Calculus 221, sections 1.1–6.5 Stuff You Need to Know
notes prepared by Tim Pilachowski
I hope to have notes for each lecture posted on my math department website, http://www.math.umd.edu/~tjp,
prior to the lecture itself. Feel free to print out and/or download each of these and bring it with you to class. In
this way you can put your attention on listening and thinking, and only need to write all those little “extras” that
will come up during my presentation. Need I tell you that these notes will be an outline only, and that they
cannot replace your presence in the lecture?
Be sure to attend the discussions on a regular basis, too. You’ll find them to be valuable in cementing the topics
covered in the lecture. You’ll get the most out of the discussion if you do the assigned homework before the
discussion, and participate in all the discussion activities.
To help you get up to speed for Math 221, we’re going to spend this first class going over some things I assume
you already know, but about which you may need a little reminder. The assigned practice exercises are from the
math 220 Final Exam from Spring 2006. (Skip and partial derivative questions for now.) I leave it to you to go
back on your own to topics and exercises on which you personally need some more review. Also, you can go to
the Math Dept. Testbank (http://db.math.umd.edu/testbank/) and get some other final exams from recent
semesters of Math 220, however, DON’T use the one from Fall 2005.
The following statements are mathematically equivalent:
a) Find the slope of the line tangent to the graph of f at a point (x, y). b) Find
()
(
)
h
xfhxf
h
−
+
→0
lim .
c) Find the first derivative of f(x). d) Find
(
)
xf
′
. e) Find dx
dy .
Recall, however, that the first derivative is itself a function, which has its own domain and graph. Since it is a
function, it has its own derivative. Given a function f, we can calculate the first derivative dx
dy
for
′. We can
then calculate the derivative of , also called the second derivative of f, symbolically f′2
2
or
dx
yd
f′′ .
Important note: Just like dx
dy is not a fraction, but is a notation for the first derivative, 2
2
dx
yd is also not a
fraction but a notation. There is no multiplication involved! Rather, you need to interpret it this way:
⎟
⎠
⎞
⎜
⎝
⎛
=dx
dy
dx
d
dx
yd
2
2
which means “the derivative of dx
dy ”, the derivative of a derivative.
Example A: Given , find
()
28
3+−= xxxf
(
)
xf
′
,
(
)
xf
′
′
,
(
)
(
)(
1and ,1,1 −
′′
)
−
′
−
fff .
answers: , 6x, 9, –5, –6 83 2−x
