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Introduction to Calculus with Derivatives
- foxes 8y agoEssentially from the definition of the derivative you can write f(x+h) = f(x) + h f'(x) + O(h^2). If you use this to solve differential equations its called the Euler method.
- pedrosorio 8y ago> its called the Euler method Disclaimer: and it sucks
- FPGAhacker 8y agoWhen I was on vacation a couple weeks ago, I was inspired by the Feynman lectures on physics to review basic calculus. This time I tried my best to understand derivatives at a low/basic level. When I was in college, many years ago, I was doing my best to just to tread water in physics and the early stages of the calculus classes. I did my level best to try to explain derivatives as if I knew nothing of calculus. Used my wife as a guinny pig. What I ended up with explained both derivatives and integrals at the same time in a very basic, but correct (I believe) sort of “add up the arrows” sort of way. I really struggled to do this, but I thought it was really cool when I finished. Well finished is perhaps too strong a term. Most of it is scrawled out in handwriting on my iPad. I’ve been meaning to go back and wrap it up and put it on the web. I think I’ll try tonight and post back here or somewhere.
- digitalsushi 8y agowell i really hope you do, i'd love to flip through it. each time i read some other person's notes, another tiny angle of light hits the statue.
- FPGAhacker 8y agoJust read through the post to make sure there was still value to me doing this. Seems like it. We start similarly, which is good. The visual description seems to be pretty different so I think there will be value in another point of view.
- dvfjsdhgfv 8y agoDon't hesitate to do that!
- shawn 8y agoI did this today with neural nets and training classifiers. Looking forward to reading your notes!
- commandlinefan 8y ago> guinny pig Guinea pig. They were originally believed to have originated from west Africa (Guinea), hence the name.
- tmpmov 8y agoI like this take on introducing derivatives. My first intro to derivatives was a little less than 20 years ago, but I feel like it was very much in the "traditional" vein of: Suppose we have "f'(x) = lim(h->0) (f(x+h) - f(x))/h" and we substitute in various equations. What will f'(x) be? The difference as presented here: I (re?)learned an estimation method for decimal place mathematics while at the same point tying it to a larger/underlying principle. I think a great approach would be to then do the stuff I started with, e.g. finding f'(x) given f(x). Out of curiosity, how many of you have seen the approach as seen in the above article? I can't recall seeing it before, but again, it was a fair time ago for me.
- stevewilhelm 8y agoI was hoping this article was going to introduce calculus using financial derivatives in practical examples. Instead the article asks how would one extimate 4.1 squared without a calculator. I can't recall of any time in my life when I needed to do so. Now that I have a smart phone and watch, I don't anticipate needing to do so any time soon.
- yiyus 8y agoIf everybody was of the same opinion as you, we would not have smart phones and watches.
- deleted 8y ago[deleted]
- crispyambulance 8y agoIt looks like a very nice careful description, but where is the physical intuition? Where's the plots of tangent lines on a function at various points? There's nothing wrong with the article (I think), I am just curious about the pedagogy and motivation for not immediately introducing a graphical conceptualization of a derivative especially for such a "from the ground up" exposition.
- egonschiele 8y agoHi, author here. This is what made derivatives finally click for me (as opposed to the tangent line) so that’s how I wanted to explain it...maybe the same thing would click for others!
- 1ba9115454 8y agoI was looking for a tutorial on derivatives the other day and most of the results popping up at the top of Google were a bit dry. This one is very good though. Excellent work. How did you do the diagrams?
- egonschiele 8y agoThe Procreate app on the ipad
- shawn 8y agoDo you have any recommendations for learning the app to make what you’ve made? I don’t suppose I could bribe you into recording a video of yourself making some diagrams... Your algorithms book is excellent and fun!
- egonschiele 8y agoIMO it is more about drawing ability vs app knowledge, so I would suggest learning to draw! Just start drawing and you'll get better over time. You could join something like https://streak.club/s/8/daily-art https://streak.club/s/8/daily-art to keep motivated.
- d--b 8y agoI think this is backwards. Explaining derivatives by Physics (speed or slope) seems a lot more intuitive to me.
- ColinWright 8y agoBackwards for you - different people approach things from different points of view.
- mayankkaizen 8y agoGreat article. Though I know some basic calculas but I never really thought about calculas this way. This is an interesting and eye opening perspective. Wish to see more of such articles.
- presscast 8y agoIf one wants to learn calculus, I can't recommend Silvanus P. Thompson's Calculus Made Easy enough. It's one of those old (1920's, IIRC) books that gets right down to brass tax, and doesn't clutter your understanding with useless fluff designed to make learning maths "fun". These kinds of books are all too rare.
- FranzFerdiNaN 8y agoThis explanation actually reminded me of that book and I suspect it was an inspiration.
