COMPUTING LIMITS
When you build something you need two things:
Ingredients to use
Tools to handle the ingredients
Your textbook gives you the tools first (called the
limit laws), then the tools (called special limits.)
Just for fun I will give you the ing

The A B Cs of Graphing
(Tips to follow)
This presentation is devoted entirely to
Giving you all the necessary hints, tips, rules to
follow, and some smart advice, to allow you to
look at a given function
and have a decent chance to draw a (maybe rough,
bu

HOW DO YOU GET THOSE LOVELY CURVES?
Unfortunately the only curves we can draw (so far,
youll learn more later) are graphs of functions
Today, with any decent graphing calculator, you
dont have to draw anything, its done for you!
Quite often, however, one

DOT A FEW IS, CROSS A FEW TS
The most important thing to understand from
the last lecture (on Riemann integration) is that
The Riemann protocol is a new way to assign
areas to figures in the plane that gives the same
answer as the old method (formulas!) f

COMPUTING ANTI-DERIVATIVES
(Integration by SUBSTITUTION)
The computation of anti-derivatives is just an intellectual challenge, we know how to take derivatives, but
can we invert the process? We call this
Computing the indefinite integral
.
In the last p

NEWTONS METHOD
(OR: FINDING YOUR ROOTS)
(NOT a genealogy concept)
They say a picture is worth a thousand words.
Here is a picture (observed by Sir Isaac Newton,
I guess) which gave him the germ of an idea
for devising a method that iteratively finds
solut

Whats going on out there?
(way out there!)
Suppose you are studying the increase or decrease
P
of a population
over time.
Somebody gave you a formula that presumably
describes the amount of members of the
population at a given time ,t
that is a function
.

THE POWER OF
ADDING AND MULTIPLYING
Conceptually the idea of area is simply
the product of two linear dimensions
The notion of Riemann Sum is then an extension
of this idea to more general situations. However,
in the formula
A could be anything, and so co

HOW DO WE MEASURE A QUANTITY ?
Actually, what does it mean to measure some-thing,
or more precisely, what are the ingredi-ents needed
to be able to measure something?
One ingredient should be obvious
A UNIT !
The second Ingredient is not so obvious, it i

BETWEEN CURVES
Integration (actually calculus) is a very powerful
tool, it gives us power to answer questions that
the Greeks struggled with unsuccessfully (as did
pre-calculus Mathematicians) for a very long time.
Here is a figure whose area was not comp

THE SHELL GAME
OR
RIEMANN SUMS REREVISITED
Some solids of revolution are not amenable to the
methods we have learned so far (disks and washers).
For example, we may have to compute the volume of
the solid of revolution generated by rotating the
violet are

REVIEW
OF
THE COURSE
The series of titles of the presentations given in class
provide a good outline of the topics covered during
this semester.
In the following slides we provide snapshots of the
list and we will identify with a those topics that
require

AVERAGING THINGS OUT
Suppose you have a given function
and your boss (instructor, departmental director)
orders you to estimate an average value of the
infinitely many values (one for every point in the
interval
) of the function. How do you
proceed?
If y

THE WORKS
OR
RIEMANN SUMS REREREVISITED
Newtons Second Law of motion is generally
written as
, where m is the mass of
a body, F is the force that is applied to the
body to produce the acceleration a.
When the acceleration is no force is applied
and, conve

COMPUTING ANTI-DERIVATIVES
(Integration by PARTS )
The computation of anti-derivatives is just an intellectual challenge, we know how to take derivatives, but
can we invert the process? We call this
Computing the indefinite integral
.
In the last present

MUST AVERAGES HAPPEN?
57 students, average score in an exam 73.25.
Must one students have scored 73.25?
What goes up must come down. OK, but
must the velocity become 0 sometimes?
Takes me two hours to go to Chicago, 90 miles away.
Must I have gone at 45

OPTIMIZATION
(OR: is that the best you can do?)
Very often in life we are faced with a situation
wherein we have to decide what is the best
course of action within a certain set of conditions.
For example, consider the following situation:
A little doggie

LIMITS
We are going to make sense of vague statements
like
limiting value of slopes of secants
as A and B get close to P (remember?)
or
limiting value of the ratio
as
h gets close to
(remember?)
We are going to learn the concept of
As usual the English

CONTINUITY
The man-in-the-street understanding of a
continuous process is something that proceeds
smoothly, without breaks or interruptions.
Consequently, for a function
to be
called continuous, we would expect its graph to
be a smooth line, without break

DEFINITION OF LIMIT
We are going to learn the precise definition of
what is meant by the statements
and
.
We must make precise our intuitive notion that
gets arbitrarily close to
close to a .
L
as
x gets
Lets begin by noticing that
gets close to L
vertica

Functions
In everyday language the word function has at least two
separate meanings I can think of:
A. The purpose of something, as in
The function of a teacher is to impart knowledge.
B. When the value of some item
somehow determines uniquely the value o

Velocity and tangents
We are going to look at two questions that, in
appearance, have nothing to do with each other
(one is geometrical, the other physical);
We will find out (not surprisingly if you admire
Mathematics) that the answer to both questions
i

DERIVATIVES
The second day of classes we looked at two
situations whose resolution brought us to the
same mathematical set-up:
Tangents and
Velocities
Both ended up being modeled by the following:
A function
is given, as well as two
numbers
and
. We are a

DERIVATIVES ARE
FUNCTIONS TOO !
First of all, let us see how many consequences are
implied by the statement
The derivative of
f at a exists.
Recall that the statement means:
1.
2.
3.
f is defined at a , that is
exists.
exists.
f is defined in an open inte

IMPLICIT DIFFERENTIATION
AND
RELATED RATES
Once again we are going to look at two separate and
apparently distinct situations that, not surprisingly,
are resolved with the same mathematical model.
Situation no. 1: Some very nice looking and useful
curves

IMPLICIT DIFFERENTIATION
AND
RELATED RATES
Recall the two separate and apparently distinct
situations that, not surprisingly, are resolved with
the same mathematical model.
Situation no. 1: Some very nice looking and
useful curves in the plane are NOT des

LINEAR APPROXIMATIONS
AND
DIFFERENTIALS
One of the beauties of Mathematics is that it is able
to provide help in all sorts of different situations and
to all sorts of people. Today we will learn how the
two distinct groups of people listed below profit
fr

RELATED RATES
Recall the second situation we still have to
address, namely:
Situation no. 2: Two quantities
and
are
related to each other via some formula
Suppose we know the rate of change of one of them
(either one, for ease of thought say
) with
respec

COMPUTING DERIVATIVES
During the last lecture we built some bricks
(derivatives of four actual functions) and some
mortar (commonly known as rules of
differentiation.)
We can now apply the mortar to the bricks and
start building a collection of known deri

COMPUTING DERIVATIVES
During the last lecture we saw that we need
some bricks (derivatives of actual functions)
and some mortar (commonly known as rules
of differentiation.)
We list and prove the rules first, they are rather
easy to prove. Let us state al