)(e), 12.1.19, 12.1.38
12.2: What the heck is a vector?, standard basis vectors, un