Mathematics 226
4 marks
Midterm 1 Solutions October 3, 2014
Page 1 of 2
1. Decide whether each of the sets below is open, closed, or neither. For this question
only, an answer without explanation is sucient.
(a) cfw_(x, y) R2 : 0 <
x2 + y 2 3
(b) cfw_(x,
Math 226 Final Exam, December, 2009
[12] 1. (a) Sketch the hyperboloid z 2 = 4x2 + y 2 1.
(b) Find all points on the hyperboloid z 2 = 4x2 + y 2 1 where the tangent plane is
parallel to the plane 2x y + z = 0.
[8] 2. A bug walks on a at horizontal metal p
Math 226 Final Exam, December, 2007
[8] 1. Compute the following limits or explain why they do not exist.
xy
(a)
lim
x2 +y 2
(x,y)(0,0)
(b)
(c)
(d)
lim
(x,y)(0,0)
|y|x
x2 +2xy 2 +y 4
1+y 4
(x,y)(1,1)
sin(xy)
lim
2
2
(x,y)(0,0) x +y
lim
[15] 3. Let f (x, y
Math 226 Final Exam, December, 2005
[12] 1. (a) Prove that the line given by the parametric equations x = 1+4t, y = 2t, z = 3t,
is parallel to the plane 2x + 5y + z = 4.
(b) Find the distance between the plane and the line in (a).
[10] 3. (a) Find
z
x
and
Solutions to Math 226 Final Exam, December, 2005
1. (a) Prove that the line given by the parametric equations x = 1 + 4t, y = 2 t, z = 3t, is parallel to
the plane 2x + 5y + z = 4.
(b) Find the distance between the plane and the line in (a).
Solution. (a)
Math 226 Final Exam, December, 2010
[10] 1. The plane P passes through the points A(1, 1, 0), B(0, 1, 1), and C(1, 3, 1).
(a) Find an equation for P .
(b) Show how to split the given vector F = 4, 5, 7 as F = u + w, where u is normal to the plane P
and w
Solutions to Math 226 Final Exam, December, 2006
1. Let z = f (x, y) be a dierentiable function on IR2 such that f (1, 2) = 3, f (1.2, 2.3) = 3.4 and f (0.9, 2.1) =
3.2.
z
z
(a) Estimate x (1, 2) and y (1, 2).
(b) Estimate the value of the directional der
Solutions to Math 226 Final Exam, December, 2010
1. The plane P passes through the points A(1, 1, 0), B(0, 1, 1), and C(1, 3, 1).
(a) Find an equation for P .
(b) Show how to split the given vector F = 4, 5, 7 as F = u + w, where u is normal to the plane
Solutions to Math 226 Final Exam, December, 2007
1. Compute the following limits or explain why they do not exist.
xy
(a)
lim
x2 +y 2
(x,y)(0,0)
(b)
(c)
(d)
lim
(x,y)(0,0)
|y|x
x2 +2xy 2 +y 4
1+y 4
(x,y)(1,1)
sin(xy)
lim
2
2
(x,y)(0,0) x +y
lim
Solution.
Math 226 Final Exam, December, 2006
[15] 1. Let z = f (x, y) be a dierentiable function on IR2 such that f (1, 2) = 3, f (1.2, 2.3) =
3.4 and f (0.9, 2.1) = 3.2.
(a) Estimate
z
x (1, 2)
and
z
5 (1, 2).
(b) Estimate the value of the directional derivative
Mathematics 226
Midterm 2 Sample, Fall 2014
Page 1 of 5
This midterm has 5 questions on 5 pages, for a total of 40 points.
Duration: 50 minutes
Read all the questions carefully before starting to work.
Give complete arguments and explanations for all yo
MATHEMATICS 226, FALL 2014, PROBLEM SET 5
Solutions1
Section 13.1, Question 25: Not marked
Let (x, y, z) be the coordinates of the corner of the box in the octant x, y, z
0. We want to maximize the function V (x, y, z) = 8xyz subject to the
x2 y 2 z 2
co
MATHEMATICS 226, FALL 2014, PROBLEM SET 3
Solutions1
Section 12.2, Question 19: 6 marks
Answer: a = c = 0, b arbitrary.
Proof: If the limit exists, it should be the same along every path as (x, y) (0, 0). Let
y = mx for some xed slope m, then
xy
mx2
m
= 2
Mathematics 226
Midterm 2 November 3, 2014
Page 1 of 5
This midterm has 4 questions on 5 pages, for a total of 40 points.
Duration: 50 minutes
Read all the questions carefully before starting to work.
