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 (calle
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 funct
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
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 tha
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
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
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 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.
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
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 v
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 s
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
lis
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 t
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.
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
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 t
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 exa
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
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 wo
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 g
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 val
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
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 f
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
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.
Sit
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
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 t
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
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 t
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 rule