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Calculus 1 & Gandhi

Blogs > Laerties
Post a Reply
Laerties
Profile Blog Joined March 2010
United States361 Posts
Last Edited: 2011-06-24 06:48:18
June 24 2011 06:11 GMT
#1
This is my first time writing about myself + blogging so please don't hesitate to let me know what sucks.

So my primary reason for starting this blog is that I don't really have anyone who I can discuss my studies with. Now that its summer and I have hardly any obligations I've decided to dedicate most of my time towards studying things I'm interested in.


Anyway, I've decided to start off my studies by developing a strong understanding of Calc 1 and reading The Story of my Experiments With Truth by Gandhi, The Art of Happiness, by the Dali Llama, and Ethical Religion by Gandhi.

I took a calc class this year but it only got up to integration by u-sub, and it skipped over major things like the fundamental theorem…..Today I spent learning integration by parts & partial fractions. By parts is pretty easy, its just applying what I already know in a specific way but partial fractions is giving me a really hard time. I pretty much dicked around in math class up until the end of 11th grade, and my weak-ish understanding of basic concepts is making it more difficult than it should be. The textbook gives a nice table of how to set up the partial fractions depending on the denominator of the fractions but even with it, it takes me a minute to figure out what I'm looking at in the denominator. Also, why the hell is it that the coefficients on both sides of the equation need to be equal? When I look at examples it makes sense but I don't understand why its true…the whole thing just confuses me. I think I have most of the execution down though.

So i've read up to around page 160 in Gandhi's autobiography. Around January of this year I re-watched the movie Gandhi made in 1982 and I watched it like 10 times over the period of a month. Every time I watched it I began to realize more and more how fucking awesome all of Gandhi's principles are. Non-violence makes so much sense to me but I also recognize how hard it is to turn into action. So I've been reading his autobiography and I've begun to realize that his lifestyle was so much more radical than the one I'm used to. First of all he lived almost half his life celibate, made all his own clothes, fasted, took vows and constantly denied himself emotional pleasures. I really want to believe, and do to a certain extent, that his ideals are brilliant and genius but I sometimes think he might have just been crazy. He says that he receives freedom and pure happiness from restricting himself. I can see how disallowing your desires to influence you can be freeing, but it seems like he is just restricting himself in other ways. I just cant understand how denying pleasures can bring anyone happiness.

I've begun a few of my own 'experiments with truth' from reading his book. The first is, that I have stopped eating meat. Its been probably a bit over a month now since I've eaten any meat. It hasn't been that hard, and I do honestly feel better about what I'm doing. I claim to be stopping for moral reasons, but before I stopped I never felt guilty about eating meat. I think that from reading his book and considering the reasons for eating and not eating meat I have developed a stronger moral conduct in general. It's only been a few days since I started to try and actually develop a disliking for meat though. "Renunciation of objects without the renunciation of desires is short lived, however hard you may try". That quote really stuck with me, and I realized that to truly comply to my experiment I had to reject not just meat, but the desire to eat meat.

The other experiment I tried was to not lie, exaggerate or manipulate what I understand as truth in anyway. It has been realllllyy fucking hard. I seriously can't go a single day without letting one lie or exaggeration slip. I never thought that I was a liar before, but now I realize that I lie all the fucking time when communicating with other people. Gandhi saw truth as the end all be all, he once said "god is truth". He dedicated his entire life to following what he saw as truth. On a side note, thats also what's kind of awesome about math, its the only place where absolute truth exists, in every other field truth is relative. I've also been trying to talk less. Gandhi said in his book, and I agree, "Experience has taught me that silence is part of the spiritual discipline of a votary of truth". Also, "proneness to exaggerate to suppress or modify the truth, wittingly or unwittingly, is a natural weakness of man and silence is necessary to surmount it."

Gandhi spent most of his life reforming the bestial elements of his nature in order to develop moral and spiritual consistency. He said that this is what brings true happiness. I can't say I understand or relate to any aspect of that statement but I feel that it is worth the attempt, and my own endeavors have centered around understanding this statement.

