How to Learn Calculus - the beautiful way

Summary

In this educational video, the presenter shares an engaging approach to learning calculus, starting with understanding inequalities and their philosophical implications. They recommend specific books for grasping concepts like limits and continuity, and emphasize the importance of problem-solving. The video also touches on the historical context of calculus, including its relation to Zeno's paradoxes, and encourages viewers to appreciate the beauty of calculus by connecting it to geometry and algebra.

Takeaways

• 📚 Calculus has two main parts: differential calculus and integral calculus.
• 📐 Differential calculus helps in finding the slope of the tangent line at any point on a curve.
• 📉 Integral calculus is used to calculate the area under a curve.
• 🔄 Differential and integral calculus are inverse operations and are related by the fundamental theorem of calculus.
• 🤔 Understanding inequalities is a crucial foundation for grasping the concepts of calculus.
• 📖 The book 'Little Mathematical Library' is recommended for learning about inequalities.
• 🌟 The sequence 1/2^n, where n is an increasing natural number, is bounded and always increasing but never exceeds a certain finite number.
• 📘 'Calculus' by Tasov is suggested for understanding the concept of limits and continuity.
• 📔 'Single Variable Calculus' by I.A. Maron is a problem-driven book that helps in understanding calculus.
• 🎨 'Art and Craft of Problem Solving' by Paul Zorn explains why differential and integral calculus are converse operations.
• 🤝 Studying ancient philosophy, like Zeno's paradoxes, can provide insights into calculus concepts.

Q & A

What is the main topic of the video script?

The main topic of the video script is learning calculus, specifically focusing on how to understand and approach the subject effectively.

What are the two parts of calculus mentioned in the script?

The two parts of calculus mentioned are differential calculus, which involves extracting the derivative of a function, and integral calculus, which involves calculating the area under a curve.

What is the geometric interpretation of differential calculus?

In geometric terms, differential calculus gives the slope of the tangent line at any point on a curve.

What does integral calculus help calculate?

Integral calculus helps calculate the area under a curve.

What is the fundamental theorem of calculus as described in the script?

The fundamental theorem of calculus, as described, states that differentiation and integration are converse operations of each other, and taking the derivative of an integral nullifies the integral.

Why does the speaker suggest starting with inequalities when learning calculus?

The speaker suggests starting with inequalities because mastering them helps understand the philosophical transformations calculus brought to mathematics.

What book does the speaker recommend for understanding inequalities?

The speaker recommends 'Little Mathematical Library' for understanding inequalities.

What is the significance of the sequence mentioned in the script?

The sequence mentioned is significant because it is increasing but bounded, which is a concept that baffled Greek mathematicians and is fundamental to understanding limits in calculus.

Which book by Tasov is recommended for understanding calculus?

The book recommended by the speaker for understanding calculus is 'Calculus' by Tasov, which introduces the idea of limits and continuity in a conversational manner.

What is the approach of the book 'Single Variable Calculus' by I. A. Marin?

The book 'Single Variable Calculus' by I. A. Marin is problem-driven, starting with problems to enhance understanding of calculus concepts.

How does the book 'Art and Craft of Problem Solving' by Paul Z connect to calculus?

The book 'Art and Craft of Problem Solving' by Paul Z helps understand why differential and integral calculus are converses of each other through a geometric perspective.

What additional study does the speaker recommend to enhance understanding of calculus?

The speaker recommends studying Zeno's paradoxes and their relation to infinite sequences and calculus to enhance understanding of the subject.

Outlines

00:00

📚 Introduction to Calculus

The speaker begins by expressing their intrigue with calculus from a young age, comparing it to other mathematical disciplines they found more accessible. They highlight the two main parts of calculus: differential calculus, which involves finding the derivative of a function and can be visualized as the slope of a tangent line at any point on a curve; and integral calculus, which calculates the area under a curve. The speaker simplifies these concepts for beginners but also mentions the fundamental theorem of calculus, which reveals the deep connection between differentiation and integration, essentially showing that they are inverse operations.

05:00

🧐 Inequalities and the Foundations of Calculus

The speaker emphasizes the importance of understanding inequalities to grasp the philosophical underpinnings that calculus introduced to mathematics. They recommend starting with a book from the 'Little Mathematical Library' series, which covers various types of inequalities and their algebraic results. A specific sequence is discussed to illustrate the concept of a number sequence that is always increasing but bounded, a concept that puzzled ancient Greek mathematicians. The speaker suggests that understanding this paradoxical sequence is key to understanding limits, a fundamental concept in calculus.

