Of course there are
also
other important mathematical tools. Algebra, discrete
mathematics,
probability and statistics, topology, and computer science all have
roles
to play as research areas and as tools in applications, and these tools
complement each other. The increasing power
and availability of technology enhances their usefulness and doesn't
replace
the need for any of them. Technology makes it possible to explore
calculus
numerically
and graphically in ways
which were impractical
even
a decade ago, but technology cannot replace understanding the
subject.
A calculator or computer is only an assistant that needs an intelligent
user. Otherwise it may be unable to find an answer, or may
produce
an "answer" which is misleading or even completely incorrect !
The
technology needs a user who understands and can tell it exactly
what it's supposed to do.
There are lots of details and techniques for us to learn, but in the big picture there are only two "great ideas" in calculus. Both of them are illustrated on the dashboard of your car.
1) The first idea concerns "how fast is a quantity changing?" For example, if you're drivingNearly all of Math 131-132 consists of
down the highway and s represents the distance you've traveled from home, then you might be interested in
how fast s is changing (measured, perhaps, in km/hr). How fast s is changing at a time t is your velocity v at
that time. It's analogous to the speedometer on your car.Studying rates of change involves a concept from Calculus I called the derivative. The velocity v is the derivative of the position function s. If we think of s = f ( t ) as a function of time, then some of the ways of writing the derivative are v = f ' ( t ) or v = ds / dt.
Of course, instead of how fast s is changing, you might be interested in the rate of change of some other quantity. For example,
Whether a rate of change occurs in biology, physics, or economics, the same mathematical concept--the derivative--is involved in each case.- y = the size of a population of bacteria
- y = the amount (mass) of a radioactive isotope in a sample
- y = the price of a gallon of gasoline.
2) The other idea is the "opposite" of the first one. If you know how fast some quantity is changing, then how big is it at a certain time? On the highway again, you could imagine trying to figure out how far s you are from home at time t by studying the velocity v (speedometer information).
This is easy to do if the car moves at constant velocity: in that case, s = vt , that is distance = (rate)(time). But finding s from v is harder if the velocity is always changing during the trip. The calculation of a total amount s from a rate v =ds/dt involves a concept from calculus called the integral. It's analogous to your car's tripmeter which (if you set it to 0 when you start out) measures s on your dashboard.
Instead of a distance, you might be interested in trying to compute the total amount of money in an account if you know how fast it's growing (the interest rate), or the total number of people infected with some disease at time t if you know the infection rate. The same mathematical concept, the integral, is involved in each case. The study of the integral and how it relates to the derivative, is the main theme of Calculus II.
All of this is hard
work.
But what did you expect in learning about "one of the great
intellectual
accomplishments of the human mind?" The day-to-day work may seem
tedious at times, but it's essential, like finger exercises for the
future
pianist. Or, to change the analogy, it's like learning a new
language:
it can open new vistas but only if you're willing to memorize
vocabulary,
learn to conjugate verbs, and practice, practice, practice!
We'll see indications
of
some of the diverse applications of calculus during the course. But
it's
not a course in physics, biology, economics, or business. Many of
the most interesting and significant applications you will have to meet
elsewhere. That should be a relief! It's certainly nice to
get ideas about what the material is good for, but students who want
"more
applications" in math courses often don't realize that applications,
generally,
are much harder: a little like "story problems," only worse. That's
because
applying math to a concrete situation involves taking a complicated,
messy
real-life situation, sorting out what's relevant to the problem and
what
isn't, creating a mathematical approximation ("model") to reality, and
then setting up a mathematical formulation of the problem. Only at that
point are you ready to apply the "tools" from calculus. Setting up a
mathematical
model of a complicated real-world situation is often not easy, and it
sometimes
requires detailed knowledge of another subject such as physics, biology
or economics. In a calculus course you learn the tools and see
them
applied in some "tidy" applications which only hint at the real
usefulness
of the subject. The biologists, chemists, physicists, engineers,
architects,
economists, and others who have recommended that you take a calculus
course
will have to show you the reasons why it's useful in their own fields (please,
put
them on the spot and ask!! ) For now, try to learn to
appreciate
the subject itself, its beauty, and how the pieces fit together.
Expectations and Advice
You'll probably find that the material in most university courses, including this one, is covered much more quickly than it would be in high school. You'll probably also be asked to have a greater command of the material than before, especially in understanding the ideas (not just the techniques) and applying them in new situations. This may take some adjustments in your style of learning.
The instructor and TA are here to help you understand and learn the material, but actually learning it is your responsibility. The lectures are designed to try to highlight the important ideas and give you some perspective on the material, so that you can digest and really learn the material as you think, read the text, discuss the material with other students, and practice on problems. You should expect that more learning will occur outside the classroom than in it. On the practical side, the average student should expect to spend at least a couple of hours on calculus for every hour in the classroom. If you do this for all your courses, then being a student is easily the equivalent of a full-time job.
