How to Learn Calculus - the beautiful way
Summary
TLDRIn 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
📚 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.
🧐 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.
📘 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.
🌟 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.
Mindmap
Keywords
💡Calculus
💡Differential Calculus
💡Integral Calculus
💡Derivative
💡Tangent Line
💡Inequalities
💡AM-GM Inequality
💡Limit
💡Book Recommendations
💡Xenophanes' Paradoxes
💡Fundamental Theorem of Calculus
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
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