A Little Logic
There Exists and For All
The symbol is read there exists and the symbol is read for all (or for each
or for every, if it reads better). Let S () be a statement that contains the parameter .
For example, S () might be 5 < . Then
the stateme
Math 226 Final Exam, December, 2003
[10] 1. A nonzero vector c IR3 is given, along with constants a, k obeying |k| < a|c|. Show
that the plane c x = k and the sphere |x| = a intersect in a circle. Find the centre
and radius of the circle in terms of a, k,
Math 226 Midterm 1 Solutions
October 5, 2011
1a) The vector 1, 3, 1 is normal to P . So, calling the angle , cos =
1,3,1 1,2,2
| 1,3,1 | 1,2,2 |
=
7
3 11
.
b) x = 1 + s, y = 1 + 2s, z = 2s is in P if and only if (1 + s) 3(1 + 2s) + (2s) = 5. That is, s =
Math 226 Midterm 2 Solutions
November 2, 2011
1) The gradient of h is h(x, y) = (3+x280 2 )2 (2x, 4y). In particular, for x = 3, y = 2,
+2y
80
1
h(3, 2) = (3+9+8)2 (6, 8) = 5 (6, 8). The stream ows in the direction of maximum rate
of descent, which in thi
Math 226 Midterm 2
Name:
Student #:
Page 1 of 4
Mathematics 226
Midterm 2
November 2, 2011
Note: The quiz contains four questions worth a total of 50 points.
16
50
1) The height of land in the vicinity of a hill is given, in terms of horizontal coordinate
MATHEMATICS 226 December, 2011 Final Exam Solutions
1. Consider the ellipsoid
x2 +
y2
4
+
z2
9
= 21
(a) Find an equation for the tangent plane at the point (a, b, c) on the ellpsoid.
4
(b) At which points on the ellipsoid is the tangent plane parallel to
Solutions to Math 226 Final Exam, December, 2003
1. A nonzero vector c IR3 is given, along with constants a, k obeying |k| < a|c|. Show that the plane
c x = k and the sphere |x| = a intersect in a circle. Find the centre and radius of the circle in terms
Math 226 Midterm 1
Name:
Student #:
Mathematics 226
Midterm 1
October 5, 2011
Note: The quiz contains four questions worth a total of 40 points.
12
40
1) Consider the plane P given by the equation
x 3y + z = 5
and the line L with parametric equation
(x, y
MATHEMATICS 226, FALL 2016, PROBLEM SET 3: SOLUTIONS
Section 12.1, Question 35: (8 marks, 2 for each
p part)
2
2
2
(a) f (x, y) = C when x + y = C , so f (x, y) = x2 + y 2 .
(b) f (x, y) = C when x2 + y 2 = C 4 , so f (x, y) = (x2 + y 2 )1/4 .
(c) f (x, y
MATHEMATICS 226, FALL 2016, PROBLEM SET 2
Solutions
Section 10.3, Question 13: 6 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: 6 marks
The tetrahedron is spanned by the vectors
the fo
MATHEMATICS 226, FALL 2016, PROBLEM SET 6
Solutions
Section 13.2, Question 11: 10 marks: 2 for the critical points inside the
ball, 2 for setting up Lagrange equations, 6 for solving them.
Let f (x, y, z) = xy 2 + yz 2 . We first look for critical points:
Final Examination December 6th 2014
Mathematics 226
Page 1 of 10
This final exam has 8 questions on 10 pages, for a total of 100 marks.
Duration: 2 hours 30 minutes
Name (last, first, all middle names):
Student-No:
Course Section:
Signature:
Question:
1
2
Mathematics 226 Midterm 1, Fall 2016 Sample Page 1 of 4
This midterm has 7 questions on 4 pages, for a total of 50 points.
Duration: 50 minutes
0 Read all the questions carefully before starting to work.
0 Give complete arguments and explanations fo
MATHEMATICS 226, FALL 2016, PROBLEM SET 1
Solutions
1. (24 marks: 8 for each set) Sketch the sets in 3-space whose points
(:13,y, z) satisfy the given conditions. Specify the boundary and the in
terior of each set. Are these sets open, closed, or neither?