10:02

📘 Books for Deepening Understanding of Calculus

The speaker recommends several books for a deeper understanding of calculus. 'Calculus' by Tasov is praised for its conversational style and use of dialectics to explain limits and continuity. 'Single Variable Calculus' by I. A. Maron is highlighted for its problem-driven approach, which helps to internalize mathematical concepts. Lastly, 'Art and Craft of Problem Solving' by Paul Zeitz is mentioned for its chapter that geometrically explains the inverse relationship between differential and integral calculus. The speaker also suggests studying Zeno's paradoxes to connect philosophical thought with calculus.

15:04

🌟 The Beauty of Calculus and Encouragement to Learn

In the final paragraph, the speaker expresses their admiration for the beauty of calculus and encourages learners to enjoy the process of studying it. They invite questions in the comments section and promote their institution's programs for students interested in problem-solving, research, and similar fields. The speaker concludes with a positive note, encouraging continued learning and problem-solving.

Keywords

💡Calculus

Calculus is a branch of mathematics that studies how things change. In the video, calculus is described as a subject that involves two main parts: differential and integral calculus. Differential calculus deals with the rate at which quantities change, while integral calculus is concerned with the accumulation of quantities over time. The video aims to demystify calculus by explaining these concepts in a more accessible way.

💡Differential Calculus

Differential calculus is a part of calculus that focuses on the concept of limits and the rates at which quantities change. It is used to find the derivative of a function, which can be thought of as the slope of the tangent line to a curve at a specific point. In the script, the speaker uses the example of a curve on an XY plane and explains how differential calculus can determine the slope of this curve at any given point.

💡Integral Calculus

Integral calculus is the other main part of calculus and is concerned with accumulating quantities. It is used to calculate areas under curves, which is a way to understand the accumulation of change over a period. The video script mentions integral calculus in the context of calculating the area under a curve, which is a fundamental concept in understanding the accumulation of quantities.

💡Derivative

A derivative in calculus represents the rate at which a function is changing at a particular point. It is a fundamental concept in differential calculus. The video script uses the tangent line at a point on a curve to illustrate the concept of a derivative, explaining how it gives the slope of the curve at that point.

💡Tangent Line

The tangent line to a curve at a given point is the best linear approximation of the curve at that point. It is a key concept in differential calculus and is used to understand the rate of change of the curve. The video script describes how differential calculus can be used to calculate the slope of the tangent line at any point on a curve.

💡Inequalities

Inequalities are mathematical statements that compare expressions or values. In the context of the video, inequalities are suggested as a starting point for understanding calculus because they introduce the philosophical transformations that calculus brings to mathematics. The speaker recommends studying inequalities to grasp the foundational concepts that underpin calculus.

💡AM-GM Inequality

The Arithmetic Mean-Geometric Mean (AM-GM) inequality is a fundamental result in mathematics that relates the arithmetic mean and the geometric mean of non-negative real numbers. The video script mentions this inequality in the context of sequences that are increasing but bounded, which is a concept that leads to the idea of limits in calculus.

💡Limit

A limit is a fundamental concept in calculus that describes the value that a function or sequence 'approaches' as the input or index approaches some value. The video script discusses the concept of limits in the context of sequences that increase without bound but are always less than a certain number, illustrating the philosophical underpinnings of calculus.

💡Book Recommendations

The video script includes several book recommendations for learning calculus. These books are suggested to help viewers understand the philosophical and mathematical foundations of calculus. The books mentioned include 'Little Mathematical Library' for inequalities, 'Calculus' by Tasov for understanding limits and continuity, 'Single Variable Calculus' by I.A. Maron for problem-driven learning, and 'Art and Craft of Problem Solving' by Paul Zeitz for understanding the relationship between differential and integral calculus.

💡Xenophanes' Paradoxes

Xenophanes' Paradoxes are philosophical questions that relate to the concept of infinity and are mentioned in the video script as a way to understand the philosophical implications of calculus. The paradoxes challenge the notion of infinite sequences that are increasing but bounded, which is a concept that calculus helps to resolve.

💡Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a theorem that links the concept of differentiation and integration. It states that the integral of the derivative of a function is the original function. This theorem is briefly mentioned in the script as a way to illustrate that differential and integral calculus are, in a sense, inverse operations.

Highlights

The student's question about learning calculus intrigues the speaker, reminding them of their own childhood curiosity.

The speaker explains that calculus has two main parts: differential and integral calculus.

Differential calculus is about finding the slope of the tangent line to a curve at any given point.

Integral calculus is about calculating the area under a curve.

The speaker emphasizes that differential and integral calculus are essentially inverse operations.

