MATH 237 Spring 2014
Assignment 4
Topics: Dierentiability, chain rule
Due: Friday June 6th
1. Determine all of the points where the following functions are dierentiable.
x sin2 (y) (x, y) = (0, 0)
x2 +y 2
(a) f (x, y) =
0
(x, y) = (0, 0)
|x| + |y|
(b) f
UWaterloo Email:
Last Name.
First Name.
Student ID
Date Tuesday. February 23'“. 2016
Time Period 4:30 pm - 6:15 pm
Duration of Exam 105 minutes
Number of Test Pages 9 pages (5 double-sidml sheets}
Total Possible Marks 5
Additional Materials Allowui None.
Math 237, Winter 2015
237S0L
1.
2.
Assignment 5 solutions
Page 1 of 4
Math 237, Winter 2015
Page 2 of 4
f = (2x, 2y, 6z) and so the tangent
Assignment 5 solutions
2
2
2
3. We have f (x, y, z) = x y + 3z = 0. Thus,
plane is given by
0 = fx (1, 2, 1)(x 1) +
MATH 237-001
ASSIGNMENT 3: Linear approximation, di bility
Winter 2016
Submit your work at the usual time and place on Friday, January 29th in the correct drop
slot across from MC 4066/4067. Late assignments or those put into the wrong drop slot will
not
MATH 237
ASSIGNMENT 4: Chain Rule
Winter 2015
Submit your work at the usual time and place on Friday, February 6th in the correct drop
slot across from MC 4066/4067. Late assignments or those put into the wrong drop slot will
not be marked and will receiv
MATH 237-001
ASSIGNMENT 2: Continuity, partial derivatives
Winter 2015
Submit your work by the instructor-specied time (10:23am) on Friday, January 23rd in
the correct drop slot across from MC 4066/4067. Late assignments or those put into the
wrong drop s
MATH 237
ASSIGNMENT 9: Jacobians, Double Integrals
Winter 2014
Submit your work at usual time/place on Friday, March 28th .
1. Find the Jacobian of the mapping
(u, v) = F (x, y) = (x2 sin y, y 2 cos x)
2. Consider the map dened by
(u, v) = F (x, y) = (y +
MATH 237
Assignment 9 - Solutions
1. Let F : R2 ! R2 . Briey describe the mapping geometrically. Compute the Jacobian of each mapping,
and interpret in terms of how F transforms the area. Sketch the image of the unit square in the rst
quadrant under the g
MATH 237 Spring 2015
Assignment 5
Topics: Chain rule, directional derivatives, gradient vector
Due: Friday June 12th
1. The partial dierential equation
2u 2u 2u
2u
= c2
+
+
t2
x2 y 2 z 2
()
is the three dimensional wave equation.
In the case of spherical
Math 237, Winter 2015
237S0L
1.
2.
Assignment 5 solutions
Page 1 of 4
Math 237, Winter 2015
Page 2 of 4
f = (2x, 2y, 6z) and so the tangent
Assignment 5 solutions
2
2
2
3. We have f (x, y, z) = x y + 3z = 0. Thus,
plane is given by
0 = fx (1, 2, 1)(x 1) +
MATH 237-001
ASSIGNMENT 3: Linear approximation, di bility
Winter 2015
Submit your work at the usual time and place on Friday, January 30th in the correct drop
slot across from MC 4066/4067. Late assignments or those put into the wrong drop slot will
not
MATH 237
ASSIGNMENT 11: Triple Integrals
Winter 2015
This assignment is not due in, however you are responsible for the material for the nal
exam. Full solutions will be posted at least a week before the nal.
x2 + y dV where D is the region bounded by x +
Math 237 - Sample Midterm 2
1. Short Answer Problems
a) State the precise denition of
lim
f (x, y) = L.
(x,y)(a,b)
b) If fx and fy are both continuous at (a, b), then what are two things you can say about f at
(a, b)?
c) Let f (x, y) = x2 + xy + y 3 . Wha
Math 237
Tutorial 1 Problems
1. Sketch the level curves and cross sections for each of the following functions, and
use them to sketch the surface z = f (x, y).
a) f (x, y) = x2 + y 2
b) f (x, y) =
x2 + y 2
c) f (x, y) =
|1 x2 y 2 |
d) f (x, y) = x2 2xy
1
Math 237
Tutorial 5 Problems
1: Let g be a dierentiable single-variable function and let f (x, y) = g(u2 v), where u = ex
and v = x2 + y 3 .
