Welcome to Math 171…
Mathematics
171 is part of a two-semester sequence, 171 and 217, and most students
would not be taking 171 unless planning to go on to 217. The two together amount
to ten UW credits, both for precalculus (equivalent to mathematics 114 or 112
and 113) and the first semester of the standard calculus sequence, mathematics
221. Some parts of 171 will probably seem familiar: You may have had a high
school course that covered some or all of the precalculus material (The names
of the high school courses are continually changing: it might have been called
precalculus, or functions, or algebra and trigonometry, …) but the UW placement
tests did not show that you were fluent enough in it that you would be well
prepared for calculus. Notice that word “fluent”: You may know this material
somewhat, but in a calculus course this material is the language that is spoken
as calculus is taught. So if you often had to stop for long to think out what
something means, you would fall further and further behind on the new material
and would soon be in deep trouble.
The
goal of 171 is to be first and foremost a calculus course, and together with
217 it will cover all of the calculus material of 221, but with a pace that
lets us fill in from time to time precalculus material as needed. One advantage
to this is that we can try to see why it is needed as we proceed: One
reason people don’t always remember enough from their high school classes to go
directly to a full-time calculus class is that they were never given reasons,
“I need to know this because …” Most of us are not good at working hard at and
remembering things just because somebody says we have to learn them, we learn
better when we can see a reason to care. (Now there is still a problem here:
Why are you taking calculus in the first place? For most of you it is because
somebody said you had to! I will try as we go along to show some of the many
places where calculus has proved invaluable in solving real world problems, but
I won’t be able to do examples all the time or to cover all the many different
interest areas represented with about 450 students this semester!)
What is calculus? Why has it been so amazingly successful at solving real-world problems for over three hundred years now? To start with: What is mathematics? That could generate a long answer, but the only part I want to consider right now is that mathematics is not doing calculations! It is a language, and like any other language it is used for saying things. Mathematics is better than other languages at saying some things, and worse at saying other things. (Try writing a love poem in mathematics… People have done it as a joke.) Mathematics has proved outrageously good when talking about almost anything that can be described precisely. Within the things we can talk about using mathematics, calculus is the science of describing how things change. We know that as the seasons move from winter to summer, sunrise time gets earlier. That is the kind of change we mean, but “move from winter to summer” and “gets earlier” are not very precise and are not stated in the language of mathematics. In the part of calculus we get through this semester, we will generally (a) think of one thing as changing, (b) think of some other thing as determined by the first, and (c) if the first thing is changing, analyze how rapidly the second will be caused to change. So to make that mathematical: (a) becomes “We assign numerical values to some ‘thing’, and call that thing a variable. (b) becomes: “We find a relation between that variable and another variable that depends on the first”, i.e. is a function of the first, like the time of sunrise might be a function of where we were in the year. (c) does not even need rewording. (Note that many things depend on more than one other variable: Sunrise depends on where you are on the earth, as well as where in the year. But in this class we will almost always ignore all but one causal factor. Later classes get around this limitation!)
So here is what we need from what is generally called “Precalculus”: (i) the ability to work fluently with functions. That is it! But note that word “fluently”. In a calculus class algebra is the language spoken. You need to be able to use algebra to say things and to understand what someone else (e.g. me, or your TA) says, and if you have to stop and work out the algebra you will fall too far behind very quickly. (You probably would not expect to do well in a class on French literature that was taught in French, if you were not fluent in French!) Most of the precalculus we study will be either about functions specifically or about the parts of algebra (e.g. polynomial arithmetic, exponents, working with fractions) that are frequently used to define particular functions. We also will pay some attention to the specific functions called “trigonometric functions”. While we will need to be able to do right-triangle calculations with them (e.g. “how tall is that flag pole if the sun is so many degrees above the horizon and the shadow of the pole is this long?”), the emphasis is on them as functions: The trig functions are especially useful in describing virtually any real-world measurements that vary in a somewhat repeating fashion, such as the way sales or profit change with time when selling swimwear: You know there will be certain big seasons each year, as well as some variation from year to year depending on fashion, weather, etc.
Now enough of that general discussion of what we are embarking on. Here are some specific facts about our class.