Okay, so thats it. Sorry if this is terrible/boring/snobbish…

Happiness is when what you think, what you say, and what you do are in harmony.
Blisse
Profile Blog Joined July 2010
Canada3710 Posts
Last Edited: 2011-06-24 06:41:21
June 24 2011 06:40 GMT
#2
When you start actually learning about these people, like really studying, you sort of realize how amazing they are. Like Einstein or Newton. It still astounds me how much better they are than me, and how insignificant I will always be compared to the both of them.

Math is hard, but fun once you sort of understand it. And then you have to question, how the hell did these people figure out how to do integration by substition or integration by parts?

There are moments when you realize people are ridiculous and there are many many people that will stomp your ass.


Gandhi sounds like an amazing person, but don't press yourself to follow his words completely. Sure a lot of it is true and makes a lot of sense to you, but realize everyone is different, and if that's the way he wants to live his life, then so be it. Don't spend your existence dwelling on being inferior to them.
There is no one like you in the universe.
Laerties
Profile Blog Joined March 2010
United States361 Posts
June 24 2011 06:45 GMT
#3
On June 24 2011 15:40 Blisse wrote:
When you start actually learning about these people, like really studying, you sort of realize how amazing they are. Like Einstein or Newton. It still astounds me how much better they are than me, and how insignificant I will always be compared to the both of them.

Math is hard, but fun once you sort of understand it. And then you have to question, how the hell did these people figure out how to do integration by substition or integration by parts?

There are moments when you realize people are ridiculous and there are many many people that will stomp your ass.


Gandhi sounds like an amazing person, but don't press yourself to follow his words completely. Sure a lot of it is true and makes a lot of sense to you, but realize everyone is different, and if that's the way he wants to live his life, then so be it. Don't spend your existence dwelling on being inferior to them.


Thanks for the advice. I'm always thinking, how the hell did someone figure this out when learning a new concept. I guess thats what happens when you spend ALL YOUR TIME examining tangent lines and areas. Most of the time I feel like i'm just understanding how to execute the tasks of finding areas rates of change volumes etc rather than understanding what I am really doing and why it really works.
Happiness is when what you think, what you say, and what you do are in harmony.
GERMasta
Profile Joined October 2010
Germany212 Posts
June 24 2011 07:08 GMT
#4
Gandhi was an incredibly radical critic of modernity and most people don't really quite realize that, since he's mostly known for his advocacy of non-violence and his political successes against the British. But I would say that he was probably the most radical 20th century critic of modernity that I know of and I'm sure you will come to a similar understanding once you finish the books.

Having said that, many of his ideas I can certainly agree with (despite his criticism of liberalism), since he calls for being deeply self-reflective, being aware of our own values and motivations, leading a life of social and political activity (!), a life of relentless non-violence, consistency of values and austerity. He also emphasises our duties to others and to society, which I especially dig.

The other experiment I tried was to not lie, exaggerate or manipulate what I understand as truth in anyway. It has been realllllyy fucking hard.
You have to be careful with that once you hit a moral situation, since not telling lies could be overridden by higher order moral principles like, in this case, the principle of non-violence. You would certainly not want to tell the Nazis that you're hiding Jews in your cellar because that, even if that means you're telling a lie.
Abductedonut
Profile Blog Joined December 2010
United States324 Posts
Last Edited: 2011-06-24 08:08:28
June 24 2011 07:57 GMT
#5
On June 24 2011 15:40 Blisse wrote:
Math is hard, but fun once you sort of understand it. And then you have to question, how the hell did these people figure out how to do integration by substition or integration by parts?


Actually, things just seem out of reach when you have no idea how they came about. Discovering how to do integration by parts was trivial, because it's just the reverse of the chain rule. Check this out.

Let p(x) = f(x)*g(x)

Then p'(x) = f '(x)*g(x) + f(x)*g'(x) ( Simply by the chain rule, right?)

Solving for f(x)*g'(x)...

-f(x)*g'(x) + p'(x) = f '(x)*g(x)

-f(x)*g'(x) = - p'(x) + f '(x)*g(x)

f(x)*g'(x) = p'(x) - f '(x)*g(x)

Integrating both sides...

