Study Guide for the 1st midterm
Stuff from Chapter 1
- Make sure you know function notation and understand inverse
functions (problems 6.2, 6.3, 6.8--6.10)
Limits
- Be able to prove that lim f(x) = L using the
ε—δ definition of the limit, as in problem 12.1.
- Use the limit properties to compute limits. Read all examples in
§15, problems 20.1—20.5, 21.1, 21.2, 23.1. (The last one is
about derivatives, but it boils down to computing limits.)
- Use the limit properties to show that a limit cannot exist, as in
examples 16.3, 16.4 and 16.5.
- Continuity: know the definition, for I might ask you to state it.
Be able to do problems 20.6 and 21.3 (21.4 is the same as 20.6).
Derivatives
- be able to compute a derivative by using the definition and finding
the corresponding limit, as in problems 23.1.
- show a function does not have a derivative at some point by showing
that the corresponding limit does not exist, as in problem 23.2.
Notation
When you write something it should mean something and you should mean it.
Often people hand in a sheet filled with apparently unrelated formulas where
the reader (i.e. the grader) has to figure out what the formulas have to do
with each other.
For instance, when asked to find a limit like
limx to 3 (x-x2)/(x-1)
some will simply write the following
limx to 3 (x-x2)/(x-1)
x(1-x)/(x-1)
-x -3
Whoever wrote this did not say that any of these formulas represent
quantities that are equal to each other. It is hard to tell in which order
these formulas were written down. We, the readers, have to guess what
is meant.
One could write
limx to 3 (x-x2)/(x-1)
= limx to 3 x(1-x)/(x-1)
= limx to 3 -x
=-3.
If you write this then anyone can see how you got your answer and that is
-3. Both are important.
Read §17 to see what's wrong with the following
limx to 3 (x-x2)/(x-1)
= x(1-x)/(x-1) = -x.