# Why is calculus so widely taught? **URL:** \ **Category:** Factual Questions\ **Created:** [March 27, 2005, 5:35pm UTC](https://boards.straightdope.com/t/why-is-calculus-so-widely-taught/296472 "2005-03-27T17:35:21Z")\ **Posts on this page:** 20\ **Page:** 2
**Author:** ![wolf\_meister](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/wolf_meister/32/15202_2.png) [@wolf\_meister](https://boards.straightdope.com/u/wolf_meister)\ **Post date:** [March 28, 2005, 4:19am UTC](https://boards.straightdope.com/t/why-is-calculus-so-widely-taught/296472/21 "2005-03-28T04:19:00Z")
**klaatu** Yes, quite often you can get a _very_ good answer to a problem _without calculus_ and just using trial and error. I posted that link just to show that calculus is applicable in the real world. * * * Going a little off-topic here: Since you mentioned carpentry, quite often I get E-Mails from carpenters (and general “do-it-yourselfers”) who have found my calculators to be very helpful especially: [http://www.1728.com/trig.htm](http://www.1728.com/trig.htm) [http://www.1728.com/trig4.htm](http://www.1728.com/trig4.htm) and especially the [Pythagorean Theorem Calculator.](http://www.1728.com/pythgorn.htm) * * * ---
**Author:** ![bitwise](https://avatars.discourse-cdn.com/v4/letter/b/ea5d25/32.png) [@bitwise](https://boards.straightdope.com/u/bitwise)\ **Post date:** [March 28, 2005, 7:01am UTC](https://boards.straightdope.com/t/why-is-calculus-so-widely-taught/296472/22 "2005-03-28T07:01:02Z")
> [@NattoGuy](#): > > OK, here’s a dumb question, since we’ve got all the math people together… > > Can you suggest any online texts where I can re-learn math? Is it even possible to self-teach math? > > I went all the way up to AP calculus in high school and did fairly well, but never took any more math after that. Now I am interested in queue theory due some problems that need solving at work. But looking over the web sites, there is some notation that I’d swear I have never seen before. I thought we’d seen all the symbols by the time we’d reached calculus, if not all the concepts. I don’t know anything about queueing theory, but from the table of contents in the PDF I just downloaded, it seems to require some knowledge of probability. It’s not really something that is taught in a calculus class. ---
**Author:** ![Sleel](https://avatars.discourse-cdn.com/v4/letter/s/73ab20/32.png) [@Sleel](https://boards.straightdope.com/u/Sleel)\ **Post date:** [March 28, 2005, 7:36am UTC](https://boards.straightdope.com/t/why-is-calculus-so-widely-taught/296472/23 "2005-03-28T07:36:22Z")
Calculus was when it all started to come together for me. I’d studied algebra, trigonometry, geometry, and some basic statistics and really none of it seemed to have any use or reason (besides some trig) for being taught at all until my first real calculus class. Suddenly all that boring crap I’d learned could actually be **used** for something. “Cool, I can figure out the volume of this vase? I can figure out how much pressure there is at the bottom of this window in the Monterey Bay Aquarium? Sweet!” Many of the formulae we learned in chemistry or physics classes were derived from using calculus in the first place. I personally wish they’d taught us the why and the how instead of just giving us a formula to memorize. ---
**Author:** ![Eleusis](https://avatars.discourse-cdn.com/v4/letter/e/f4b2a3/32.png) [@Eleusis](https://boards.straightdope.com/u/Eleusis)\ **Post date:** [March 28, 2005, 8:38am UTC](https://boards.straightdope.com/t/why-is-calculus-so-widely-taught/296472/24 "2005-03-28T08:38:23Z")
Is there something wrong in that I passed college calculus but still don’t even know wtf it is? ---
**Author:** ![nivlac](https://avatars.discourse-cdn.com/v4/letter/n/3bc359/32.png) [@nivlac](https://boards.straightdope.com/u/nivlac)\ **Post date:** [March 28, 2005, 8:46am UTC](https://boards.straightdope.com/t/why-is-calculus-so-widely-taught/296472/25 "2005-03-28T08:46:23Z")
> [@NattoGuy](#): > > … Now I am interested in queue theory due some problems that need solving at work. But looking over the web sites, there is some notation that I’d swear I have never seen before. I thought we’d seen all the symbols by the time we’d reached calculus, if not all the concepts. So you want to learn queuing (sometimes spelled _queueing_) theory? Well, you’ve picked a tough subject. You’ll need more than