There are two textbooks for this
class. The calculus text Thomas’ Calculus, 12th edition, will
also be used in subsequent calculus classes. (The new 12th edition
is very similar to the 11th. If you have a copy of the 11th
you might be able to get by if you can refer some times to somebody’s copy of
the 12th: The page numbers and problem numbers for assigned work
won’t be the same.) The precalculus text Precalculus; A Prelude to Calculus
by Axler was used last year, so there will be used copies around.
The schedule for the class tells what
sections to read before class. It is a good idea to read any math book
with pencil and paper handy, to try out things and to note questions, and if
you do that before class you can actively look for what it was you needed.
The schedule also lists (many!)
problems from the texts. You should make sure you can do all of those: Exam
problems won’t necessarily be taken from among those, but if you can do all the
listed problems you should be in good shape for the exams. You are not expected
to write them all up to hand in! But you should be sure you understand them,
which is more than just being able to find an answer. (See below re grading:
Your TA might ask you to write up some of them to turn in for part of the
discussion section grade.)
On the schedule page, numbers preceded
by “P” refer to sections in the Axler Precalculus book, while numbers preceded
by “C” refer to sections in the Thomas Calculus book.
Grades will be based on (i) your
scores on the midterm exams, (ii) your score on the final exam, and (iii) your
work in your discussion section. There will be three “midterm” exams: At UW
that term is taken to mean any exam before the final, not necessarily in the middle
of the term! They are shown on the class schedule. Each midterm exam will count
100 points. The final exam will count 200 points, and the final exam will be to
some extent cumulative, including work from earlier in the semester. Each
midterm will mostly focus on work since the last midterm. (But in any math or
science class, what you learn next always uses what you did earlier.) You will
get a discussion score of up to 100 points from your TA. (Your TA will discuss
with me how that is derived, and how we will make sure at the end of the
semester that scores from different TAs are used equitably. TAs may ask you to
hand in homework, do work at the board, take quizzes, write a journal, or
combinations of these and other measures. So at the end of the semester you
will have a total of up to 600 points, from all the exams and the discussion
section, and we will construct final grades from the point totals. I will give
you a ``curve’’ after each midterm exam (this many points for an A, etc.) but
that is just to tell you how you are doing: The final grades will not be
just an average of the exam letter grades. Those letter grades for midterm
exams will be on the A-B-C-D-F scale, but final grades will be on UW’s
A-AB-B-BC-C-D-F scale.
You will be allowed to use calculators
on exams, and of course there is no way we could enforce any rule on calculator
use on homework. Your TA might or might not allow calculators on quizzes: There
are good arguments either way. The exam problems will be constructed to be
“calculator neutral” so far as possible: In particular I don’t want one person
who can afford a fancy calculator to have an advantage over another who cannot.
Exam problems will thus all be “doable” in a straight-forward way without use
of a calculator, and you will be expected to show exactly how you arrived at
your answer using the mathematics we have been studying, so just writing down
an answer from a calculator will get little if any credit anyway.
At http://www.math.wisc.edu/~wilson/Courses/Math171/171fall11/index.htm,
the course website, I will post general announcements, sample problems, etc. In
addition I will send email messages to the class: Make sure your email address
is correctly entered in the UW database. The university maintains the email
list based on your email address on record with the registrar. (I cannot add to
or modify that automatically constructed list!)
My schedule, including office hours,
is posted at my web site, http://www.math.wisc.edu/~wilson.
My office is 807 Van Vleck Hall. That office is shared with many other people,
so it is better to email me than to try to reach me by telephone. My email
address is wilson@math.wisc.edu.
A personal note: I officially retired
a couple of years ago. I still enjoy teaching and enjoy the material we will be
covering in 171. But I have suffered several health problems (one knee has been
replaced twice and the doctor says it needs a third time, one shoulder has been
operated on four times and replaced twice, and my back had the bottom section
beefed up with titanium last summer…) so I may be in some pain in class. I
apologize in advance if I sit down during lecture, I never used to do that! And
I have for now only listed a limited set of office hours at my
website. I will see as we go along whether I can extend those, and I will try
when possible to see you at other times if you email me to make arrangements. (I cannot
promise to come in at just any time. For one thing, UW has not nearly enough
disabled parking places, and there are times when I know I won’t be able to
find a place.)
Bob
Wilson