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MAT223H1 Study Guide - Problem Set, Joule


Department
Mathematics
Course Code
MAT223H1
Professor
Robert Brym

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University of Toronto, Faculty of Applied Science and
Engineering
MAT291F, Calculus III, Fall 2011
——————————————————————————————————
Problem Set 1
DUE DATE: Monday, October 3rd, 2011, at 4PM.
Where to Submit: The drop boxes for MAT291F are located on the first floor
of the Sanford Fleming building (directly above SFB600, the ECE undergrad-
uate office). There are eight labeled boxes, one for each tutorial section. You
must submit your problem set to the drop box for your tutorial section.
Rules for Submission: Please be reminded of the following rules:
Required Information: The front page must include your name, student
number, your tutorial code (which will be assigned to you when tutorial rooms
are announced), and the name of your teaching assistant. Failure to put your
name and/or your student number will result in a zero in your assignment.
Failure to put the name of your TA or your tutorial code will result in a 20%
reduction of your assignment mark. A cover page is not required as long as the
necessary information is on the top of the first page.
Paper Size and Requirements: Assignments must be submitted on letter-
sized (8.5 x 11 inch) paper. Using ripped notebook paper is unacceptable and
will result in a zero in your assignment mark. Assignments that are more than
one page in length must be stapled in the top left corner. Failure to staple such
assignments will result in a 20% reduction of your assignment mark. Do not
use clear plastic binders.
Hand in solutions to the following problems:
1. Specify the domain of the function f(x, y, z) = ln(zx2y2+ 2y+ 3).
Describe the domain in words or with a sketch.
2. §12.1#86.
3. Find the equations of all spheres that pass through the points (2,2,1), (0,0,1)
and (0,4,1), and intersect the plane Π :
2x+y+z=1 at only one point.
HINT: Consider using the formula for the least distance from a point to a plane
found in §12.1#86(b) (even if you could not derive this formula).
4. Evaluate the following limits, if they exist. If the limit does not exist, prove
it using the two-path test.
(i) lim
(x,y,z)(1,1,1)
yyz xy +
xz
yyz +xy
xz
,
1
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