# Calculus Notes

**I. Questions
**

The most commonly asked question in a mathematics course is:

**Will this be on the exam?**

Well not everything will be on the exam, and so this is certainly a

legitimate question.

On the other hand, there are lots of things that will not be on the

exam that work in your favor if you understand them and can work

with them. The calculus course is based on a few very simple and

powerful ideas .

In order to make use of ideas and make them work for you it is

necessary to think and recall and comprehend.

You need to use curiosity and imagination. You need to be willing to

look for meaning and logic . You need to appreciate structure.

You need to be willing to calculate and work and ask questions.

The question with which we began this section is not sufficient!

In a very real sense , you need to question everything.

Why does mathematics work the way it does?

What is the nature of number and geometry ? How does mathematics

manage to model complex phenomena in the world? What is the

internal structure of mathematics? How do questions in

mathematics become interesting? What sort of mathematical

problems do you find easy? Which ones are hard? What is

mathematics? Is it calculation, reasoning, problem solving, a

language, an art form, a mental discipline, a form of philosophical

investigation?

What are the largest mathematical ideas that you know?

What is a number?

What is the meaning of continuity?

What is the nature of the discrete and the continuous?

What beliefs are in back of mathematics as we do it?

What beliefs are in back of the physical modeling that is so

tightly related to mathematics?

What is the nature of time and space?

How does mathematics have anything to do with time and space?

We want you to ask questions.

We want you to ask questions whose answers really mean a lot to

you. We want you to ask questions that have nothing to do with the

exams and everything to do with understanding.

What should I do in order to ace this course?

Ask questions.

Exercise curiosity, attention, intelligence and imagination.

Work hard.

Use logic.

Think about the material in the course.

Teach the material in the course!

Teach it to another student.

Teach it to yourself.

Teach it to the professor by asking him really tough questions about

how calculus works and how it is applied!

**II. Integration and Differentiation**

Calculus is based on a small number of key ideas. The purpose of

this section is to give you a quick introduction to these ideas that

will be useful as a start and as a reference as we progress in the

course.

**Three Big Ideas**

Calculus is all about three basic mathematical notions.

**1. Deriviatives = DIFFERENCES**

and

**2. Integrals = SUMMATIONS**

and

**3. LIMITS.**

Please come back to this remark after you have read the rest of the

section!

Certainly differences and sums are related to one another.

For example,

(2-1)+(3-2) +(4-3) + (5-4) + (6-5) + ... +(99 - 98) + (100-99)

= 100 -1 = 99.

A sum of differences can be a bigger difference because all the little

differences cancel each other out.

Taking differences can reveal a rule.

What is the rule for the following sequence?

1,6,11,16,21,26,31,36,41,...

Limits are interesting to think about. For example, you may be able

to see at once that

**1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ...
**has limiting value equal to 2 as the number of terms in this sum

becomes arbitrarily large. What is the geometric interpretation of

this claim?

Another example:

π

^{2}/6 = 1/1 + 1/4 + 1/9 + 1/16 + 1/25 + 1/36 + 1/49 + ...

π

^{2}/6 is equal to the sum of the reciprocals of all the squares.

This last example is a deeper result than the first series. Try

checking it on your calculator.

**Differentiation**

Consider the following problems:

1. You drive a car for one hour and cover a distance of 60 miles.

What is your average velocity?

Answer: 60 miles/hour.

2. In driving a car for one hour you cover

**21**miles in the first

**1 5**

minutes. How fast should you drive in the remaining

**45**minutes in

order to cover a total of

**60**miles in the hour? What is your average

velocity for the whole hour?

**Answer:**You need to cover

**39**miles in

**45**minutes. That is a

velocity of

**39**miles/(

**3/4**hour) = (

**4/3**)

**39**miles/hour =

**5 2**

miles/hour.

