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An uncareful mathematician would always mistake this for love photo

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

 

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