The fundamental theorem of calculus is introduced as a key concept.

The speaker suggests starting the study of calculus with inequalities.

The book 'Little Mathematical Library' is recommended for understanding inequalities.

The concept of a sequence that is always increasing but bounded is introduced.

The AM-GM inequality is mentioned as a tool to prove that certain sequences are bounded.

The historical context of calculus development is touched upon, including the contributions of Archimedes and Madhava.

The book 'Calculus' by Tasov is recommended for its approach to teaching limits and continuity.

The use of dialectics in teaching calculus is highlighted.

The book 'Single Variable Calculus' by I.A. Marin is recommended for its problem-driven approach.

The book 'Art and Craft of Problem Solving' by Paul Zorn is suggested for understanding the relationship between differential and integral calculus.

The speaker recommends studying Zeno's paradoxes for a philosophical understanding of calculus.

The importance of understanding basic geometry and algebra in the context of calculus is emphasized.

The speaker shares their personal journey of learning calculus and appreciating its applications.

The speaker invites questions in the comment section and promotes their educational programs.

Transcripts

hello children I hope you are doing

beautiful

mathematics a few days ago a student of

mine asked me can you tell me how to

learn

calculus and that question immediately

intrigued me it took me to my childhood

when I started learning calculus and I

used to ask my father tell me in one

line in two lines what is calculus

because I was so intrigued I mean I was

in I was learning Algebra I was learning

geometry and all those things I seem to

understand but calculus seems like a

very weird different thing mysterious

thing that only adults know and he tried

to explain a few things but I could not

understand that very well uh so I

understand that children are you know

intrigued by the subject and in this

video I will share with you a few things

that can really help you to learn really

learn really understand

calculus so uh the first thing I would

say that there are two parts of calculus

one is differential

calculus differential calculus which

involves extracting derivative of a

function I'll explain what that means in

a second but geometrically

speaking suppose you have an XY

coordinate plane and and you have a

curve like

this whatever the equation of the curve

is and you choose a point on the curve

let's say a point

P then what differential calculus does

is gives you the the tangent

slope so this is the tangent line at the

point P it gives you a way to calculate

the tangent line at that particular

Point p in fact at every Point p on the

curve wherever it is possible to draw a

tangent line it gives you the slope of

the tangent line how fast the tangent

line is rising or how fast is it falling

that's what differential calculus

is and there is another part of calculus

which is called integral

calculus which again if I draw the same

picture let's say I draw an XY plane I

draw a curve like this then integral

calculus gives me a way to calculate the

area under the

curve so that's it differential calculus

gives you a way to calculate slope of a

tangent line differential integral

calculus gives you a way to calculate

the area under the

curve the this is one way to think about

it and to be very honest this is not how

you should be thinking about calculus

when you are starting with it but if

someone like me who is like really knif

who is very rushed that please tell me

please tell me what it is what it is so

you say that okay so differential

calculus you calculate the slope of the

tangent line at every Point integral

calculus you calculate the area under

the Curve and behold they are actually

the same thing in a way they are

Converse of each

other each

other so like if you take the square

root and if you take the square of the

square root the square root goes away

similarly if you take the derivative of

the

integral then the integral goes away so

they are Converses of each other they

nullify each other in some sense so that

what I just said is the fundamental

theorem of

calculus so this is a very very naive

way of saying what is calculus but that

is not the purpose of this video and to

be very honest this is not how you

should be approaching the subject so I

will give you the way I think is most

effective to approach the

subject of course there are two parts I

already talked about this differential

integral and they developed

separately so it's better to approach

them separately in the beginning so I

would suggest that you start with

inequality it's a very strange place to

start if you if someone just talks about

calculus you immediately don't think

about inequalities but trust you me if

you know how to handle inequalities

really well then you will understand

some of the basic philosophical

Transformations that calculus brought

into the world of

mathematics so how from where do you

study in inequality my favorite is

little mathematical

Library this is a very beautiful book we

actually have a reprint of the book at

chenta so if you want to purchase it for

a very small cost I would say we just

take the printing cost so you can check

the link in the description I just want

our kids to get a handle on this

particular

book learn how to understand

inequalities really well there are

different types of inequalities

arithmetic mean geometric mean

inequalities cish words inequalities

there are a ton of interesting results

algebraic

results one of the results which is very

fundamental to the study of calculus is

this if you take a

sequence of numbers let me write the

sequence if you take a take this

particular sequence of

numbers I've written some of the terms

of this sequence

something strange happens I've written