∫f(x)*g'(x) = ∫p'(x) - ∫f '(x)*g(x) ( But since ∫p'(x) = p(x) )

∫f(x)*g'(x) = p(x) - ∫f '(x)*g(x) ( But since p(x) = f(x)*g(x) )

∫f(x)*g'(x) = f(x)*g(x)- ∫f '(x)*g(x)

That's exactly the formula for integration by parts. This isn't a rigorous proof of course, but should give you an idea that discovering things sometimes only seems far if you don't have any clue as to how the people who discovered it might have discovered it. After you read that pseudo-proof it should be like, "wow that's easy!!"

On June 24 2011 15:11 Laerties wrote:
The textbook gives a nice table of how to set up the partial fractions depending on the denominator of the fractions but even with it, it takes me a minute to figure out what I'm looking at in the denominator. Also, why the hell is it that the coefficients on both sides of the equation need to be equal? When I look at examples it makes sense but I don't understand why its true…the whole thing just confuses me. I think I have most of the execution down though.


I'll teach you a secret about solving partial fractions that almost nobody knows. I'm not even sure if it's a legitimate method, but it works about 99% of the time. I have no idea where i learned this. Let's look at the partial fraction on this page:

http://www.purplemath.com/modules/partfrac.htm

So it says:

(3x + 2) / x*(x + 1) = A / x + B / ( x + 1)

(I'm not a teacher so I can't explain to you the ideas behind why this works.. but here we go...)

Look simply at...

(3x + 2) / x*(x + 1) = B / ( x + 1) ( so we're ignoring the A part. We will call this EQUATION1 )

Multiply both sides by (x+1) so it's...

(3x + 2) / x = B ( We will call this EQUATION2)

Now ask yourself, how could we have made the denomenator of B zero so that it was B / 0 in EQUATION1. By setting x = -1 so that it's B / ( -1 + 1 ). Now, let x = -1 in EQUATION2. If we do that then it's

( 3*(-1) + 2 ) / (-1) = B

So B = ( -3 + 2 ) / (-1) = 1 (You can check the website to make sure it's true.)

Now let's solve for A... Ignoring B, it becomes:

(3x + 2) / x*(x + 1) = A / x (EQUATION1)

Multiply both sides by x... now its:

(3x + 2) / (x + 1) = A (EQUATION2)

How could be have made the denominator of A zero in equation 1 zero? By letting x = 0. So, let x = 0 in equation2... we get...

(3*(0) + 2) / ( (0) + 1) = A

A = ( 0 + 2 ) / (1) = 2

So, A = 2 and B = 1... Finally we get...

(3x + 2) / x*(x + 1) = 2 / x + 1 / ( x + 1)

Partial Fractions will be VERY easy once you learn to use my method! And lastly, I shall part you with this amazing song!!

+ Show Spoiler +


*edit* Haha, I'm an idiot! "My" method is just the second method in that site I linked you..
Laerties
Profile Blog Joined March 2010
United States361 Posts
June 24 2011 08:06 GMT
#6
On June 24 2011 16:57 Abductedonut wrote:
Show nested quote +
On June 24 2011 15:40 Blisse wrote:
Math is hard, but fun once you sort of understand it. And then you have to question, how the hell did these people figure out how to do integration by substition or integration by parts?


Actually, things just seem out of reach when you have no idea how they came about. Discovering how to do integration by parts was trivial, because it's just the reverse of the chain rule. Check this out.

Let p(x) = f(x)*g(x)

Then p'(x) = f '(x)*g(x) + f(x)*g'(x) ( Simply by the chain rule, right?)

Solving for f(x)*g'(x)...

-f(x)*g'(x) + p'(x) = f '(x)*g(x)

-f(x)*g'(x) = - p'(x) + f '(x)*g(x)

f(x)*g'(x) = p'(x) - f '(x)*g(x)

Integrating both sides...