calculus. For queuing theory, you’ll need a strong grounding in probability theory. I flip to page 1 of my old textbook on probability and stochastic processes, and what do I see? Derivatives and integrals. So the road to queuing is: basic calculus, advanced calculus, elementary probability, basic analysis, functional analysis, probability and stochastic processes, queuing. Of course, it’s possible to solve queuing problems without any knowledge of the underlying theory – you can just stick your numbers into an existing queuing model. But understanding the theory will help you understand the model, its applicability to your problem, and the results that are generated. Sure, you can teach yourself math. It’s easier, less painful, and quicker to have someone who already knows the subject explaining things to you and for answering questions. Part of learning math is by doing problems and the more advanced the math, the harder the problems. When you’re learning on your own, can you really stay motivated to do the problems when you really don’t have to? That’s what you need to ask yourself. ---
**Author:** ![Stranger\_On\_A\_Train](https://avatars.discourse-cdn.com/v4/letter/s/13edae/32.png) [@Stranger\_On\_A\_Train](https://boards.straightdope.com/u/Stranger_On_A_Train)\ **Post date:** [March 28, 2005, 9:31am UTC](https://boards.straightdope.com/t/why-is-calculus-so-widely-taught/296472/26 "2005-03-28T09:31:53Z")
> [@Eleusis](#): > > Is there something wrong in that I passed college calculus but still don’t even know wtf it is? Yeah. It means someone went thorugh the routine of showing you how to do the trained-monkey operations of integrating and differentiating polynomial and trigonometric functions but never explained to you their applications and utitlity. In principle, calculus is quite simple. Any ten year old of average intelligence can understand the basic principles behind integral calculus _if it is presented correctly and with application to the real world_. The fact that most calculus texts and classes present it in the abstract, without reference to application, makes it seem esoteric. Math is simple, really. (Well, except for complexity theory. _snark_) Interpersonal relationships? Now that’s genuis-level stuff. Stranger ---
**Author:** ![Chefguy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chefguy/32/138_2.png) [@Chefguy](https://boards.straightdope.com/u/Chefguy)\ **Post date:** [March 28, 2005, 2:39pm UTC](https://boards.straightdope.com/t/why-is-calculus-so-widely-taught/296472/27 "2005-03-28T14:39:40Z")
I remember taking a dynamics class wherein the teacher asked the class if we had all taken differential equations. When we indicated that we had not, he told us that the class was going to be long and difficult. I struggled through three semesters of calculus without really ‘getting’ it. Then I took differential equations, which seemed to simplify everything and wondered why calculus was even necessary. Is it that these don’t apply to the same physics or accomplish the same things? ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [March 28, 2005, 2:51pm UTC](https://boards.straightdope.com/t/why-is-calculus-so-widely-taught/296472/28 "2005-03-28T14:51:35Z")
> [@NattoGuy](#): > > I thought we’d seen all the symbols by the time we’d reached calculus, if not all the concepts. This is an understandable mistake given the awe/fear this culture gives the calculus. Calculus is the beginning of mathematics, not the end. As the reformers have been saying for decades now, “Calculus should be a pump, not a filter”; it should motivate and spur the student through higher math rather than act to hold back those who “can’t hack it”. There are whole fields you’ve never even touched by the time you first learn the calculus, and you’ve got at least two more passes at (essentially) the calculus itself before getting a master’s degree, let alone the forefront of mathematics that is the almighty Ph.D. Truly there are more things in mathematical heaven and earth. ---
