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Calculus for the Curious

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Re: Calculus for the Curious

#21

Post by dxdy »

soVeryTired wrote: post #19
braindeath wrote: post #16

Earlier quoted context omitted.

Euler lived in the 1700s. Not sure how that would have affected your training unless you’re much older than average. Cauchy and Taylor both formalized many concepts, again in the 18th and 19th century. What are you thinking is ad box?

In high school mathematics, you're not really given the definition of a limit. Consider the definition of the derivative limit as h -> 0 of (f(x+h) - f(x)) / h. That's well-defined on (0, inf) but not on [0, inf). So you can't just evaluate at h=0 and be done with it. The intuition is 'as h gets smaller and smaller, the ratio gets closer and closer to a new function of x'. But many high-school students aren't given a…

Let n = 0.999... Then multiply both sides of the equality by 10 so that we have 10n = 9.999...Then subtract n from both sides of the resulting equality to get 9n = 9.000...Finally, divide both sides by 9 and voila we have n = 1 which is what we wanted to show.

Re: Calculus for the Curious

#22

Post by mav3rick »

yessenia1 wrote: post #17

I used to always associate 'math smarts' with 'code smarts'. Spent most of my life telling myself that since I was bad at math I had no hope learning how to code. Now I'm 2 weeks into a coding bootcamp after losing my job and am realizing they are complete different parts of the brain. I believe in myself more with every push to heroku. maybe I will do a Calculus class next and prove to myself I can learn anything. A…

What did you do before if you don't mind me asking ? Would you have done an investment banking bootcamp if it guaranteed higher pay ?

Re: Calculus for the Curious

#23

Post by vostok »

jacobolus wrote: post #6
vostok wrote: post #4

Earlier quoted context omitted.

Spivak has a pretty decent calculus textbook that you might find interesting. I used a different textbook, but I believe that Spivak is more popular.

The point of Spivak’s book is to be rigorous and offer hard problems, not teach the basics of how calculus is used. Doing epsilon–delta proofs can be fun, but it’s mostly useful for aspiring pure mathematicians, and not really relevant per se for the grandparent poster.

I took this to mean that they're interested in a more rigorous treatment.

> [I] am now in a graduate CS program, I feel like I’m missing the deeper understanding of many of the formulas that are presented

> My plan is to get through it to get some background on the main ideas of calculus

Re: Calculus for the Curious

#24

Post by yessenia1 »

mav3rick wrote: post #22
yessenia1 wrote: post #17

I used to always associate 'math smarts' with 'code smarts'. Spent most of my life telling myself that since I was bad at math I had no hope learning how to code. Now I'm 2 weeks into a coding bootcamp after losing my job and am realizing they are complete different parts of the brain. I believe in myself more with every push to heroku. maybe I will do a Calculus class next and prove to myself I can learn anything. A…

What did you do before if you don't mind me asking ? Would you have done an investment banking bootcamp if it guaranteed higher pay ?

Apple Retail. Didn't qualify for Genius position after trying very hard and never felt like I fit in. it inspired me to play with swift playgrounds, which led me to the bootcamp program which I am LOVING

Re: Calculus for the Curious

#25

Post by chobytes »

lordnacho wrote: post #15

"See for example Euler, who made great strides in the development of calculus without any really defined concepts of convergence, divergence or limits, but who doesn’t appear here at all." Can anyone say more about this? I always suspected this was so, because the end of high school / start of uni is roughly that sort of time period, and I always thought there was something about convergence/limits that was missing,…

As I understand it our modern notion of what rigorous mathematics is didn’t exist back then. The justification for analysis was basically physical intuition and simply that it worked.

Euler and company just worked directly with the intuition of infinitesimal quantities. To be fair though, infinitesimals are the essential intuition for how calculus works anyway.

Personally I don’t really care about constructions of the real numbers or technical details of calculus. It’s enough for rigorous mathematics to know that models exist of the theories which produce calculus; that is models of complete ordered fields and even plain old ordered fields with infinitesimals.

Re: Calculus for the Curious

#26

Post by soVeryTired »

dxdy wrote: post #21
soVeryTired wrote: post #19

Earlier quoted context omitted.

In high school mathematics, you're not really given the definition of a limit. Consider the definition of the derivative limit as h -> 0 of (f(x+h) - f(x)) / h. That's well-defined on (0, inf) but not on [0, inf). So you can't just evaluate at h=0 and be done with it. The intuition is 'as h gets smaller and smaller, the ratio gets closer and closer to a new function of x'. But many high-school students aren't given a…

Let n = 0.999... Then multiply both sides of the equality by 10 so that we have 10n = 9.999...Then subtract n from both sides of the resulting equality to get 9n = 9.000...Finally, divide both sides by 9 and voila we have n = 1 which is what we wanted to show.

My gut feeling is that that proof isn't quite correct, since you haven't used the notion of a limit anywhere. There's a fundamental fact about convergence of geometric series that you need to use.

I think your proof goes wrong since you haven't justified how arithemtic operations work with infinite decimals. AFAIK the only way to add non-terminating decimals is to convert them to fractions (or sequences of fractions as with pi, e, etc), add the fractions, and convert them back. So if you convert 0.999... and 9.999... to fractions, you've assumed the conclusion.

To play devil's advocate, I can try to rephrase your proof without infinite decimal arithmetic as follows.

