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This is fantastic! After nearly 20 years as a self-taught programmer I've been thinking about going back to school to get a CS degree (or master's) Since I ha

• ben_jones 11y ago
  Also an interesting place to apply calculus is in computer graphics. Unlike a lot of math calculus is extremely practical here and sometimes its more fun to make a game then to beat your head on practice problems.
• pakled_engineer 11y ago
  MIT also has supplementary lectures like "Big Picture of Calculus" with Gilbert Strang. http://ocw.mit.edu/resources/res-18-005-highlights-of-calculus-spring-2010/highlights_of_calculus/big-picture-of-calculus/ http://ocw.mit.edu/resources/res-18-005-highlights-of-calcul... He also put his text online for free http://ocw.mit.edu/resources/res-18-001-calculus-online-textbook-spring-2005/textbook/ http://ocw.mit.edu/resources/res-18-001-calculus-online-text...
• matt_wulfeck 11y ago
  Same! I took one college level math class and never looked back. At this point I think I'd get a marginal boost to my programming career, mostly I want to learn because it looks fun and I've never studied it.
  • maaku 11y ago
    You don't know what you're missing without a background in calculus. Calculus is the mathematics underlying human-scale physics, and therefore all of engineering, art, and design. It is the fundamental mathematical theory alongside statistics underlying most of economics and biology. I tell kids going into calculus to pay extra attention: it will change the way they view the entire world. And it will. It is absolutely no accident that the enlightenment, industrial revolution, and modern era began with the discovery of calculus. If you don't have calculus under your belt, I'd definitely suggest starting there.
• el_benhameen 11y ago
  Hackaday also had a great series that nicely answers the question "what is calculus?". There are some really elegant explanations in there. Start here: http://hackaday.com/2015/12/15/calculus-is-not-as-hard-as-you-think/ http://hackaday.com/2015/12/15/calculus-is-not-as-hard-as-yo...
• mindcrime 11y ago
  Two other resources I've used are: this OSU / Coursera Calculus course: https://mooculus.osu.edu/ https://mooculus.osu.edu/ and these Youtube videos: https://www.youtube.com/user/mathbff https://www.youtube.com/user/mathbff https://www.youtube.com/watch?v=UcWsDwg1XwM https://www.youtube.com/watch?v=UcWsDwg1XwM
  • fapjacks 11y ago
    Yes!!! MathBFF is awesome!
• kalid 11y ago
  Kalid from BetterExplained here, glad you like it! I had similar confusions; despite years of engineering classes, I didn't build an intuition for Calculus until a decade later. It felt like such a waste. (Yes, I could memorize rules; but could I visualize the product and quotient rule?) I decided to Elon Musk It™ and reason from first principles (http://betterexplained.com/articles/honest-learning/ http://betterexplained.com/articles/honest-learning/). What would a learning plan look like, assuming limited motivation, a need for genuine understanding, and goal of starting with appreciation and moving to proficiency? Just like learning music, I'd listen to the song, then clap along to it, then play a few chords along with the band, then learn a bit of music notation, and then practice playing it. I wouldn't start with scales (er... limits) for weeks on end. Not if I was honest about my motivations. Ultimately, we have to acknowledge the approach that helps us. For me, investing a few hours of exploration before the formalities is well worth it.
  • petra 11y ago
    Kalid, thanks for your amazing content! Reading this from the link you provided:"In my ideal world, every Wikipedia topic would have a guide that took you from the 1-minute version to a full technical understanding. Go as far as you wish, make meaningful progress at each step, and have fun along the way.". Have you though how to build/scale it ? Because it would seem like an amazing site, and maybe a good startup.
    • defen 11y ago
      > "In my ideal world, every Wikipedia topic would have a guide that took you from the 1-minute version to a full technical understanding. Go as far as you wish, make meaningful progress at each step, and have fun along the way." This is my biggest gripe with Wikipedia for mathematical topics - as it stands, it's only useful if you already know the material, in which case you probably shouldn't be using wikipedia as a reference / refresher except as a last resort.
      • kalid 11y ago
        Exactly. Wikipedia helps me remember things I've forgotten, but it's slow going when trying to learn a new topic. The first paragraph includes a set of links that you recursively follow.
• fapjacks 11y ago
  If you want to try a hardcopy book, a mathematician friend of mine suggested "The Cartoon Guide To Calculus" which is available very cheaply on Amazon. Her recommendation has helped me immensely to learn calculus.