- snthd 8y agoFound an html conversion - http://calculusmadeeasy.org/ http://calculusmadeeasy.org/
- shawn 8y agoThis was one of the books Feynman used to learn Calculus. The other was Advanced Calculus by Woods: https://libgen.pw/download/book/5a1f04723a044650f5fbaf91 https://libgen.pw/download/book/5a1f04723a044650f5fbaf91
- mmcclellan 8y agoI have seen this discussed online several times. Here is a HN thread: https://news.ycombinator.com/item?id=14192894 https://news.ycombinator.com/item?id=14192894 that indicates the text Feynman first used was Calculus for the Practical Man by J. E. Thompson. Both are quality texts and as the linked HN thread notes, can be found on archive.org.
- shawn 8y agoMy mistake. Thanks!
- jaclaz 8y ago1914 to be picky: https://news.ycombinator.com/item?id=14161876 https://news.ycombinator.com/item?id=14161876 (link to pdf) A similar one: https://news.ycombinator.com/item?id=14178174 https://news.ycombinator.com/item?id=14178174
- toolslive 8y agoI'm from Belgium, and here kids learn this in high school (5th year). It's basic knowledge. So why does this show up on HN?
- hackandtrip 8y agoI'm from Italy and it's the same here, but the problem remains: it seems that a big part of population (devs are not excluded) have problems and hate relationship with math.
- Sharlin 8y agoYep. In Finland, derivatives are discussed (in a superficial manner) in high school even if you take as little math as you can, but from what I’ve heard most people who don’t elect to take more in-depth courses don’t really grok what they’re about and promptly forget about them after the exam.
- Regardsyjc 8y agoI graduated college from the USA but with a liberal arts degree and the highest math I learned was precalc.
- tinco 8y agoOnly kids who got into higher education tracks get calculus. No need to be smug about it, many kids don't get in those tracks for any reason and might find later in life that they are interested in calculus. I feel if anything the US education system exposes to more children than the European systems, even though I feel the system is misguided. In any case, I had calculus in highschool, then more of it in University and I still feel I could do with a fun refresher every now and then.
- toolslive 8y ago> I feel if anything the US education system exposes to more children than the European systems What do you mean?
- Patient0 8y ago"That change c never truly goes away, but it is so small that it doesn't really matter: if you make c small enough, the derivative is basically 8. So we're just going to pretend that all the cs are so small they have become zero." My first calculus teacher taught us derivatives in a similar way - but I have to say that, for me, this imprecise language confused me. When and why is it OK to pretend that c is zero!? Further on, the official definition using a limit claims that the "c->0" means "make c as small as possible". But that's not what it means. Again, this is imprecise language that confuses more than it explains. What does "as small as possible" even mean? If we want to make it as small as possible, why not set it to 0!? How small is small enough? I think somewhere in this article there should be a precise definition of what a limit is, using epsilon and delta. It should be saying something using the words "arbitrarily close" and "sufficiently small". Something like : "we can make the approximation on the left arbitrarily close to the expression on the right by choosing any value of c sufficiently close to 0, even though the expression might be undefined when c=0. Epsilon: you tell me how close you want the approximation to be. Delta: I tell you how small c needs to be". There used to be a great website that was precise but also informal: karlscalculus.org - but sadly it appears to be down now.
- foxes 8y agoI think it was Weierstrass who first gave a proper definition in terms of epsilon / delta (e,d). To say it more precisely: We say that the limit lim x -> c f(x) = L, if for all e > 0, there exists a d, such that for all x, if | x - c | < d, then | f(x) - L | < e.
- Patient0 8y agoThe history seems to be summarized well in this wikipedia entry: https://en.wikipedia.org/wiki/(%CE%B5,_%CE%B4)-definition_of_limit https://en.wikipedia.org/wiki/(%CE%B5,_%CE%B4)-definition_of...
- marsrover 8y ago> Further on, the official definition using a limit claims that the "c->0" means "make c as small as possible". Yeah.. I’m pretty sure the official definition of a limit says nothing like that. Any definition of a limit that I’ve ever seen (official or not) has been careful to note that you cannot set c to 0, but only approach it (whether in mathematical terms or English).
- koalala 8y agoJust have to add K.A. Stroud's Engineering Mathematics to this list. Easiest maths book I've ever known, with only the slightest amount of hand waving. If you're having any trouble with it, this is your go-to book.
- billfruit 8y agoIsn't there a better, more clearer, and more intuitive notation for calculus than dx/dy, or even f'? I feel these notations are somewhat ambiguous and bordering on abuse of notation.
- bsznjyewgd 8y agoYes. There is an operator notation for the derivative where you can both avoid the not-quite-division and specify the variable you're dealing with. https://en.wikipedia.org/wiki/Notation_for_differentiation#Euler's_notation https://en.wikipedia.org/wiki/Notation_for_differentiation#E...
- z2 8y agoUnrelated to calculus, a simple mental trick for quickly calculating squares close to round numbers is to just break it into (a+b)^2=a^2+2ab+b^2. E.g. 51^2 = 2500+100+1=2601.
- SOLAR_FIELDS 8y agoThis is a slight reduction of the FOIL method, a classic in any first year algebra course. A decent trick to remember on tests when you are trying to quickly solve a lot of algebra problems.