Calculus In 20 Minutes

If you went to engineering school, you probably remember going to a lot of calculus classes. You may or may not remember a lot of calculus. If you didn’t go to engineering school, you will find that there’s an upper limit to how much electronics theory you can learn before you have to learn calculus. Now imagine Khan Academy, run by an auctioneer and done without computers. Well, you don’t have to imagine it. Thinkwell has two videos that purport to teach you calculus in twenty minutes (YouTube, embedded below).

We are going to be honest. If you need a refresher, these videos might be useful. If you have no idea how to do calculus, maybe these are going to whiz by a little fast. However, either way, the videos have some humor value both from the FedEx commercial-style delivery to the non-computerized graphics (not to mention the glass-breaking sound effects). Of course, the video is about ten years old, but that’s part of its charm.

Although calculus has a reputation of being tough, the concept is quite simple. If you have a function that represents something, the derivative of that function describes the rate of change at any point. The integral describes the area under the curve of the function. At first glance, those things don’t appear related, but in fact, they are inverses of each other. That is if you take a derivative and then integrate the result, you’ll wind up with the original function (well, at least, that’s one of the answers you’ll get). Conversely, if you integrate a function then take the derivative you will wind up back where you started.

This is useful in electronics when dealing with AC circuits. The behavior of capacitors and inductors depend on rates of change and while you can memorize certain common formulas, they are just answers to calculus problems that someone else worked for you.

One thing we did notice is that the video covers the theory behind taking a derivative but doesn’t cover much about the rote rules most people remember for common cases. Once you notice that the derivative of 2x is 2 and the derivative of 4x squared is 8x, for example, you can probably deduce the generic rule that would tell you that the derivative of 3x to the fifth power is 15x to the fourth power.

Hackaday has covered this topic, too. If you want something more leisurely than these videos, we might suggest Khan Academy. If you prefer a fast read, the book Quick Calculus is short, inexpensive, and a good refresher. A hundred-year-old classic book is available free if you are too cheap to drop a few dollars. We love that book’s subtitle: “Being a very-simplest introduction to those beautiful methods of reckoning which are generally called by the terrifying names of the differential calculus and the integral calculus.” If you are starting from scratch and want the college experience, try MIT.

56 thoughts on “Calculus In 20 Minutes

  1. I missed calculus as well and it has plagued me all through my career. I decided recently I was going to try and work it out so borrowed a book from the library called something like calculus for the totaly confused.

    By pg 13 I was totally confused ( not Helped by major error in an equation in the book)

    I will now have to look at these videos and see if they help.

      1. Be careful giving that advice, different people learn differently.

        I knew lots of people in college who would barely even crack their books open. Probably like you, they derived everything they needed from the in-class lectures. The books did nothing for them. Most of the time those were the students with the best grades. If they had a class where the professor expected the students to learn certain things from the book however, without having explained it in class… they were in trouble.

        I was the opposite. Half way through a lecture I would realize that I had missed most of it because I was still pondering something about the first thing the professor said or some implication of it. If there were no surprise quizzes I could probably have skipped most of my classes and done just as well. From the books however… I could learn anything.

        My favorite classes were those where the professor wrote the book. Those were the classes where I could trust that everything the professor talked about was also in my textbook. I could get away with toning the lecture out, reading the book and doing great!

        Needless to say, I kind of liked the internet better before video became so popular.

      1. It depends on your work. There are some — maybe even most — jobs where you just use the results. But if you do things like try to find RMS of arbitrary waveforms or do many kinds of modeling, simulation, or optimization, it is useful. Of course, like most things now, you don’t HAVE to know how to do it. The computer can spit it all out. But you do have to know that you WANT to do it and how to go from the word problem (how do we reduce harmonic distortion?) to the actual formulation to solve and how to interpret the results.

  2. While I guess your statement is correct, saying that not knowing calculus sets a limit on your ability to learn electronics is like saying you have to be a chemist to fully grok cooking food. Technically, sure, it helps, but not many people will find that lack of calculus knowledge is a barrier. But of course if you find yourself in need of calculus it’s good to have a resource like this :-).