• drostie 11y ago
  I've often thought that you could give people a huge leg up on calculus if you could first explain to them the discrete calculus, possibly with a programming language like Haskell. It would look like this: We're going to start thinking of infinite sequences. A sequence starts with a value, and then there is another value, and so on, and so on. We might number these sequences and then write down rules for them; for example we might write $x_n = (n + 1) ^2$ to denote the sequence of square numbers starting with 1, `map (^2) [1..]`; if you're not familiar with these then we'll write out the first five as: squares = 1 : 4 : 9 : 16 : 25 : map (^2) [6..] I'm going to teach you one list-operator, it's a function which subtracts each number in a sequence from the one that comes before it in that sequence. For the first element, it just subtracts 0, so it doesn't do anything. This is written in Haskell as: delta seq = zipWith (-) seq (0:seq) When we use the above we get: Prelude> take 30 (delta squares) [1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,51,53,55,57,59] (Here the `take 30` just stops the computer from trying to give us all infinity of the results!) Interestingly, the difference between square numbers is the set of odd numbers. This is not too hard to understand after you know it -- $(n + 1)^2 = n^2 + 2n + 1$, so of course after you subtract $n^2$ you get $2n + 1$. Similarly when we start looking at the various counting sequences like [1..] (which again if you don't know Haskell is the same as 1:2:3:4:5:[6..], the infinite sequence of successive integers starting with 1) and [2..] we find: Prelude> take 30 (delta [1..]) [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] Prelude> take 30 (delta [2..]) [2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] Prelude> take 30 (delta [3..]) [3,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] We notice that they're all the same except for that very first element, which basically just tells us the first value. But actually, `delta` has a special property which is that it is invertible: just like you can invert `\x -> x - 3` with `\x -> x + 3`, you can invert this `zipWith (-)` beast with `scanl1 (+)` Prelude> let sigma seq = scanl1 (+) seq Prelude> take 30 (sigma (delta [3..])) [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32] Prelude> take 20 (sigma (delta squares)) [1,4,9,16,25,36,49,64,81,100,121,144,169,196,225,256,289,324,361,400] What is this magical beast `sigma`? What is `scanl1`? This is a function which processes a sequence by passing on an "accumulating value", but unlike `foldr` and `foldl` where you get that accumulating value at the end of the sequence, `scanl` gives it to you at each step of the journey, including the first. So like with foldl1, we start with the more general case of foldl or scanl: scanl fn val list = val : case list of [] -> [] x : xs -> scanl fn (fn val x) xs So we always emit a sequence whose next element is `val` and whose follow-up sequence is either empty (if the source sequence is empty) or consists of scanl'ing the rest of the sequence with the same function but with val updated to `fn val x`. Each new element gets combined into the existing value with `fn` and the results are getting steadily emitted as it goes. Then the "1" in `scanl1` means, just like the "1" in `foldl1`, "read the first value from the first element of the list": scanl1 fn list = case list of [] -> [] x : xs -> scanl fn x xs So when we defined `sigma` as `scanl1 (+)` we just specified this `fn` to be one which adds the two values together. The above recomputation of the squares from the odd numbers actually just works like this: 1 = 1 1 + 3 = 1 + 5 = 4 1 + 3 + 5 = 4 + 5 = 9 1 + 3 + 5 + 7 = 9 + 7 = 16 1 + 3 + 5 + 7 + 9 = 16 + 9 = 25 1 + 3 + 5 + 7 + 9 + 11 = 25 + 11 = 36 Notice as the second column here indicates, we don't need to recompute all of these pluses each time: we can just hold onto the last number we've computed, the previous item from the last column, and add that to the new item to get the new value. I'm asking you to notice the last point because if you study it carefully enough, that we generate each new value by adding the sequence to the last value, you'll start to ask, "wait, is `delta . sigma` also an identity transformation, just like we saw `sigma . delta` is?" And the answer is a resounding yes. This is actually a very common property for inverses, usually after you've proven that `f . g = id` then there's also an argument that `g . f = id`. So, we've learned that just like `(+)` undoes `(-)` and vice versa, summing up a bunch of terms from some start point to a current place undoes taking differences between nearest-neighbors, and vice versa. I haven't actually proven either side of this, but it's not too difficult: to see that delta undoes sigma, notice that for n > 0 we have $$ \sum_{k=0}^{n} x_k - \sum_{k=0}^{n-1} x_k = x_n $$ (and the x_0 case is just x_0 - 0 = x_0, so that works too). To see that sigma undoes delta is a little more complicated, but just write out the n'th term to see:$$ (x_0 - 0) + (x_1 - x_0) + (x_2 - x_1) + \dots + (x_{n-1} - x_{n-2}) + (x_n - x_{n-1}). $$ Notice that except for the first and the last term, each term contains two parts: a +x_k which will cancel with the next term and a -x_{k-1} which will cancel with the previous term. This is called a "telescoping" series because it's really long but it collapses like a telescope due to perfect cancellations between adjacent terms. The only thing that's left at the end is $(-0) + x_n,$ which of course is just $x_n.$ So `sigma` has indeed undone `delta`. To follow up, first I would probably define: instance Num n => Num [n] where (+) = zipWith (+); (*) = zipWith (*); (-) = zipWith (-) abs = map abs; negate = map negate; signum = map signum fromInteger = repeat . fromInteger k !* seq = map (k *) seq so that we have sequence algebra and we can multiply them by constants; then I would introduce people to basic discrete-calculus results like `delta (f + g) = delta f + delta g` and `delta (k !* f) = k !* delta f` and `delta (f * g) = f * delta g + g * delta f + delta f * delta g`. Finally comes the big reveal: derivatives, to difference a continuous function, treat that function as if it were a straight line when you zoom in close enough, and to do this we have to divide `delta f` by some $d x$ to make it work; similarly integrals, to be the proper inverses to derivatives, contain a `(dx * )` operation. So if you start to define a grid of size dx over the x's that you care about, u < x < ∞, then each function corresponds to a sequence $f_k = f(x_k) = f(u + k * dx).$ Then `map (/ dx) . delta` generates a version of `delta` whose results converge, but not (in general) on the trivial answer 0; rather it converges on the "instantaneous slope" at x. Meanwhile `map (dx *) . sigma` perfectly undoes this, and calculates the area under a curve from u to x. From this, observe that the product rule gets easier due to these limits and introduce the formal definitions (limits, largest element of a partition going to zero for integrals; simple "h" parameter going to zero for derivatives).
• Buttons840 11y ago
  I would recommend the book "What is Mathematics" to you. It covers a wide range of math topics from first principles. The beginning of each chapter is light reading, but gets more rigorous as the chapter progresses. It talks about mathematical notation a good amount, which is surprisingly helpful because many math ideas are intuitive but hidden behind extremely inaccessible notation. The Kindle version has bad formatting, but the Google Books version works well on all devices supported by Google Books.