1.
real life:
we know
how can we?
a couple
will draw some consequences
Suppose the largest
couldn’t be the highest
Warnings must be given right away
need not be
special
let
let
let
Say this question
(At some later time)
What does this exploit?
It does not move.
the constant should be no surprise
(xxx)
for all, a constant
i
increase
you
i
bisect
you
root, pin down
root
i
root
you
negative, only one solution
(a given)
2.
we assume
we consider
we call it
If you were to walk
almost like walking
the summit of a single mountain
that is critical
glance at how the derivative behaves
Say
you want it
rough
i
use
you
We must keep increasing
or decreasing
Similarly
(xxx)
Again
a little larger
a little smaller
We complete
closer and closer
We turn
closed
In short, pay special attention
3.
we are defined
by such symbols
(xxx)
A falling rock
Find it
Translate this news item:
let
the bad news
promise
then switch
in the crystal ball
we see
no force
antiderivative
antiderivative
antiderivative
(but not)
Where is later?
When does it reach?
How high above?
it follows
us
4.
The changing
The changing
(one is changing another)
(xxx)
time yielding
from the fish’s point of view
the very large fish
moving at huge speeds
a tactical mistake
there is nothing left free
time is safest if broken
A woman on the ground
is the telescope
the woman?
is the jet
the woman?
(the woman must change)
We know.
We wish.
(we want)
the tank is filled
the tank is filled
the tank
does not rise
5.
We provide
neither
it follows
(xxx)
(xxx)
So begin by finding
Setting
Using
we only have
the same sign
Up Down Up
(Keep in mind)
6.
Some are very hard
(roots)
If we don’t know
we make a guess
this guess resembles
the point
Advisory: Don’t
Call it
the one
The 1 is not an exponent
many people
root
(xxx)
One more
quite close
you “wander all over the place”
[figure: There is no root]
get closer and closer
HOW GOOD?
you get a feel
7.
Too low, and you earn too little;
too high, and you lose
Too small a harvest, and you starve;
too large, and you destroy
and you starve
1. Get a feel
The most important step is
To become skillful
A couple
They wish
to be small
the garden
the garden
the garden
(xxx)
it makes no sense
First we get a feel for the problem
a couple
Say we are 1 inch
1 inch
1 inch
let’s try cutting
we get
very small
It is
meaningful in our problem
The smallest
can be as we please
let us allow cuts
cut
Remark:
restrict
avoid the extra work
(in case the problem
may be of aid)
which requires the least metal?
can can can
Does it provide?
the graph lies
if you want to know
8.
Sometimes
(xxx)
and
(xxx)
the bottom satisfies
roots would appear
which requires the least metal?
which requires the least metal?
Warning: a derivative
will usually be needed
9.
A small piece
suggests
the change
Consider
constants
Find the
good
(xxx)
small
small
small
very close
as was to be expected
instead
It is meaningful
vanishingly
we wish
we know
we use
we have
we are not interested
error
error
error
(xxx)
three times
3
we expect
we wish
we already have
we find
at most
It is
It is small
10.
The basic idea is this.
Say that you know
how quickly
we
press
to the floor
Where will you be?
you
you
you had neglected to set your stopwatch
you would have traveled
But what if?
What if you have a nervous foot?
Imagine
Let’s
Assume
How large
(without reference to automobiles)
How good is this?
We want to know
we want to find out
say something about the size of itself
(xxx)
very near
roughly
rapidly
nearly constant
near
near
near
you shrink
From: CALCULUS AND ANALYTICAL GEOMETRY:
“4. THREE THEOREMS ABOUT THE DERIVATIVE”
Stein/Barcellos, McGraw-Hill, 1992