**Author:** ![Stranger\_On\_A\_Train](https://avatars.discourse-cdn.com/v4/letter/s/13edae/32.png) [@Stranger\_On\_A\_Train](https://boards.straightdope.com/u/Stranger_On_A_Train)\ **Post date:** [March 28, 2005, 3:16pm UTC](https://boards.straightdope.com/t/why-is-calculus-so-widely-taught/296472/29 "2005-03-28T15:16:11Z")
> [@Chefguy](#): > > I struggled through three semesters of calculus without really ‘getting’ it. Then I took differential equations, which seemed to simplify everything and wondered why calculus was even necessary. Is it that these don’t apply to the same physics or accomplish the same things? Er, the class “Differential Equations” is just an extension of integral calculus pertaining to the solution of, well, differential equations. You use plenty of calculus in diffy-q, of course, but there are particular solutions to various types of time or position (or frequency, or whatever) varying processes to which one must apply the appropriate form. The first semester if diffy-q is usually involving in solving first- and some second-order **ordinary differential equations** , with some basic coverage of certain forms of higher order equations, matrix solution of linear systems of equations, and the Laplace Transform (very useful in circuits and controls.) There should be a bit of coverage of numerical methods as well (Euler’s, Taylor’s, and Runge-Kutta, at least) though in practice most instructors seem to pass over this lightly or skip it altogether, which is a real shame because in practice most solutions to diffy-qs in real life involve open form approximations. So, it isn’t that they don’t apply to the “same physics” as the calculus you learned in semesters 1, 2, and 3, but rather they are ways of classifying and codifying the forms into which those phenomena fit. Then integrating the function over time (or path, or whatever) gives you a summation of its property (energy loss, say, or number of bunnies crushed to death while bulldozing Watership Down) over that domain. It is sometimes possible to “solve” a differential equation without directly integrating (using Laplace’s Method to solve for the static deflection of a beam, for instance) but the calculus is still taking place within the transform; it’s just turning a problem of integration into one of transform, and solving for initial conditions. There is an assumption that you know the initial conditions of the derivatives of the function instead of just the range over which it is to be integrated. You still need the tools of calculus in order to derive and understand the Transform. Looking through my diffy-q book (_A First Course in Differential Equations with Applications_ by Dennis G. Zill) I can see what an terribly written book it is. The chapter on the Laplace transform, for instance, gives virtually no explaination of what the transform can be used _for_. Oh, there are a couple of examples (a simle RCL circuit and a beam deflection problem) but at no time does the author sit down and have a simple, round-the-fire conversation with the reader about what the tranform does and how trippy it is that it lets you turn difficult to integrate trig and logrithmic functions into simple algebra on the frequency domain. Although I did fine in the class (an A, which was a relief after my unexpectedly poor performance in Calc III) I didn’t really _get_ differential equations until my linear systems (ODEs applied to mechanical and electrical systems). I kicked arse in classical controls and vibrations, though. Stranger ---
**Author:** ![Stranger\_On\_A\_Train](https://avatars.discourse-cdn.com/v4/letter/s/13edae/32.png) [@Stranger\_On\_A\_Train](https://boards.straightdope.com/u/Stranger_On_A_Train)\ **Post date:** [March 28, 2005, 3:24pm UTC](https://boards.straightdope.com/t/why-is-calculus-so-widely-taught/296472/30 "2005-03-28T15:24:47Z")
> [@Stranger On A Train](#): > > There is an **assumption** that you know the initial conditions of the derivatives of the function… That should be _requirement_, not assumption. :smack: And actually, you need to know some boundry conditions, but they need not necessarily be initial conditions as long as the function is of order two or higher. Stranger ---
**Author:** ![F.U.Shakespeare](https://avatars.discourse-cdn.com/v4/letter/f/aca169/32.png) [@F.U.Shakespeare](https://boards.straightdope.com/u/F.U.Shakespeare)\ **Post date:** [March 28, 2005, 4:01pm UTC](https://boards.straightdope.com/t/why-is-calculus-so-widely-taught/296472/31 "2005-03-28T16:01:00Z")
One of my professors had a sign on his desk that read: _An engineer can do for $2 what any fool can do for $20_ IOW, the theoretical approach isn’t always best. > [@wolf\_meister](#): > > Yes, quite often **you can get a _very_ good answer to a problem _without calculus_ and just using trial and error.** I posted that link just to show that calculus is applicable in the real world. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [March 28, 2005, 9:38pm UTC](https://boards.straightdope.com/t/why-is-calculus-so-widely-taught/296472/32 "2005-03-28T21:38:09Z")