    1. Just depends on what you are trying to do and how much you want to “black box”. For example, if you want to know the RMS voltage of a strange waveform you can empirically measure it or you can compute it. If I want to know how long it takes a capacitor to charge to 40% of the supply voltage, I can measure it, look it up, use a formula, or I can work it out easily from calculus and the fundamental equation that governs capacitors.

      So I don’t know if a chemist is quite the right analogy. I would say, using only canned ingredients will limit you as a chef.

          1. Quite true. I used to see it all the time with engineers who had been out of school for some time. They couldn’t do calculus without cracking open the books and refreshing themselves on it.

            It is a perishable skill.

            The thing is they didn’t use it every day if at all. Most of their work didn’t need calculus – just basic pre-calculus. Even in electronic design you can get away without needing calculus to a large extent. Which is a good thing otherwise the field would remain a black box for most of humanity.

      1. Might be one of those people who can only apply academic knowledge in a rigid context…. I’ve caught engineers staring at a mechanical issue totally bemused that only requires application of leverage or equally simple principle to solve.

  3. I have an engineering degree and took a number of calculus classes. I passed, but it was mostly by rote memorization, not actually grokking the subject. What DID finally give me a fuller picture was that hundred-year-old book, Calculus Made Easy. I think what it did was match my learning style; the college texts were just awful at imparting the subject matter whereas I’d read a passage in C.M.E and think, “Oh, so THAT’S what that meant.

    If you’re just not getting it, maybe the problem isn’t you, maybe it’s an impedence mismatch between the particular instruction material and your brain. Try another source.

    1. “Being myself a remarkably stupid fellow, I have had to unteach myself the difficulties, and now beg to present to my fellow fools the parts that are not hard. Master these thoroughly, and the rest will follow. What one fool can do, another can.”

      This is the exact opposite of every other textbook I’ve read. The author’s self effacing style is very refreshing, 100 years after its last printing, and makes me wish this author had done a whole series. The ability to be able to present a difficult subject is a clear and concise manner is a rare gift.

    1. If you have a curve that represents your speed over time (speed on y axis, time on x axis), then the derivative of this curve is another curve that represents your accelleration over time (acceleration on y axis, time on x axis).

      So you’re in a car, driving at a constant 100mph. Now you push the accelerator begin accelerating, then the first curve shows a ramp and the second curve shows a step as the acceleration sets in.

  4. Calculus is, in fact, a stunningly simple concept that’s been around for 400 years. The hard part is the algebra that goes with it which is often very long and tedious. In actual working practice you usually look up the formulae most of the time. But in some jobs you have to do a lot of work to reduce the problem to a form you for which a solution has been tabulated in a book of integrals.

    I taught myself algebra in the 6th grade and trigonometry in the 8th. After getting a BA in English lit, I went back for an MS in geology which required calculus. The first day we were given a 100 point quiz on algebra and trig. The next day the instructor said, “Students, if you made a 17 or below on this quiz you need to drop this course and take precalculus. It can be shown statistically that you won’t pass the course.” I was sitting looking at a 17 out of 100. Five yeas earlier I’d gotten a C in precalculus to meet my BA requirements because I was bored and didn’t do homework. I was NOT doing that again.

    So I worked calculus problems 3-4 hours a day for 5-6 days a week. By the end of the semester I had caught up with the class hotshot on the 20 point weekly quizzes. Sometimes he had the top score and sometimes I did. He was also in my physics I class the next semester. We got to chatting and I learned that he often spent 3-4 hours or more working on a single problem. On reflection I concluded that what mattered was not giving up. People who worked until they solved the problems got A’s, those who went for help after 2-3 hours got B’s, and so forth. The sooner someone went for help, the lower their grade.

    I enjoyed calculus I I took II, III and differential equations. I enrolled in advanced applied, but bailed after hitting eigenvalue problems. As a geologist I had no idea what to do with them. After I moved into geophysics I learned a lot more advanced applied math.

    In one of my petroleum geology class we studied electromagnetic logging tools. It was very interesting to see how easy the subject was for those who had taken physics with calculus vs how difficult it was for those who had not been taught physics using calculus.

