grade cutoff percent_receiving percent_rank A 70 27 73 B 60 28 45 C 50 24 21 D 40 11 10 F 0 10 0
Here is a practice exam, to which we will add a problem involving Taylor expansion of a function of several variables. For a function of 3 variables, up to degree 2; but f(x,y) could be also deg 3 or 4.
Errata:
To review Taylor expansion, study section 14.10 in the book and make sure that you can do the example problem (without peeking at the solution!).
For practice I'll suggest problem 9 (or problems 1 and 5 if you have not already done them) in the exercises for section 14.10. Here is a note to help you in your review:
The Taylor series
approximation T
of the function f
expanded around r0=(0, 0) is
just the polynomial whose partial
derivatives agree with those of f at (0, 0):
T(x,y) = f(0,0)
+x fx
+y fy
+(x2/2) fxx
+xy fxy
+(y2/2) fyy
+(x3/6) fxxx
+(x2y/2) fxxy
+(x y2/2) fxyy
+(y3/6) fyyy
+ ...,
where the partial derivatives are all evaluated at (0, 0).
To check your expansion (or the result above),
all you have to do is to check that each partial
derivative of the expansion agrees with the
corresponding partial derivative of the function.
For example, if in the generic expansion above
you want to check the fxxy
term, just differentiate
(x2y/2) fxxy
twice with respect to x
and once with respect to y
(remembering that fxxy
stands for fxxy(0,0)
and is therefore just a constant), and you will
discover that
Txxy(0,0)= fxxy(0,0),
as required.
For a Taylor series expanded around the point
(a, b), in the discussion above
replace x with (x-a)
and y with (y-b)
and evaluate all functions and derivatives at
the point (a, b) instead of the point (0, 0).
So the Taylor series
approximation T
of the function f
expanded around r0=(a, b) is
just the polynomial whose partial
derivatives agree with those of f at (a, b):
T(x,y) = f(a,b)
+(x-a) fx
+(y-b) fy
+((x-a)2/2) fxx
+(x-a)(y-b) fxy
+((y-b)2/2) fyy
+((x-a)3/6) fxxx
+((x-a)2y/2) fxxy
+((x-a) (y-b)2/2) fxyy
+((y-b)3/6) fyyy
+ ...,
where the partial derivatives are all evaluated at (a, b).
(usually at the origin a=0, b=0).
To check your expansion (or the result above),
all you have to do is to check that each partial
derivative of the expansion agrees with the
corresponding partial derivative of the function.
For example, if in the generic expansion above
you want to check the fxxy
term, just differentiate
((x-a)2(y-b)/2) fxxy
twice with respect to x
and once with respect to y
(remembering that fxxy
stands for fxxy(a,b)
and is therefore just a constant), and you will
discover that
Txxy(a,b)= fxxy(a,b),
as required.
Here are problems for chapter 16 that you can start working on:
Here are statistics and grading levels for the midterm:
We will be returning the examinations tomorrow at the
beginning of lecture. Please sit according to the following
seating chart to assist us in the quiet and orderly return
of the examinations (unless you have a vision or hearing
problem, in which case come up and ask for your exam when
we are returning the exams for
your section).
Solution of the problems in Chapter 15 requires the following basic skills and background:
Your TA will assign some specific problems from Chapter 15,
if you ask for specific problems. I have been doing selected
problems from pp. 1138–1141 in class.
Amir Assadi
Thursday April 9th overlaps with Passover. There will be no test
scheduled for Thursday April 9th.
Midterm II date, time and location: April 7th, in Class, during
the regular lecture hour. Closed Book, 2 index card and standard
calculators are allowed, but no notes, no laptops, no other
materials.
Test Problems will have one problem from Lagrange's Multipliers
with one constraint (as in Midterm I) and from Ch 15, sections
15.1–15.6 (last section will not be in the exam, also the
sections that cover moments of inertia will not be on this test).
A Practice Midterm Exam will be made available to students who
attend the TA Problem Sessions, starting on Monday March 30.
The students must go to their TA session, or office hour (or
make direct arrangement with their TA) to receive a copy of the
practice Midterm II.
Thursday April 9th will be spent on the solution of Midterm II.
NOTE: Your reading and solution of HW and WebWork should be on
WEEKLY basis. When the WebWork closing time is extended, the
longer time-interval intends to bring more flexibility for the
students who wish to work more often on the problems and have
more time to get a perfect score on all problems. The WebWork
deadline is NOT intended for delay in your work on the problems.