the first four terms you can actually

calculate them something strange there's

an infinite sequence of course somewhere

here there will be 1 by 2024 ra to the^

2024 like this this is an infinite

sequence there is a very beautiful

property of this

sequence the property is this that all

the terms of this sequence is less than

three even if you plug in instead of

2024 if you plug in 2

million the number would be less than

three that's the first property so it

this is called

bounded bounded means it's less than a

certain finite number the second thing

is that this is always increasing

so just it it keeps on increasing so if

you if you plug in instead of 2024 if

you plug in 2 million you will get a

larger

number so this is

increasing and this is this is where the

revolution is something that keeps on

increasing and yet that is bounded

it is increasing but it is

bounded the Greek mathematicians uid

Archimedes they were greatest of

all but somehow they could not wrap

their head around this sort of sequences

it seemed unreal to them the Greeks were

unable to figure out or really

understand that a sequence of numbers

can be

increasing and yet bounded it keeps on

increasing but it never goes it always

increases but it never goes beyond a

certain finite

number this is something

strange and this to prove that this

sequence is

bounded you have to use something called

amgm

inequality and that my friend you have

to go and learn from this particular

book and it incre and it introduces you

to such

sequences which baffled the Greek

mathematicians for

centuries and then Arya bataa came

in in India and then madavara came in

around 12th century BC ad sorry 12th

century in the Kerala School of

mathematics and he literally I think was

the first person who really understood

what sort of things we are dealing with

here in finite sequences increasing

always increasing but bounded that is a

story of a different video but as I

mentioned this is the beginning of the

idea of a

limit if you understand what's Happening

Here you understand what is a limit

therefore it is important to understand

inequalities therefore I suggest that

you go and get your hands on this

particular book you can also download a

PDF copy online it's freely available or

you can purchase one

okay next Once you have done this I

suggest you go to a very beautiful book

by

tasov the name of this book is

calculus the way tasov introduces the

idea of limit and continuity is truly

remarkable

it's like a conversation between a

student and the

teacher and he uses a very

interesting technique that was in used

by ancient Greek philosophers this is

known as

dialectics you can also read more about

it mathematics and philosophy went hand

in hand in ancient world tasov is not

that ancient but he uses dialectic

in his con conversation between a

student and a teacher and

outcomes the definitions of limit the

understanding of limit the understanding

of continuity all of that stuff it's

absolutely remarkable so that is the

second book I suggest you study calculus

by terasoft even if you do not

understand any calculus you can start

with this book the third book that I

would suggest test is single variable

calculus by I

aarin single variable calculus it's a

very beautiful problem driven

book never

read never read big articles on some

mathematical

idea solve a problem that is always

better it's always always more fun it

gets you inside the subject and Marin in

single variable calculus actually starts

with a

problem there are bunch of problems and

as you go through the problems your

understanding of the subject becomes

even

better and finally I would say there is

a book called art and craft of problem

solving by Paul zits I think I'm writing

the spelling incorrectly remember I told

you a few minutes ago that differential

and integral calculus are like Converses

of each

other Paul Z's book art and craft of

problem solving has a beautiful chapter

which geometrically makes you understand

why this is the case why these are

Converses of each other so these are the

the four books that I would

suggest from the philosophical point of

view I would also suggest you to study

xenos

paradoxes these are some of the most

well-known Paradox of the world and if

you want to understand this a little bit

more how it is connected with

Calculus you can you can Google it of

course but once you read the Paradox you

can think think about it how these

paradoxes might be related to an

infinite

sequence infinite sequence sequence of

numbers that is always increasing but

bounded it is always increasing but

bounded how xenos paradoxes are related

to that this is a fascinating part of

ancient philosophy and I'm I think that

if you are studying mathematics

occasionally you should go back to

philosophy and see what's going on there

okay so that's how I learned whatever

calculus I have learned and gradually as

I moved ahead of this learning process I

begin began to appreciate basic geometry

how that comes into play how basic

algebra comes into play how

the Advent of calculus allowed us to

extend the binomial theorem to

fractional Powers so suppose if I say 2

to the

power 3 you immediately know it is 2 * 2

* 2 that has a meaning but if I say 2

the^ 3.2 if you plug this in your

calculator you'll get an answer but you

cannot multiply two with itself 3.2

times that's not possible possible so to

even understand what's going on here you

have to use

calculus so I hope you would uh you

would enjoy learning calculus it's a

very very beautiful subject and if you

have any questions put in the comment

section and if you are interested in

Olympiad Pro problem solving student

research or similar programs then check

the link in the description we have

beautiful program at CH I'm sure you'll

love them all right take care bye keep

on doing great problems