Mathematics 133 -- Intensive Calculus for Science 1
Discussion 1 -- Functions, continued
September 4, 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.
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. The idea is that, 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,
some of you may have to adjust some of your preconceptions.
In particular:
- 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
We have seen that linear functions can be described by equations
y = mx + b or y = m(x - x0) + y0.
In the first question, we will explore some of the information that
is contained in these equations, and how that relates to the graphs
of the linear functions.
The domain of a function is the set of "input" values that can
be substituted into the function; the range of the function is
the corresponding set of "output" values. Our last questions
will deal with determining the domain and range of functions
defined by formulas.
Discussion Questions
- A) (No graphing calculators on this question!) Do problem 14
from Section 1.1 (page 7) of the text.
Give full explanations of how you are matching the formulas with the
graphs, in one or more complete sentences.
- B) Recall that saying a quantity y is (directly)
proportional to another quantity x is the same as saying
y = kx for some constant k.
- Explain why saying that y is (directly)
proportional to quantity x says that y
is a linear function of x.
- For example, the circumference of a circle is
proportional to the radius of the circle. What is the constant
of proportionality in that case?
- Write as a formula: "the energy E expended by
a swimming dolphin is proportional to the cube of its speed v."
- C) The domain of a function defined by a formula f(x)
is usually taken to be the set of all x that can be
"plugged into" the formula without causing an algebraic problem
like trying to divide by zero, or trying to take the square root
of a negative number. According to this, what is the domain of
each of the following functions?
- f(x) = 1/(x2 - x)
- g(x) = sqrt(3 - x) (sqrt means "square root")
- D) The range of a function defined by a formula
can often be determined by trying to solve the equation y = f(x)
for x. If this is possible for a given y (that is,
if you can find an x that makes the equation
y = f(x) true, then
y is in the range. What is the range for each of the following
functions:
- f(x) = x2
- g(x) = 1/(1-x)
Assignment
One write-up per group of solutions for these problems,
due Monday, September 10.