If you postpone work on the WebWork until the last day and hours,
your delay could have a negative effect on the rate and extent
of learning the materials in the lectures and the TA sessions.
Please act responsibly and keep up with the course lecture pace.
Amir Assadi
Here are statistics and grading levels for the midterm:
Here are
my solutions to the midterm.
Here are
George's explanations of some of the most common
mistakes people made on the exam..
Here is a copy of the
practice exam for first midterm that was distributed
in class on Thursday.
We have postponed the due date of webwork 5 until
Wednesday, March 25.
We will be having an in-class open book practice
exam before the midterm, probably on Thursday, March 5.
The practice problems will have samples that are very similar
to the assigned problems, the worked examples in the text,
and the webwork. The practice exams are open book.
There will be no grades assigned to practice exams, and also,
as a matter of practicality, there will be no solutions
manual to them.
The midterms and final tests will be similar
to the practice exams, but closed book. We
will allow
We recommend that you do the following exercises to make
sure that you really understand the material.
Your teaching assistant will decide how to hold you accountable
for doing them. (My students should expect to be quizzed! —Alec.)
Don't forget that the first midterm
is during the lecture period on
Tuesday, March 10, as indicated on
the evolving syllabus.
We have not had any assigned written work lately.
My sense is that some of you do not have a clear
idea of how to focus your studying.
My studying philosophy is
to begin (and end) with the goal.
I suggest the following natural points of focus for your study:
An excellent way to consolidate your understanding
and give yourself an overview of what we have covered is
to write up your own summary of the
essential definitions and formulas (basically
the boxed material).
You should be able to work each example problem in the
sections that we have covered without looking at
the solution.
For each example problem that you have difficulty with,
you should work similar problems from the exercises for
that section until you have built up confidence.
(If you choose odd-numbered problems you can check your
answers with the back of the book or the student
solutions manual.)
It is crucial that you get enough practice
in solving problems; my experience is that
most students do not. Merely doing the webwork problems
is not enough practice. For each section
you should attempt
at least one problem from each nontheoretical section of
the exercises!
Finally, doing problems from old examinations will
give you a good sense of the level of facility
that will be expected of you.
I have inserted notes on Chapter 12 in the
Course notes.
If you are still stuck on problems in webwork HW2, see
hints for WeBWorK HW2.
If you want to do your assignments in an environment where
there is a TA to help you when you get stuck, try going to
the math lab
(B227 Van Vleck) any time
Monday—Thursday, 3:30–8:30pm or
Sunday 3:30—6:50pm.
Please read
Instructor Remarks on 14.4–14.6.
We are climing a ladder.
Before coming to the lectures you must read and know
the material from last week (§§14.1–14.4)!
A couple comments of my own:
As you can see, we now have a
web page for the lecture.
To see an overview of what we have done
and what we are going to do consult
the evolving syllabus for this course.
I have posted
hints for WeBWorK HW1.
Questions 8–10 require careful logical reasoning and
a thorough understanding of lines and planes.
He plans to start straight off with section 14.2-14.3
on Tuesday, which means that you will need to study
section 14.1 on your own before coming to lecture on
Tuesday.
He also asks you to review chapter 2 as preparation.
Why? Because chapter 2 defines limits and continuity
of single-variable functions, and section 14.2
extends the definition of limit and continuity to
functions of multiple variables.
Limits are the foundation of calculus. Most people
struggle with the concept of a limit, but investing the
time to understand limits well will pay dividends.
The four important things to know are:
I elaborate on each of these four points as follows.
(A) Limits for a function of two variables, e.g. f(x,y),
are defined very similarly to limits of functions
of one variable. The primary difference is that
the absolute value is replaced by the distance:
Definition of limit: We say that the limit of f(x,y)
as (x,y) goes to the limit point (x_0,y_0) equals L
if for every desired positive upper bound epsilon
(no matter how small) on the "error" |f(x,y)-L|,
there exists a (small enough) neighborhood size delta
so that if (x,y) is within a distance delta
from (x_0,y_0) (but is not equal to (x_0,y_0)
then f(x,y) will be within a distance epsilon of L.
(Remark: the value/definition of f(x_0,y_0) is completely
irrelevant in this definition.)
(B) Definition of continuity: We say that
f(x,y) is continuous at (x_0, y_0) if
the limit of f(x,y) as (x,y) goes to (x_0,y_0)
equals f(x_0,y_0). This says two things:
(1) the limit exists, and
(2) the limit agrees with the value of the function.