2f
Use the chain rule to nd
.
xy
2: The temperature of a metal sheet as a function of position (x, y) is given by
Math 237
Tutorial 8 Problems
1: The following region in R2 is given in Cartesian coordinates. Convert the description
to polar coordinates.
cfw_(x, y) : y x, x2 + y 2 4
2: The following region in R2 is given in polar coordinates. Convert the description t
MATH 237
ASSIGNMENT 6: Taylor, Critical points
Winter 2015
Submit your work at the usual time and place on Friday, March 6th in the correct drop
slot across from MC 4066/4067. Late assignments or those put into the wrong drop slot will
not be marked and w
Math 237, Winter 2015
Assignment 2 selected solutions
Page 1 of 2
237S0L
1. Since x2 + y 2 = 0 if (x, y) = (0, 0), we have that f is continuous for all (x, y) = (0, 0)
by the continuity theorems.
For f to be continuous at (0, 0) we must have
lim
f (x, y)
MATH 237
ASSIGNMENT 8: Coords., Mappings
Winter 2015
Submit your work at the usual time and place on Friday, March 20th in the correct drop
slot across from MC 4066/4067. Late assignments or those put into the wrong drop slot will
not be marked and will r
Math 237, Winter 2015
Page 1 of 5
Asst. 11 Solutions
1. Integrating wrt to z rst gives 2 x y z 2 and then the region Dxy is bounded by
x = 1, y = x and the intersection of x + y + z = 2 and z = 2 so x + y = 0. Drawing the
region in the xy-plane we see we
Math 237, Winter 2015
Page 1 of 4
Assignment 10 solutions
237S0L
1. We observe that we need to change the order of integration. The region is x y x,
0 x 1, thus we draw y = x and y = x for 0 x 1 to get the region below. We
now want to integrate with respe
MATH 237
ASSIGNMENT 7: Optimization
Winter 2015
Submit your work at the usual time and place on Friday, March 13th in the correct drop
slot across from MC 4066/4067. Late assignments or those put into the wrong drop slot will
not be marked and will receiv
UNIVERSITY OF WATERLOO
SAMPLE MIDTERM EXAMINATION # 1
Student Name (Print Legibly)
(family name)
(given name)
Signature
Student ID Number
COURSE NUMBER
MATH 237
COURSE TITLE
Calculus 3 for Honours Math
COURSE SECTION(s)
001-007
DATE OF EXAM
Today!
TIME PE
MATH 237
ASSIGNMENT 10: Double I., Iterated I., Ch. of Var.
Fall 2014
SECTION:
001
002
003
004
005
3:30 pm F.V.
11:30 am J.V.
11:30 am L.K.
10:30 am F.V.
1:30 pm A.V.
LAST NAME:
FIRST NAME:
GRADE:
Student Id. #:
Submit your work by the time specied by you
MATH 237
ASSIGNMENT 7: Optimization
Fall 2016
Submit your work by noon on Tuesday, November 8th in the correct drop slot across from
MC 4066/4067. Late assignments or those put into the wrong drop slot will not be marked
and will receive a grade of zero.
MATH 237
ASSIGNMENT 10: Double Integrals
Fall 2016
Submit your work at the usual time and place on Tuesday, November 29th in the correct
drop slot across from MC 4066/4067. There will also be a quiz on this material at
the usual time and place on Monday,
Math 237, Fall 2016
Assignment 2 selected solutions
Page 1 of 2
This file is not displaying properly. It will work if you use Adobe Acrobat Reader (freely downloadable)
and make
1. sure that JavaScript is enabled. You will likely need to download the file
MATH 237
ASSIGNMENT 11: Triple Integrals
Fall 2016
This assignment is not due in, however you are responsible for the material for the final exam.
Full solutions will be posted at least a week before the final. The tutorial on Monday,
December 5th will ha
MATH 237
ASSIGNMENT 0: using Crowdmark
Winter 2017
This assignment is due before 11:30pm on Friday, January 6th. This assignment has
the same weight as any assignment. It is very important that you carefully read all the instructions below.
Instructions
1