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I love math and science. In fact paid for $1/2$ my college tuition by tutoring algebra and trigonometry. But when it came to calculus, I became blocked. I understand the concepts of speed and rate of change, but when I start seeing all the symbols like $f'(x)$ and $dx$, my mind goes all fuzzy.

My learning style is that if I can picture something and comprehend it in real life, then I get it in symbols. Algebra, geometry, and trig are all easy enough to picture. But what does it look like in real life to 'take the derivative' or the integral of something? Does it look like a sphere becoming a circle?

Does anyone else's brain work like this? How did you 'get' calculus?

Joaquin Liniado
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Bertha
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  • Well it sounds like you were doing well at math before, why should your mind go fuzzy from the function symbol for example? – Quality May 14 '15 at 21:50
  • Taking the rate of change is like measuring the speed of a car at close instants, then continuing to do so with closer instants until you can see a limiting value of the speed and thus the derivative. As for integrating, it's just like calculating the distance covered if you know the speed at every instant and the duration of motion. – Hasan Saad May 14 '15 at 21:50
  • I have read many posts talking about "algebra people" and "analysis people". It appears that people usually have their own specialization. Some people are good at analysis, and some people are good at algebra, and few are good at both. So you may want to take courses like linear algebra, abstract algebra, algebraic topology, (number theory maybe?) to see whether you are truly the "algebraic person". But honestly speaking, having trouble with calculus in college doesn't sounds impressive. If you are neither good at analysis nor algebra, probably the bad news is that you are not the math guy. – MonkeyKing May 14 '15 at 22:08
  • Or perhaps you mean you are a high school student selflearning college level math? My English is bad and I don't really understand the first sentence. :( Also additional comment: it appears that people will suddenly realized things they learned before after learning advanced material, although they struggled a lot before. It is also possible that in the future you will suddenly understand. – MonkeyKing May 14 '15 at 22:10
  • For similar reasons I have enjoyed reading "CALCULUS An Intuitive Physical Approach" by Morris Kline – Stephen P May 14 '15 at 22:12
  • maybe community wiki? :-) – Ant May 14 '15 at 22:20
  • Thank you all for the thoughtful replies. Hasan, your answer is the kind of constructive example I am looking for. It has given me something to digest. Stephen, I will certainly look up that book. Thanks for the suggestion. Quality, to answer your question, put differently, when I look at the equation for a line, I can easily picture the line in my head, same with a parabola when I see the equation. Same with trigonometry. Because of this, I thought I was math smart. Then I hit calculus, and the equations/symbols don't translate to anything in my mind. They just go all fuzzy. Nothing is there. – Bertha May 14 '15 at 22:46
  • First step, is understand the reason and motivation for limits. I would say that this is a concept not covered very well in most intro calc classes but is a huge part of calc – Kamster May 14 '15 at 22:53
  • A lot of Calc I would say is also a mathematical beautiful way of approximating things with things you know well. approximate a function with polynomials (taylor series expansion), approximate rate of change with line (derivative), approximate area of curve with area of squares (integration). the beautiful things about these approximation is that they are close infinitesimally – Kamster May 14 '15 at 22:56
  • I imagine the derivative as the slope of the graph, or the rate of change, or the instantaneous speed. – Akiva Weinberger May 15 '15 at 03:02
  • Calculus starts making sense when you realize that $$\lim_{\Delta x\to 0}\frac{\Delta y}{\Delta x} = \frac{dy}{dx}$$ and $$\lim_{\Delta x\to 0} \sum f(x)\Delta x = \int f(x)dx$$ – John Joy May 15 '15 at 20:53
  • Reading Calculus Made Easy by Silvanus P. Thompson might just solve your problem. – Joshua Meyers Oct 12 '17 at 16:13

4 Answers4

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Im going to share with you my (short) experience regarding mathematics. It will not be a direct answer to your question but maybe it will help you! So when I was in school, I loved maths. I was actually good at them. Trigonometry, algebra and even pre-calculus. I even thought that was all what maths were about. However, when 4 years ago, I entered the university, I started realizing maths wasn't even about numbers (in many cases). I suddenly found my self dealing with concepts that were absolutely imposible to picture them somehow. Not only that, suddenly I was solving a problem that I would have never thought that it could even be called maths. This made me feel seriously mixed up. When I decided to study mathematics, I am sure I wasn't $picturing$ my self doing what I am doing today. However, I am still studying, and every time, more eagerly. And that's the important thing I want to share with you. Don't think about maths as something you can picture in reality. That will only stop you. Actually, the great thing about maths, which makes it $ perfect $ is that it doesn't have to even explain reality!