(C) I would say that the most important property of continuous
functions is the composition property. It basically says
that composition of continuous functions is continuous, i.e.
you can bring the limit inside the parentheses. In particular,
the properties of limits in section 14.2 essentially say that
addition, subtraction, multipliction, division by a nonzero
number, and taking of most powers are all continuous functions.
(D) Showing that a limit does not exist can be a bit trickier for
multivariable functions than for one-variable functions.
For one-variable functions, for the limit to exist it is enough
for the limit from the left and the right to agree,
but for a multivariable function you must get the same limit no
matter what path you use to approach the limit point.
To begin doing the homework, select "Homework Sets"
(again under the "Main Menu"). Then click on "HW1".
The numbers in the problems are randomly chosen
especially for you.
When I do the first problem, I find that I can enter an
answer like "2/sqrt(33)". (WeBWorK will calculate
the decimal value of the expression you enter and compare
it with the decimal value of the correct answer.)
Note that you are not done when webwork tells you, "The answer
above is correct." You must hit "Submit Answers".
(Otherwise when
you look at the problem list for HW1 you will be dismayed to
discover that webwork thinks you haven't done the problem yet!)
I urge you to get started. The first ten problems
are review from chapter 12. The last five problems
cover chapter 13. After that you should be ready to
tackle the four more challenging written problems.
HW2 is related to chapter 14.
section problem_numbers
16.1 1–8, 13, 15, 23, 25
16.2 11, 13, 21, 23(a), 29(a,b), 33
16.3 1, 3, 5, 7, 9, 13, 17, 19, 25, 34, 37
16.4 3, 5, 7, 15, 17, 22, 29, 35, 39, 40
Your TA will decide if you should submit these.
possible: 100 points
median score: 78
average score: 76
grade cutoff percent_receiving percent_rank
A 92 19 81
AB 88 6 75
B 80 21 54
BC 77 8 46
C 63 23 23
D 50 17 6
F 0 6 0
___________________________________
| front stage |
|___________________________________|
___________________________________
| 321 M 8:50 325 M 11:00 | 329 M | (rows 1-4)
| 322 W 8:50 326 W 11:00 | 329 M | (rows 5-7)
| 323 M 9:55 327 M 13:20 | 330 W | (rows 8-10)
| 324 W 9:55 328 W 13:20 | 330 W | (rows 11-14)
|(George_Brown Yuan_Fang | Alec) |
|___________________________________|
possible points: 140
median score: 106
average score: 104
grade cutoff percentage percent_receiving percent_rank
A 125 89 16 84
AB 120 86 8 76
B 109 78 22 54
BC 104 74 7 47
C 92 66 20 27
D 81 58 16 11
F 11
section 15.1: 1, 5, 7, 11, 15–25(odd), 31, 43, 47, 59
section 15.2: 3, 11, 16
section 15.3: 3, 5, 7, 21, 29, 40
section 15.4: 7, 8, 21, 23, 25, 43
section 15.5: 17
section 15.6: 1, 11, 13, 31, 39
section 15.7: 1, 9, 21
one two index cards (both sides) for writing formulas
and other important information so you will not need
the book or notes. Nothing else will be allowed except for
your calculators (no laptops, no fancy small-format PCs or
electronic books or work-pads, etc.) A calculator could be
programmable, and do graphics etc, but only at the level of a
calculator like those they used in other calculus courses!
section 14.1: 3, 9, 13–18
section 14.2: 1, 9, 13, 17, 21, 27, 29, 31, 33, 35, 37, 51, 57
section 14.3: 7, 13, 19, 33, 34, 41, 43, 51, 57
section 14.4: 1, 5, 9, 13, 15, 25, 27, 29, 30
section 14.5: 2, 5, 9, 15, 17, 21, 27, 29, 36
section 14.6: 1, 3, 9, 18, 22, 37, 38
section 14.7: 3, 17, 39, 43, 44, 53
section 14.8: 1, 5, 7, 9, 17, 18, 26
On the second problem I find that I can enter an answer such as
"arccos(-12/(4*sqrt(13)))".
For the third problem, the scalar projection is the
length of the projection.
Math 234: 4:30-5:30 p.m. Room 6203 Social Sciences
* Wednesday January 21 : Vectors, Dot and Cross Products
* Thursday January 22 : Parametric Equations, Lines and Planes
* Monday January 26 : Polar, Cylindrical, and Spherical Coordinates