Math 226 Final Exam, December, 2002
[10] 2. When x, y, u, v are related by the pair of equations x = u3 + v 3 3, y = uv v 2 , the
symbol u/x has two possible interpretations. Explain what these are, and calculate
both of them at the point corresponding to
Solutions to Math 226 Final Exam, December, 2002
2. When x, y, u, v are related by the pair of equations x = u3 + v 3 , y = uv v 2 , the symbol u/x has two
possible interpretations. Explain what these are, and calculate both of them at the point correspon
An Example With Two Lagrange Multipliers
In these notes, we consider an example of a problem of the form maximize (or minimize) f (x, y, z ) subject to the constraints g (x, y, z ) = 0 and h(x, y, z ) = 0. We use the
technique of Lagrange multipliers. To
A Limit That Doesnt Exist
In this example we study the behaviour of the function
(2xy )2
xy
if x = y
0
f (x, y ) =
if x = y
as (x, y ) (0, 0). Here is a graph of the level curves, f (x, y ) = c, of this function for
various values of the constant c.
1
1
y
Roots of Polynomials
Here are some tricks for nding roots of polynomials. These tricks work well on
exams and homework assignments, where polynomials tend to have integer coecients and
roots that are integers, or at least fractions.
Trick # 1
If r or r is
Taylor Expansions in 2d
In your rst year Calculus course you developed a family of formulae for approximating a
function F (t) for t near any xed point t0 . The crudest approximation was just a constant.
F (t0 + t) F (t0 )
The next better approximation in
Equality of Mixed Partials
2
2
f
f
Theorem. If the partial derivatives xy and yx exist and are continuous at (x0 , y0 ),
then
2 f
2 f
(x0 , y0 ) = yx (x0 , y0 )
xy
Proof: Here is an outline of the proof. The details are given as footnotes at the end of
th
Approximating Functions Near a Specied Point
Suppose that you are interested in the values of some function f (x) for x near some xed point x0 .
The function is too complicated to work with directly. So you wish to work instead with some other function
F
The Chain Rule
The Problem
You already routinely use the one dimensional chain rule
d
f
dt
x(t) =
df
dx
x(t)
dx
( t)
dt
in doing computations like
d
dt
sin(t2 ) = cos(t2 )2t
In this example, f (x) = sin(x) and x(t) = t2 .
We now generalize the chain rule
Directional Derivatives
The Question
Suppose that you leave the point (a, b) moving with velocity v = v1 , v2 . Suppose further
that the temperature at (x, y ) is f (x, y ). Then what rate of change of temperature do you feel?
The Answers
Lets set the beg
Limits
Notation.
IN is the set cfw_1, 2, 3, of all natural numbers
IR is the set of all real numbers
is read for all
is read there exists
is read element of
is read not an element of
/
A B is read the set of all A such that B
/
If S is a set and
The Binomial Theorem
In these notes we prove the binomial theorem, which says that for any integer n 1,
n
n
n
(x + y ) =
+m
x y n =
=0
x ym
n
where
n!
!(n )!
=
(Bn )
,m0
+m=n
The proof is by induction. First we check that, when n = 1,
n
n!
!(n )! x
y n =
Solutions to Math 226 Final Exam, December, 2001
3. Find the equation of the plane which contains (1, 2, 3) and (4, 6, 7) and is perpendicular to the plane
3x + 2y + z = 1.
Solution. Since the plane contains the points (1, 2, 3) and (4, 6, 7), the vector
Math 226 Final Exam, December, 2000
[10] 1. Let S be the surface x2 + 4y 2 9z 2 = 1.
(a) Sketch the surface.
(b) Find the equation of the tangent plane to the surface at (1, 3, 2).
(c) Suppose that a smooth curve C lies on the surface and passes through t
Math 226 Final Exam, December, 2001
[8] 3. Find the equation of the plane which contains (1, 2, 3) and (4, 6, 7) and is perpendicular
to the plane 3x + 2y + z = 1.
[12] 5. Assume temperature (in degrees Celsius) is a C 1 function T : IR2 IR and T (0, 0) =
MATHEMATICS 226, FALL 2016, PROBLEM SET 5
Solutions
Section 12.8, Question 15: (Not graded)
x x
cos r sin
r
y
=r
y =
sin
r
cos
r
The transformation is invertible except when r = 0.
Section 13.1, Question 3: (10 marks) We have f1 (x, y) = 3x2 3y an