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