Mathematics 134 -- Intensive Calculus for Science
Discussion 1 -- Definite Integrals and the Fundamental Theorem
January 17, 2001
Working in a Group
Since this is the first of the discussion class of the semester,
a few words about this way of working are probably in order for
those joining the class
In the discussion meetings of this class, we will be aiming
for collaborative learning -- that is, for an
integrated group effort in analyzing and attacking the discussion
questions. The ideal is for everyone in each of the groups to be
fully involved in the process. By actively
participating in the class through talking about the ideas
yourself in your own words, you can come to a better first
understanding of what is going on than if you simply listen
to someone else (even me!) talk about it.
However, to get the most out of this kind of work,
you may need to adjust some of the way you think
about class work. For instance,
- This is not a competition.
You and your fellow group members are working as a
team, and the goal is to have everyone understand what the
group does fully.
- At different times, it is inevitable that
different people within the group will have a more
complete grasp of what you are working on and
others will have a less complete grasp. Dealing with
this a group setting is excellent preparation for real
work in a team; it also offers opportunities for significant
educational experiences:
- If you feel totally "clueless" at some point
and everyone else seems to be "getting it,"
your job will be to ask questions and even pester your fellow
group members until the point has been explained to your full
satisfaction. (Don't forget, the others may be jumping to
unwarranted conclusions, and your questions may save the group
from pursuing an erroneous train of thought!)
- On the other hand, when you think you do see
something, you need to be willing to explain it patiently to
others. (Don't forget, the absolutely best way to make
sure you really understand something is to try to
explain it to someone else. If you are skipping over an
important point in your thinking, it can become very apparent
when you set out to convey your ideas to a team member.)
In short, everyone has something to contribute, and
everyone will contribute in different ways at different times.
Background
The Fundamental Theorem of Calculus says that the definite integral
of f from a to b is equal to the
difference F(b) - F(a) if F is an
antiderivative of f. (If F'(x) = f(x),
then the integral of f gives the total change of F.)
As we will see, once
we have a collection of short-cut rules for finding antiderivatives,
this will give a way to compute exact values for a number
(though far from all!) definite integrals of interest. Before
we get to that, however, we want to spend a day or two really
concentrating on the meaning of this important theorem
in graphical and conceptual terms.
Discussion Questions
All of these are problems from our textbook:
- Section 3.4, problem 2
- Section 3.4, problem 3
- Section 3.4, problems 10-13 (a rough hand sketch of the graph is OK,
you don't need to reproduce it exactly -- just get the right general
shape)
- Review Problems for Chapter 3 (page 177), problem 19
Assignment
One write-up per group of solutions for these problems,
due Friday, January 19.