Your velocity in the first

**15**minutes was

**21**miles/(

**1/4**hour) =

**84**miles/hour. (You were speeding!)

Your average velocity for the hour was

**(1 / 4 ) (8 4**mph) +

**(3 / 4 ) (5 2**mph) =

**(84 + 156) /4**m p h

=

**240/4**mph =

**60**mph.

Notice that this average velocity is the same as the total number of

miles covered divided by the total time. Can you figure out why it

worked out this way? I will leave the question of why it works in

general to class discussion, but the arithmetic of the situation

reveals a pattern:

AvgVelocity

**= (1/4)(84) + (3/4)(52 )**

= ( 1 / 4 ) (2 1 / (1 / 4 )) + (3 / 4 ) (60 -21) / (3 / 4 ) )

=21 + (60-21)

=6 0 .

= ( 1 / 4 ) (2 1 / (1 / 4 )) + (3 / 4 ) (60 -21) / (3 / 4 ) )

=21 + (60-21)

=6 0 .

Discuss what happened here. Will it work in other problems about

average velocity?

3. In driving a car you start off by accelerating your car.

The distance you travel while you are accelerating is given by

the formula

**X(t) = t**where

^{2}**t**is in minutes and

**X(t)**is in miles.

You only accelerate in this way for two minutes . How far do you

travel in one minute? If you kept on accelerating, how far would you

go in 10 minutes?

What is your velocity at time

**t = 1**minute?

Answer:

**X(1) = 1**, so you travel one mile in one minute.

**X(10) = 100**miles, so you would go 100 miles in ten minutes.

Now to see approximately how fast your are going at one minute,

lets first see how far you go in a time Δt after 1 minute. We have

**X(1) = 1**and

**X(1 + Δ t) = (1 + Δ t)**

This means that the distance you travel in time

^{2}= 1 + 2Δ t + (Δ t)^{2}.**Δ t,**after already

traveling one mile is

**Δ X = X(1 + Δ t) - X(1) = 1 + 2Δ t + (Δ t)**

So your average velocity in the time interval

^{2}-1 = 2Δ t + (Δ t)^{2}.**Δ t**is

**Δ X/Δt = (2Δ t + (Δ t)**miles per minute.

^{2})/Δ t = 2 + Δ tThis is

**120 + 60Δ t**miles per hour.

Your velocity is varying as a function of time, but if we make the

time interval Δt very short, we can estimate your "instantaneous

velocity" at time

**t = 1**as the limiting value of

**120 + 60Δ**t as

**Δt**goes to zero. This limiting value is clearly

**120**. So we can say that

you are going at

**120**miles per hour after one minute.

4. You would be crazy to keep on accelerating at the rate given in

the previous problem. But lets calculate some other limiting

velocities. Given that your distance function is

**X(t) = t**,

^{2}what is your instantaneous velocity at an arbitrary positive time

**t**?

Answer:

**Δ X(t) = X(t + Δ t) - X(t) = (t + Δ t)**

= 2tΔ t + (Δ t)

^{2}- t^{2 }= (t^{2}+ 2tΔ t + (Δ t )^{2}) - t^{2}= 2tΔ t + (Δ t)

^{2}.Thus

**Δ X(t)/Δ t = 2t + Δ t .**

We conclude from this that the instantaneous velocity of

our car at time

**t**is

**2t**miles per minute, which is equal

to

**120t**miles per hour where

**t**is measured in minutes.

At the

**30**second point you are going

**60**miles per hour.

If you keep accelerating at this rate, you will be going

**240**miles per

hour after two minutes and

**1200**miles per hour after

**10**minutes!

The method by which we have solved these problems is called

differentiation. The general pattern is just the same as in our

example for velocity. We have a function

**F(t)**(it was

**X(t) = t**in

^{2}our example) and we define the difference quotient by the formula

Δ F(t)/Δ t = (F(t + Δ t) - F(t))/Δ t

Δ F(t)/Δ t = (F(t + Δ t) - F(t))/Δ t

The derivative of

**F(t)**at

**t**is defined by taking the limit of

**Δ F/Δt**as

**Δ t**goes to zero. We denote this limit by

**dF(t)/dt**

and sometimes by

**F'(t)**. Thus we write

**d ( t**

^{2})/dt = 2tvia the calculations that we did in the problem solving above.

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