Quoth **Klaatu** : > [@](#): > > Hmmm, I don’t know much about calculus, but as a carpenter/remodeler, I deal with this every day. I would simply figure the angles with my handy tools (squares, angle finders and such), measure them out, and figure it much more easily than using equations. I would then compare the cost/mile prices with the distance. I don’t know about that… You’d presumably have to do the calculations for a few different angles, to pick which one had the lowest price, and then maybe interpolate to find an angle that’s even better. That’s pretty slow. I could solve that same problem with an application of Snell’s Law, and get the exact answer probably just as quickly as you could get an approximate answer. Using the same mathematical background, I could also solve roof-pitch problems or many other practical problems without needing to consult tables in a book. And even if you’re going to use tables in a book, someone had to write that book in the first place. Meanwhile, I would say that it’s impossible to teach physics without calculus. If your students are not already familiar with calculus, then you’re going to need to introduce at least some of the notions of calculus in your physics class. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [March 28, 2005, 10:26pm UTC](https://boards.straightdope.com/t/why-is-calculus-so-widely-taught/296472/33 "2005-03-28T22:26:40Z")
> [@F. U. Shakespeare](#): > > One of my professors had a sign on his desk that read: > > _An engineer can do for $2 what any fool can do for $20_ > > IOW, the theoretical approach isn’t always best. Where do you think the engineers are getting their tricks from? ---
**Author:** ![Hyperelastic](https://avatars.discourse-cdn.com/v4/letter/h/a3d4f5/32.png) [@Hyperelastic](https://boards.straightdope.com/u/Hyperelastic)\ **Post date:** [March 29, 2005, 2:23am UTC](https://boards.straightdope.com/t/why-is-calculus-so-widely-taught/296472/34 "2005-03-29T02:23:16Z")
I took a macroeconomics course as an elective when I was an undergaduate. I was an engineering major and had taken plenty of calculus. The basic concepts in the economics course, like the market-clearing price, maximization of profit, marginal utility and the like were all essentially very simple calculus problems. But the course was for nontechnical people and was presented calculus-free. Thus, while most of the class was struggling with concepts like extrema and how the slope of a curve can be different at every point, never mind the economic applications of these ideas, the engineering majors were getting perfect scores on every test without even cracking the textbook. The odd thing is, I know most of the people in the class were also required to take calculus, albeit a different course than the one the engineers took. I don’t know what they were teaching in that course, but it sure couldn’t have been calculus. ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [March 29, 2005, 2:37am UTC](https://boards.straightdope.com/t/why-is-calculus-so-widely-taught/296472/35 "2005-03-29T02:37:15Z")
> [@Hyperelastic](#): > > I took a macroeconomics course as an elective when I was an undergaduate. I was an engineering major and had taken plenty of calculus. The basic concepts in the economics course, like the market-clearing price, maximization of profit, marginal utility and the like were all essentially very simple calculus problems. But the course was for nontechnical people and was presented calculus-free. Thus, while most of the class was struggling with concepts like extrema and how the slope of a curve can be different at every point, never mind the economic applications of these ideas, the engineering majors were getting perfect scores on every test without even cracking the textbook. The odd thing is, I know most of the people in the class were also required to take calculus, albeit a different course than the one the engineers took. I don’t know what they were teaching in that course, but it sure couldn’t have been calculus. There’s a commonly-offered course at most schools called “business calc”, which is intended to be just what you need to work with the tools and not have any clue why they work. Needless to say I recommend as strongly as I can against anyone anywhere ever taking this course. ---