I like to think of maths as this indescriptible magic stuff that can make you love it or hate it. And if you do love it, just learn from it, don't look for any reality in it. You have physics for that!

Joaquin Liniado
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  • And yet...(http://it.wikipedia.org/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences) :-) However I agree with you, intuition is important but don't fixate on trying to visualize it first. You'll start visualizing it after you become acquanted with the concept :) – Ant May 14 '15 at 22:21
  • Thanks for the thoughtful response Joaquin. For clarification, I do have a post-graduate degree in Chemistry, for which I had to take enough math classes to get almost a minor in math. With your insight, and that from 'Ant', it does seem to make sense that while algebra and trig are easy to draw or visualize, Calculus might be the transition where it gets much harder to visualize the reality being described by the symbols and numbers. – Bertha May 14 '15 at 22:41
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If you want to "picture" something in calculus, here's one example:

  • The size of a boundary times the rate at which the boundary moves, equals the rate of change of size of the bounded region.

(Privately I think of this as "the boundary rule". Does anyone know a better name for it?)

Examples:

  • A region is enclosed by a sphere. The change in the radius is how far the boundary moves. Hence the rate at which the boundary moves is the rate of change of the radius. The size of the boundary is the surface area of the sphere. Multiply those to get the rate at which the volume changes. (This explains why the formula for the surface area of a sphere as a function of the radius is the derivative with respect to the radius, of the volume of the sphere.)

  • An $n$-cube has sides of length $x$, one corner right at the origin, and $n$ edges on the coordinate axes. As $x$ changes, $n$ of the faces move, and they move at the rate at which $x$ changes. Each of them has size $x^{n-1}$. So the rate of change of volume, i.e. the rate of change of $x^n$, is $$\underbrace{x^{n-1}+\cdots+x^{n-1}}_{n\text{ terms}}$$ times the rate at which $x$ changes.

Michael Hardy
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I can understand why your mind goes fuzzy when you see things like $x^2 dx$ and $\frac{dy}{dx}$, because these are tricky concepts that are difficult to formalize and frequently used incorrectly, and their geometric meaning is pretty complicated. In fact, even after 5 or so years of university mathematics, I honestly still don't get the geometric meaning of $\frac{dy}{dx}$. However, the meaning of the notation $f'(x)$ should be perfectly clear, and you should be able to visualize it with only a little bit of thought. If you can't do so at the moment, perhaps the problem is that maybe you are missing is the concept of a function. I won't try to explain it here, but I suggest searching online for an explanation of the function concept. Some key words:

  • Set, function
  • Domain, codomain; make sure you know what the notation $f : X \rightarrow Y$ means, where $X$ and $Y$ denote sets.
  • Injective (one-to-one), surjective (onto)
  • Higher-order function
  • Lambda abstraction

Another important point is that, given a function $f : \mathbb{R} \rightarrow \mathbb{R},$

  • $f$ can usually be visualized as a curve in the plane.
  • the notation $f(x)$ ("$f$ evaluated at $x$") can be visualized as the height of $f$ at $x$.

Make sure you understand this.

Once you've got these concepts, calculus shouldn't make your mind go numb anymore. A simple way to understand the notation $f'(x)$ is that it really means the slope of $f$ at $x$. In other words, its the derivative of $f$, evaluated at $x$. Try to think of derivatives as higher-order functions; the notation $f'(x)$ really means something more like $$(D(f))(x),$$ where $D$ is the derivative function (which is higher-order). The expression $(D(f))(x)$ means: start with the function $f$, then apply $D$ to it, thereby obtaining the corresponding "slope function" $D(f)$, and then evaluate this new function $D(f)$ at $x$; you can visualize this as the height of the slope function $D(f)$ at $x$.

goblin GONE
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  • I do think you've hit upon a good point, and that's that calculus does invite one to take another step up the the abstraction ladder. To put one of your points another way, differentiation and integration (as long as they're not being evaluated/definite) are indeed "higher-order" functions: They eat functions and spit out other functions, much like "regular" functions eat numbers and spit out other numbers. – pjs36 May 14 '15 at 23:33
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The concepts you mention were historically derived from geometrical and physical problems, like curves and areas, motion etc.

You can get them explained very close to everyday conception. And we might try this here if you break it down in single questions. However the real power of mathematics is to abstract, leaving out everything except the essential properties. This allows applications to a broader range.

E.g. you can work out geometry in 10 dimensions, which comes in handy if you have an optimization problem in 10 sorts of different products.

Also everyday conception can be misleading, that is why later calculus got more abstract methods to achieve more rigour.

So you might use conception to get an initial grasp of the subject, but the step to abstraction can not be fully avoided if you want to do modern mathematics.

mvw
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