Read the text. Your instructor can't possibly say everything that's there. An hour's lecture wouldn't be enough even to read the day's assignment aloud, let alone try to highlight or clarify. Your instructor's job is to complement the text so that you can learn from it. Simply relying on your lecture notes will (and should!!!) probably only give you the highlights of the material. To ignore the text is to start out looking for a minimal understanding.
Reading a math
text requires
active involvement and is often a slow process. You might spend a
few
hours reading an assignment of 100 pages in another subject. A
reading
assignment of 10 pages of mathematics might take just as much
time.
Read it over once quickly just for an overview. Then read
carefully
with scratch paper at hand and fill in missing steps. When the author
simply
asserts "...this expression simplifies to...", simplify it!
When you read "you should check that...", check it! Try to work
some
of the examples in the text before reading them. Play with the
material:
ask yourself "what if I changed the problem to...?". Write
down questions that come to mind. If you can't answer them, ask
your
friends. Try to understand why the assertions in the text are true.
Ideally, you should read
ahead. The lectures will be much more valuable if you have
tried
to digest some of the material ahead of time. Also, you'll be
able
to judge that some things on the board needn't be written down.
For
example, you'll know that those formulas the instructor is writing on
the
board are just the ones listed in the text. That frees you to
think
about what's being said and not be always frantically transcribing
material
into your notebook.
If you choose to ignore the advice about reading ahead, then at least read the material in the text before trying to do the assigned problems. Some students start out with the problems and when they get stuck, they page back through the text looking for a formula or example to help. This puts the focus in the wrong place. Simply being able to "do the problems" represents at best a minimal strategy for the course. You do have to do problems, and you do want to have certain technical skills that the problems sets can help develop, but you also want to understand the material. Carefully reading the text is essential for that.
You can't learn to speak French by merely sitting and listening, and you can't learn mathematics by watching someone else do problems. You need to do as many problems as you can realistically find time to do: the more, the better. It's useful to work with other people on homework as long as you're involved as an active partner. Compare solutions; ask each other questions; make up quizzes for each other; agree on some extra problems to try together--such as the review problems at the end of each chapter. Prod each other to do the assignments! Do whatever helps! But all of this is to improve your understanding: your study group isn't there at quizzes, tests, or when you need your math in another course. See the syllabus section Homework Policy and Suggestions.
There are facts and
formulas
you need to have at your fingertips: that is, there are some things
you simply need to memorize. But that's only the
beginning.
Formulas are just tools. Some routine problems, admittedly, are
designed
merely to be sure you really can handle the tools, but when attacking a
non-routine problem, don't just try to hunt around for a formula. To
quote
an old but accurate text,
Many problems on tests
will be "like" problems you've seen before. But some won't
be.
This isn't grounds for complaining that "we never saw a problem like
that
before." If you can only do problems like ones you've
seen
before, what's the point? The purpose of the problems is to
help
you understand the ideas and techniques, not merely to learn how to do
more similar problems.
Stay on top. Sometimes it's hard to keep up with everything, but it's important not to fall behind in the course. A lot of the material we cover is interdependent, and if you're not comfortable with the material from two days ago, today's lecture might be totally incomprehensible. It's better to be a bit ahead--the lectures will mean more and you've got a buffer when an emergency strikes.
Also, there's a limit to how fast you can digest material and learn actually to use it in your thinking. Avoid cramming just before tests by staying on top of things. It makes life much more pleasant.
Make use of your instructor and the TA in your discussion session. Office hours are set aside specifically to see students; those times are for you! Most instructors are also willing to set up special meetings with students as time permits. Your responsibility is not to waste these times. Try to solve your own problems first, together with your friends. What you learn that way will be more valuable than the same answer "given" by your instructor or TA. Come to office hours or your discussion section prepared with specific questions. Bring questions to your TA or instructor quickly, as soon as you realize there's some difficulty you can't resolve. (It's hard for any instructor to cope with "I haven't understood anything for three weeks"; we aren't magicians.)
Finally, one last word. Some of you have had some calculus already: this can be a blessing or a curse. It's a curse if it makes you think you know more than you do and traps you into slacking off until you suddenly realize you're in trouble. Even if you do already understand what's being covered in class and the homework, this shouldn't make you think there's nothing to learn. Try instead some of the harder problems in the text so that the time's not wasted. If you're bored and want to talk about some harder problems or find some extra reading, talk to your instructor. If you're serious, your instructor will probably be delighted! And, if you're on top of the current material, then you have a great opportunity to be helpful to some of your friends who are still working on it!