**Author:** ![Chefguy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chefguy/32/138_2.png) [@Chefguy](https://boards.straightdope.com/u/Chefguy)\ **Post date:** [March 29, 2005, 3:17pm UTC](https://boards.straightdope.com/t/why-is-calculus-so-widely-taught/296472/36 "2005-03-29T15:17:07Z")
> [@Stranger On A Train](#): > > Er, the class “Differential Equations” is just an extension of integral calculus pertaining to the solution of, well, differential equations. You use plenty of calculus in diffy-q, of course, but there are particular solutions to various types of time or position (or frequency, or whatever) varying processes to which one must apply the appropriate form. The first semester if diffy-q is usually involving in solving first- and some second-order **ordinary differential equations** , with some basic coverage of certain forms of higher order equations, matrix solution of linear systems of equations, and the Laplace Transform (very useful in circuits and controls.) There should be a bit of coverage of numerical methods as well (Euler’s, Taylor’s, and Runge-Kutta, at least) though in practice most instructors seem to pass over this lightly or skip it altogether, which is a real shame because in practice most solutions to diffy-qs in real life involve open form approximations. > > So, it isn’t that they don’t apply to the “same physics” as the calculus you learned in semesters 1, 2, and 3, but rather they are ways of classifying and codifying the forms into which those phenomena fit. Then integrating the function over time (or path, or whatever) gives you a summation of its property (energy loss, say, or number of bunnies crushed to death while bulldozing Watership Down) over that domain. It is sometimes possible to “solve” a differential equation without directly integrating (using Laplace’s Method to solve for the static deflection of a beam, for instance) but the calculus is still taking place within the transform; it’s just turning a problem of integration into one of transform, and solving for initial conditions. There is an assumption that you know the initial conditions of the derivatives of the function instead of just the range over which it is to be integrated. You still need the tools of calculus in order to derive and understand the Transform. > > Looking through my diffy-q book (_A First Course in Differential Equations with Applications_ by Dennis G. Zill) I can see what an terribly written book it is. The chapter on the Laplace transform, for instance, gives virtually no explaination of what the transform can be used _for_. Oh, there are a couple of examples (a simle RCL circuit and a beam deflection problem) but at no time does the author sit down and have a simple, round-the-fire conversation with the reader about what the tranform does and how trippy it is that it lets you turn difficult to integrate trig and logrithmic functions into simple algebra on the frequency domain. Although I did fine in the class (an A, which was a relief after my unexpectedly poor performance in Calc III) I didn’t really _get_ differential equations until my linear systems (ODEs applied to mechanical and electrical systems). I kicked arse in classical controls and vibrations, though. > > Stranger It’s been 30 years since I took the courses, so I don’t pretend to understand all of your explanation. Diffy Q was like a light coming on for me, whereas calculus III was an unholy bitch that I had to beat into submission. It was such a revelation, in fact, that I wondered why it isn’t taught much earlier, like say in high school, since it seemed to explain (or possibly expanded) the reasons for other maths. Maybe it was something similar to when, after taking two years of Latin, English grammar was suddenly something that made sense to me. Sort of like a politician who stops speaking in pol-speak and starts talking like a normal human. ---