Give complete arguments and explanations for all you
MATHEMATICS 226, FALL 2014, PROBLEM SET 4
Solutions1
Section 12.6, Question 7: 6 marks
The dierential is dz = 2xe3y dx + 3x2 e3y dy. At (3, 0), we have z = 9, so that at (3.05, 0.02)
we have
z 9 + dz = 9 + 6dx + 27dy = 9 + 6(0.05) + 27(0.02) = 8.76
Sectio
MATHEMATICS 226, FALL 2014, PROBLEM SET 1
Solutions1
1. Specify the boundary and the interior of the sets S in 3-space whose points
(x, y, z) satisfy the given conditions. Is S open, closed, or neither?
(a) x2 + y 2 + z 2 16: The boundary is the sphere x2
MATHEMATICS 226, FALL 2014, PROBLEM SET 6
Solutions1
Section 13.5, Question 11: 8 marks: 2 for setting up the quantity to be
minimized, 3 for the correct equations for the critical points, 3 for solving
these equations.
n
(pexi + qexi yi )2 . We look for
MATH 226 SAMPLE MIDTERM 1-SOLUTIONS
Fall 2014
1. Let f (x, y) =
y x2 . Find the domain of f , and draw several level curves.
The domain is cfw_(x, y) R2 : y x2 . The level curves f (x, y) = c are parabolas y = x2 + c2 .
2. Decide whether each of the sets
MATHEMATICS 226, FALL 2014, PROBLEM SET 2
Solutions1
Section 10.3, Question 13: 4 marks
If u + v + w = 0, then
v w = v (u v) = v u v v = u v,
w u = (u v) u = u u v u = u v.
Section 10.3, Question 15: Not graded
The tetrahedron is spanned by the vectors 0,
Solutions to Math 226 Final Exam, December, 2009
1. (a) Sketch the hyperboloid z 2 = 4x2 + y 2 1.
(b) Find all points on the hyperboloid z 2 = 4x2 + y 2 1 where the tangent plane is parallel to the plane
2x y + z = 0.
Solution. (a) For each xed z, 4x2 + y
Webwork for Math 226, Section 101
September 28, 2013
1
Getting started
You need a campus-wide login (CWL).
Go to https:/webwork.elearning.ubc.ca/webwork2/. You will see
a list of courses. Scroll to the Math 226 listing and click to login using
your CWL.
Department of Mathematics
University of British Columbia
MATH 226 Midterm 1
October 17, 2012, 11:00 - 11:50
Family Name:
Initials:
I.D. Number:
Question
Signature:
Mark
Out of
1
25
2
25
3
25
4
25
Total
100
CALCULATORS, NOTES OR BOOKS ARE NOT PERMITTED.
TH
December 2009 Mathematics 226 Name
Page 2 of 10 pages
Marks
[12] 1.
(a) (4 marks) Sketch the hyperboloid z 2 = 4x2 + y 2 1.
(b) (8 marks) Find all points on the hyperboloid z 2 = 4x2 + y 2 1 where the tangent plane
is parallel to the plane 2x y + z = 0.
C
Be sure that this examination has 9 pages including this cover
The University of British Columbia
Sessional Examinations - December 2000
Mathematics 226
Advanced Calculus I
Time: 2 1 hours
2
Closed book examination
Name
Signature
Student Number
Instructor
This examination has 9 questions on 3 pages.
The University of British Columbia
Final ExaminationsDecember 2010
Mathematics 226
Advanced Calculus I (Professor Loewen)
Closed book examination.
Time: 2.5 hours
Notes, books, and calculators are not allowed.
This examination has 8 questions on 2 pages.
The University of British Columbia
Final ExaminationsDecember 2002
Mathematics 226
Advanced Calculus I (Professor Loewen)
Time: 2 1 hours
2
Closed book examination.
Notes, books, and calculators are not allowed
Be sure that this examination has 10 pages including this cover
The University of British Columbia
Sessional Examinations - December 2011
Mathematics 226
Advanced Calculus I
Time: 2 1 hours
2
Closed book examination
Name
Signature
Student Number
Instructo
December 2005 Mathematics 226 Name
Page 2 of 10 pages
Marks
[12] 1.
(6 marks for each part)
(a) Prove that the line given by the parametric equations x = 1 + 4t, y = 2 t, z = 3t, is
parallel to the plane 2x + 5y z = 4.
(b) Find the distance between the pl
This examination has 8 questions on 2 pages.
The University of British Columbia
Final ExaminationsDecember 2003
Mathematics 226
Advanced Calculus I (Professor Loewen)
Time: 2 1 hours
2
Closed book examination.
Notes, books, and calculators are not allowed