∫f(x)*g'(x) = ∫p'(x) - ∫f '(x)*g(x) ( But since ∫p'(x) = p(x) )

∫f(x)*g'(x) = p(x) - ∫f '(x)*g(x) ( But since p(x) = f(x)*g(x) )

∫f(x)*g'(x) = f(x)*g(x)- ∫f '(x)*g(x)

That's exactly the formula for integration by parts. This isn't a rigorous proof of course, but should give you an idea that discovering things sometimes only seems far if you don't have any clue as to how the people who discovered it might have discovered it. After you read that pseudo-proof it should be like, "wow that's easy!!"

Show nested quote +
On June 24 2011 15:11 Laerties wrote:
The textbook gives a nice table of how to set up the partial fractions depending on the denominator of the fractions but even with it, it takes me a minute to figure out what I'm looking at in the denominator. Also, why the hell is it that the coefficients on both sides of the equation need to be equal? When I look at examples it makes sense but I don't understand why its true…the whole thing just confuses me. I think I have most of the execution down though.


I'll teach you a secret about solving partial fractions that almost nobody knows. I'm not even sure if it's a legitimate method, but it works about 99% of the time. I have no idea where i learned this. Let's look at the partial fraction on this page:

http://www.purplemath.com/modules/partfrac.htm

So it says:

(3x + 2) / x*(x + 1) = A / x + B / ( x + 1)

(I'm not a teacher so I can't explain to you the ideas behind why this works.. but here we go...)

Look simply at...

(3x + 2) / x*(x + 1) = B / ( x + 1) ( so we're ignoring the A part. We will call this EQUATION1 )

Multiply both sides by (x+1) so it's...

(3x + 2) / x = B ( We will call this EQUATION2)

Now ask yourself, how could we have made the denomenator of B zero so that it was B / 0 in EQUATION1. By setting x = -1 so that it's B / ( -1 + 1 ). Now, let x = -1 in EQUATION2. If we do that then it's

( 3*(-1) + 2 ) / (-1) = B

So B = ( -3 + 2 ) / (-1) = 1 (You can check the website to make sure it's true.)

Now let's solve for A... Ignoring B, it becomes:

(3x + 2) / x*(x + 1) = A / x (EQUATION1)

Multiply both sides by x... now its:

(3x + 2) / (x + 1) = A (EQUATION2)

How could be have made the denominator of A zero in equation 1 zero? By letting x = 0. So, let x = 0 in equation2... we get...

(3*(0) + 2) / ( (0) + 1) = A

A = ( 0 + 2 ) / (1) = 2

So, A = 2 and B = 1... Finally we get...

(3x + 2) / x*(x + 1) = 2 / x + 1 / ( x + 1)

Partial Fractions will be VERY easy once you learn to use my method! And lastly, I shall part you with this amazing song!!

+ Show Spoiler +
http://www.youtube.com/watch?v=7eJoQWrhCtk


Very funny. Btw that method is only going to work completely for fractions that dont have a degree higher than 2 and have a lower degree numerator. So that is definitely not 99% of the time but w/e. It isn't the technical aspect or the use of proofs that goes over my head, its the principles that allow these things to work that I have a disconnect with.
Happiness is when what you think, what you say, and what you do are in harmony.
Abductedonut
Profile Blog Joined December 2010
United States324 Posts
Last Edited: 2011-06-24 08:14:38
June 24 2011 08:11 GMT
#7
On June 24 2011 17:06 Laerties wrote:
its the principles that allow these things to work that I have a disconnect with.


What does that even mean?

*edit* "have a lower degree numerator"

If the numerator is a higher degree than the denominator then you can just divide using long division and then only the remainder will be non-trivial to integrate.
Laerties
Profile Blog Joined March 2010
United States361 Posts
Last Edited: 2011-06-24 08:28:05
June 24 2011 08:26 GMT
#8
You can learn how to execute addition, but that doesn't necessarily mean you recognize or understand why its possible for you to add numbers in any order, or group them when adding more than 2 numbers *the commutative property*....same idea.

also ...your still going to run into fractions that have higher degree denominators, and your going to run into different forms of higher degree equations which require different set ups, and wont allow you to use the cover up method.
Happiness is when what you think, what you say, and what you do are in harmony.
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