    1. > electromagnetic logging tools

      I pictured the Canadian lumberjack from Monty Python in Star Trek uniform cutting trees with a phaser. Then I realized it was a different kind of logging.

  5. I found these videos years ago and shared them with my classmates! We were near the end of first semester of college and our technical math instructor apologized that she only had 2 weeks to teach us some basic calculus before the final. Of course we could all follow the steps to produce derivatives but the majority of us didn’t intuitively understand the meaning behind it all, until we watched these videos.

  6. I met calculus in the middle, then from the beginning, and it was infinitesimal slices, got it, rate of change, got it, mathsplosion, don’t got it, and spent a couple of years terminally confused, before slogging through CME and joining the dots.

  7. derivative vs integration — “they are inverses of each other”.

    f(x) = 3x^2 + 7x – 2
    f'(x) = 6x + 7

    f'(x)… is not an inverse of … f(x).

    The inverse of ( 1 / 2 ) = ( 2 / 1 ).

    It would be better to think of differentiation vs integration as:
    sqrt(2^2) = 2 ; vs 2^2 which is 4 (theses aren’t inverses).

    1. I think most people who would read this would not be confused by my common use of inverse. 5 x 2 = 10 and 10 / 2 = 5. As you say, more precisely inverse operations, but I don’t think too many people would get mixed up.

      I would have thought you more likely to take me to task for omitting the constant of integration, but I did allude ot it.

      So yes f(x)=3x^2 + 7x – 2 and f'(x) = 6x + 7… but the integral of f'(x) = 3x^2+7x+C which is a set of solutions where one of them (C=0) is f(x). That was probably more confusing if you didn’t already know it than my colloquial use of inverse.

      That’s why I said: That is if you take a derivative and then integrate the result, you’ll wind up with the original function (well, at least, that’s one of the answers you’ll get).

      1. @Al Williams
        But as you write losing the constant so its an ‘incomplete’ “inverse operation”.
        ie. Just saying an inverse operation implies reversibility which it doesn’t, well it
        does for the slope but not to restore the original function completely
        iow. Only enough to restore what could then be integrated again to give the
        same result.
        So maybe calling it a “selective inverse operation” (SIO) or that derivation
        is Non-Commutating, such as NCIO, would be better .

        It gets more interesting philosophically when you take the double derivative then
        try to recover the original function by double integration – you lose not just
        the constant but one lot of “slope” identified as the 2nd time you integrate
        you have a different slope. Fortunately in most engineering (so far as I know)
        this isn’t likely to be a problem – though it adds an area of unresolved solutions.

        a one pass derivative/integral (D to I to D) loses the constant but preserves the
        means to recover the slope
        but, a two pass DD to II to DD loses the slope information as there is a variable
        multiplied by a recovered unknown constant. So if one derivative were done again
        on the recovered first function then its slope is indeterminate.
        Maybe this could explain an obtuse niggle in Quantum Mechanics re uncertainty
        & its progression through non-commutating operators FFS “as well” – eeks !

        Little niggles like this in all sorts of places besides mere calculus has me
        appreciating the dilemma this guy faced:-

        One wonders if this is an issue in any practical sense or if it were addressed what
        value it might be as a means to preserve information all the way back to recover the
        original function, perhaps somewhat like sqrt(1-) as for phase relationships in Electronics…

        Just musing ;-)

  8. David Hilbert and Richard Courant wrote a well-regarded calculus textbook, entitled “Calculus” (in two volumes).
    As to the relationship between the fundamental operations of calculus, differentiation and integration, they are inverse operations in the sense that DIFF [INT (F(x)] is usually the same function as INT [DIFF F(x)]. Usually, but not always. Continuous second partials is good enough and in practice is usually the case. If not, bounded variation of the absolute integrand is good enough. Lacking that, check your model because you are getting into some deep weeds. There is a chain of results of increasing precision and decreasing applicability which culminates in the necessary and sufficient conditions for differentiation and integration to commute as operators. (The final result was not obtained until 1938.)

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