**Author:** ![F.U.Shakespeare](https://avatars.discourse-cdn.com/v4/letter/f/aca169/32.png) [@F.U.Shakespeare](https://boards.straightdope.com/u/F.U.Shakespeare)\ **Post date:** [March 29, 2005, 3:29pm UTC](https://boards.straightdope.com/t/why-is-calculus-so-widely-taught/296472/37 "2005-03-29T15:29:37Z")
In my experience: sometimes, from a chalkboard full of equations; other times, by trial and error; usually, a combination of both. > [@ultrafilter](#): > > Where do you think the engineers are getting their tricks from? ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [March 29, 2005, 4:55pm UTC](https://boards.straightdope.com/t/why-is-calculus-so-widely-taught/296472/38 "2005-03-29T16:55:17Z")
> [@Chefguy](#): > > It’s been 30 years since I took the courses, so I don’t pretend to understand all of your explanation. Diffy Q was like a light coming on for me, whereas calculus III was an unholy bitch that I had to beat into submission. It was such a revelation, in fact, that I wondered why it isn’t taught much earlier, like say in high school, since it seemed to explain (or possibly expanded) the reasons for other maths. Maybe it was something similar to when, after taking two years of Latin, English grammar was suddenly something that made sense to me. Sort of like a politician who stops speaking in pol-speak and starts talking like a normal human. Differential equations (and let’s drop this “diffy-q” sophomorism) are, as stated, an application of the calculus. Now, I can easily see you having had a teacher for ODEs who came down heavy on the why, while your teacher for Calc 3 didn’t emphasize the geometry. I wasn’t there, though, so I can’t really speak to that. Why is the first course in ODEs as such placed where it is? Because it needs (well, _should_ need) linear algebra first. Linear algebra and ODEs both require a certain amount of sophistication and exposure to concepts that come with the experience in a calculus sequence. In all honesty, I could see doing both without multivariate calculus (but not PDEs!). What you definitely need the second semester of calculus, especially the advanced techniques of integration and the work on series of functions. Of course if you’re just looking for a bag of tricks rather than an understanding of why they work, you really don’t get the point of mathematics. ---
**Author:** ![Chefguy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chefguy/32/138_2.png) [@Chefguy](https://boards.straightdope.com/u/Chefguy)\ **Post date:** [March 29, 2005, 10:35pm UTC](https://boards.straightdope.com/t/why-is-calculus-so-widely-taught/296472/39 "2005-03-29T22:35:43Z")
> [@Mathochist](#): > > Differential equations (and let’s drop this “diffy-q” sophomorism) are, as stated, an application of the calculus. Now, I can easily see you having had a teacher for ODEs who came down heavy on the why, while your teacher for Calc 3 didn’t emphasize the geometry. I wasn’t there, though, so I can’t really speak to that. > > Why is the first course in ODEs as such placed where it is? Because it needs (well, _should_ need) linear algebra first. Linear algebra and ODEs both require a certain amount of sophistication and exposure to concepts that come with the experience in a calculus sequence. > > In all honesty, I could see doing both without multivariate calculus (but not PDEs!). What you definitely need the second semester of calculus, especially the advanced techniques of integration and the work on series of functions. > > Of course if you’re just looking for a bag of tricks rather than an understanding of why they work, you really don’t get the point of mathematics. DIFFY-Q! ::horses neighing:: ---
**Author:** ![BobLibDem](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/boblibdem/32/3149_2.png) [@BobLibDem](https://boards.straightdope.com/u/BobLibDem)\ **Post date:** [March 29, 2005, 10:42pm UTC](https://boards.straightdope.com/t/why-is-calculus-so-widely-taught/296472/40 "2005-03-29T22:42:43Z")
I believe that if you do not understand calculus you cannot adequately understand physics. The best example I know is that principle that planets sweep equal areas in their orbits over equal times. Without knowing calculus, this seems like mumbo jumbo. Knowing calculus, it is an easily seen consequence of the principle of the conservation of angular momentum. Another benefit is that rather than memorize the equations of motion, you can simply derive them. [Previous page](https://boards.straightdope.com/t/why-is-calculus-so-widely-taught/296472.md?page=1) [Next page](https://boards.straightdope.com/t/why-is-calculus-so-widely-taught/296472.md?page=3)