Study Guide for the 1st midterm
Chapter 1
You should:
- Make sure you know function notation and understand
inverse and implicit functions (type of problems are
8–14, 16–21, 23.)
Chapter 2: Limits
You should:
- Be able to prove that lim f(x) = L using the
ε—δ definition of the limit, as in
problems in pages 25-26. Understand how they represent errors.
- Know the definitions of one-sided limits and limits at
infinity. You do not need to use the ε–δ
definition to find these limits.
- Know how to use the limit properties to compute limits.
You will need to specify properties you are using only if
asked to, otherwise you only need to show a minimum number of
steps ensuring that we understand what you are doing
(exercises are on page 38). I can ask any type of limit
appearing in the homework, including limits that do not exist.
If a limit does not exist and we have not asked you to justify
it using contradiction, you only need to give some
calculations showing you know what you are doing and state
that the limit does not exist.
- Know how to use the limit properties and contradiction to
show that a limit cannot exist, as in examples 14.4, 14.5 and
14.6.
- Know the Sandwich theorem: Know what the theorem says and
be able to use it in some trigonometric limits, like those in
page 40 (93, 94..). Find trig limits like those in page 40,
using lim x-> 0 (sin x)/x = 1, lim x-> 0
sin x = 0 and lim x-> 0 cos x = 1.
- Know the definition of Continuity, for I might ask you to
state it. Be able to do continuity problems as those in page
40.
Chapter 3: Derivatives
You should:
- Understand the derivative as both the slope of the tangent
(page 17) and the rate of change (page 18).
- Be able to compute a derivative by using the definition and
finding the corresponding limit, as in problems in page 46-47.
- Show that a function does not have a derivative at some
point by showing that the corresponding limit does not exist,
as in problems in page 47.
A rant about 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 -> 3 (x-x2)/(x-1)
some will simply write the following
limx -> 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 -> 3 (x-x2)/(x-1)
= limx -> 3 x(1-x)/(x-1)
= limx -> 3 -x
=-3.
If you write this then anyone can (1) see how you got your
answer and (2) see that it is -3. Both are important.
See if you can tell what's wrong with the following
limx -> 3 (x-x2)/(x-1)
= x(1-x)/(x-1) = -x = -3.
We will certainly subtract points for this type of writing so try
your best.