Assume

n = 0.999... = 1 - epsilon, where epsilon is 'infinitesimal' (an ill-defined version of not-quite-zero). We'd like to show that epsilon is zero.

10n = 9.999 = 10 - 10epsilon

9n = 9.999 - (1 - epsilon) = 9 - 9epsilon

9n = 8.999 + epsilon = 9 - 9epsilon

The only way to get the epsilons to cancel is to assume epsilon = 0, which is to assume the conclusion.

Re: Calculus for the Curious

#27

Post by shantly »

yessenia1 wrote: post #17

I used to always associate 'math smarts' with 'code smarts'. Spent most of my life telling myself that since I was bad at math I had no hope learning how to code. Now I'm 2 weeks into a coding bootcamp after losing my job and am realizing they are complete different parts of the brain. I believe in myself more with every push to heroku. maybe I will do a Calculus class next and prove to myself I can learn anything. A…

I have bury-the-needle good spatial reasoning and somewhat above average general intelligence. Gifted programs, all that stuff. Ought to be anywhere from somewhat good to very good at mathematics. But... I feel the way I figure dyslexics feel reading human language, when I read math.

Code? Natural and easy, even “hard” concepts. There, my measured natural abilities come out just as you’d think they would. My best guess is I find algorithmic thinking easy, but proof/equational thinking unnatural. All I can figure. I’ve had some limited success approaching math with a “what does this term _do_?” attitude, but it’s slow going.

IOW don’t worry, there are others out there. Sometimes we even get a reputation for being the ones to go with for the tricky stuff. Go figure.

Re: Calculus for the Curious

#28

Post by dxbydt »

ncfausti wrote: post #2

I just stumbled upon Infinite Powers in B&N, and after reading a bit, decided to buy it. Loving it so far. I’ve been curious about calculus for a long time now, since I’ve never taken a formal class on it and am now in a graduate CS program, I feel like I’m missing the deeper understanding of many of the formulas that are presented. My plan is to get through it to get some background on the main ideas of calculus, th…

This is a fantastic thread https://twitter.com/nntaleb/status/1163192701472428032?lang=... from Taleb. As always, a little bit over the top, but mostly on point. I've seen the exact phenomenon he describes play out so many times now its not even funny. I'm doing a PhD in mathematical statistics, which is sort of like "calculus on steroids". I basically do calculus day in day out. Many of the problems we attack are simply not of much interest to mathematicians. I used to run my problems by math profs & they would say something like yeah it can be done, have you tried mathematica etc. instead of buckling down to do it. For instance, yesterday I had to prove that Fisher information of Cauchy is half. Now that's entirely calculus. You take a function f(x,t) = (pi*(1+(x-t)^2))^-1. You then take the log of that. Then you differentiate that w.r.t t. What you get is called the Fisher score function S(t). So you take the score function and differentiate that again. Lets call that g(x,t). Stick a negative sign in front of that. So now you have a complicated looking new function -g(x,t). You multiply the -g(x,t) with your original f(x,t) and integrate that product over the reals. The result is half. Most mathematicians usually get stuck on that last step. But for (mathematical) statisticians, this os sort of our bread and butter integral. So we know a bunch of tricks. Here's one such trick - https://stats.stackexchange.com/questions/145017/cauchy-dist...

Its like riding a bicycle. Those who ride the most know how to ride. But there are some who want to know how do bikes actually work...which is not going to help you much with riding the bike.

Re: Calculus for the Curious

#29

Post by darthazad »

soVeryTired wrote: post #19
braindeath wrote: post #16

Earlier quoted context omitted.

Euler lived in the 1700s. Not sure how that would have affected your training unless you’re much older than average. Cauchy and Taylor both formalized many concepts, again in the 18th and 19th century. What are you thinking is ad box?

In high school mathematics, you're not really given the definition of a limit. Consider the definition of the derivative limit as h -> 0 of (f(x+h) - f(x)) / h. That's well-defined on (0, inf) but not on [0, inf). So you can't just evaluate at h=0 and be done with it. The intuition is 'as h gets smaller and smaller, the ratio gets closer and closer to a new function of x'. But many high-school students aren't given a…

I had trouble with the whole 0.999... = 1 thing as a young lad until I noticed something about fractions.

Any digit over 9 equals a decimal of zero point itself repeating. I.E.

1/9 = 0.111... 2/9 = 0.222... 3/9 = 1/3 = 0.333... ... 1 = 9/9 = 0.999...

Re: Calculus for the Curious

#30

Post by soVeryTired »

darthazad wrote: post #29
soVeryTired wrote: post #19

Earlier quoted context omitted.

In high school mathematics, you're not really given the definition of a limit. Consider the definition of the derivative limit as h -> 0 of (f(x+h) - f(x)) / h. That's well-defined on (0, inf) but not on [0, inf). So you can't just evaluate at h=0 and be done with it. The intuition is 'as h gets smaller and smaller, the ratio gets closer and closer to a new function of x'. But many high-school students aren't given a…

I had trouble with the whole 0.999... = 1 thing as a young lad until I noticed something about fractions. Any digit over 9 equals a decimal of zero point itself repeating. I.E. 1/9 = 0.111... 2/9 = 0.222... 3/9 = 1/3 = 0.333... ... 1 = 9/9 = 0.999...

This one is better than the other child comment, but you still need to show that the limit of a sum is the sum of the limits. Not as easy as you might think, and not usually done in high school!
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