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
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, 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) +
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 237-001
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 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
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, 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 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
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 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
Math 237, Winter 2015
237S0L
1.
Assignment 1 selected solutions
Page 1 of 4
Math 237, Winter 2015
Assignment 1 selected solutions
2
2
2
Page 2 of 4
2
2. a) The domain of f is x y 0 x y . The range is z 0.
b) Level Curves:
k = x2 y 2
x2 y 2 = k 2 , k 0
c)
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 1: Sketching & Limits
Winter 2015
Submit your work by the time specied by your instructor on Friday, January 16th in the
correct drop slot across from MC 4066/4067. Late assignments or those put into the wrong
drop slot will not be
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 5: Directional Deriv. & Gradient
Winter 2015
Submit your work at the usual time and place on Friday, February 13th in the correct
drop slot across from MC 4066/4067. Late assignments or those put into the wrong drop slot
will not be ma
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
237S0L
1.
2.
Assignment 6 solutions
Page 1 of 5
Math 237, Winter 2015
Page 2 of 5
Assignment 6 solutions
3.
4. We have
fxx = (1 + x + 2y)2 ,
fxy =
2
,
(1 + x + 2y)2
fyy =
4
(1 + x + 2y)2
So, f C 2 for x 0 and y 0. Thus, by Taylors
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 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
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
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
(Optional) Assignment 11: Ch. of Var., and Triple Int. Fall 2014
1- Chapter 14 page 180 #15.
2- Chapter 14 page 180 # 17.
x2 + y 2 dA where R is the region in the rst quadrant of the xy-plane
3- Evaluate
R
(i.e. bounded by x = 0 and y = 0) bounde
University of Waterloo
MATH 237' Midterm Examination
Calculus II for Honours Mathematics Spring 2017
Instructors: C. Wu (001), R.Garbary (002,003)
Number of exam pages: 11(includes cover page)
Additional instructions
. Fili in your username and ID number
Math 237
May 9, 2017 Tutorial #1
Sketching 3D Surfaces, Limits of Multivariate Functions
University of Waterloo
Math 237
1 / 12
Tutorial #1
Quadric Surfaces
The majority of surfaces you will have to sketch in this course will be quadric
surfaces, 2nd degr
Math 237
June 20, 2017 Tutorial #5
Taylor Approximations, Critical Points
University of Waterloo
Math 237
1 / 10
Tutorial #5
Find the Taylor polynomial P2,(a,b) of f (x, y) = (x y) sin(x + y) at (, ).
Computing P2,(,) requires all first and second partial
Brief Answers to Odd-Numbered End-of-Chapter Problems
Math 237 Course Notes, edition 6.0
Last updated: November 15th, 2016.
Purpose: to allow you to verify your answer AFTER you have tried a problem.
Note: Answers to challenging (*) questions are not incl
Math 237
Calculus 3 for Honours Mathematics
Spring 2017
Course Objectives:
To extend your knowledge of differential and integral calculus to functions of more than one variable.
Applications to science and engineering will be encountered, as this is an es