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I'm developing some instructional material for a Calculus 1 class and I wanted to know from experience for yourself, tutoring others, and/or helping people on this site where is the most difficulty in Calculus?

If you had any good methods of helping people that would be very helpful.

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Community wiki I suppose. –  J. M. Sep 23 '10 at 9:02
    
Added "soft-question" tag. –  Arturo Magidin Sep 23 '10 at 14:38
    
Who says I ever understood it? You never understand calculus,you just get used to it........LOL I'm kidding,but it speaks of a deep truth:Most of us spend the next 10 years after our first calculus class really trying to understand the subject. In fact,I'd go so far as to say that no one really understands calculus until they teach a real analysis course. –  Mathemagician1234 Mar 17 '12 at 7:57
    
I always found optimization VERY difficult. There's a lot of nit-picky stuff that you have to keep in mind. –  user39302 Sep 3 '12 at 3:19
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15 Answers

The hardest thing for me was to understand what is meant when someone writes $\mathrm d x$.

I still don't know...

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+1. Yes, this bothered me so much at the time. I mean, you have dx appearing in integration, and then dy/dx as derivatives, and some texts also mention "differentials" df = f'(x) dx (so df is a function of x and dx, and dx is just a real number). –  Jesse Madnick Sep 23 '10 at 9:38
    
Every year without fail, this is the hardest idea for my students to understand, especially in the context of anti-differentiation (as opposed to integration). –  Alex Basson Sep 23 '10 at 11:07
    
There is a really superb explanation of this in A. Ya. Khinchin's Eight lectures on mathematical analysis, in which he thoroughly explodes the notion that it has no well-defined meaning. –  MJD Jun 2 '12 at 2:33
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$dx$ is just a piece of notation, and one never needs to use this notation. For the integral you can just write $\int_a^b f$, as is done in Spivak's Calculus on Manifolds, a book that is certainly very well respected. For the derivative you can just use $f'(x)$. $dx$ is not some kind of "infinitely small number", whatever that is supposed to mean. (More advanced math subjects may give a precise meaning to $dx$, but that is not something one needs to worry about when learning calculus.) –  littleO Nov 9 '12 at 6:33
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I really struggled with the $\epsilon-\delta$ definition of limits, especially for non-linear functions. This was also my first exposure to proof, as in: Prove that $$\lim_{x \to 2} (x^2 + 3) = 7$$ and I had a hard time with it at first. To be clear, computing these limits was no problem, but using the definition to prove they were correct really confused me.

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It is probably more difficult in higher ages; my first contact with epsilon/delta limits and integration was in middle school, at the age of 14... it was slightly confusing then, but it got easier when I took calculus in high school. It was a breeze finally at the university. –  Paxinum Sep 23 '10 at 13:34
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The problem with ε/δ proofs is that initially, almost nobody teaches why one should care --- that is to say, what it means. "Formalism" is not a good enough motivation, much as I like it now. This is unfortunate, as perhaps the same one-minute challenge-response skit, performed maybe three times in the course of a semester, would solidify the concept in people's heads. –  Niel de Beaudrap Sep 23 '10 at 13:37
    
Somewhat like Niel, until I read Bressoud's treatment of $\epsilon-\delta$ as a "game", I was scratching my head for weeks at what the fuss was all about. –  J. M. Sep 23 '10 at 14:40
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@Paxinum Where did you go to school that you were learning epsilon and delta style limits in MIDDLE SCHOOL?!? Unless you were a remarkable prodigy,I can't imagine any middle school in the US teaching you that! –  Mathemagician1234 Mar 17 '12 at 7:54
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@Mathemagician1234 I was very lucky having access to a teacher willing to teach me, so I had 1 hr/week with real mathematics. Just a teacher being happy to having a student willing to learn some more advanced stuff. –  Paxinum Mar 17 '12 at 23:13
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At the school I was taught to look at the derivative as the instantaneous rate of change and that fit well with applications in physics. But later, when I was learning Economics in college, I had to learn to look at the derivative as the best linear (affine) approximation, and a differentiable function as a function which had 'good' linear approximations. That is also the intuition that generalizes to many variables. I wish it had been discussed in my early calculus classes.

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+1, for me, I gained a better geometric view of derivatives (and Taylor expansions in general) in the context of "polynomials that greatly resemble the function in the vicinity of the expansion point". –  J. M. Sep 23 '10 at 15:04
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The inverse relationship between differentiation and integration, and understanding it from the graph.

And I still have not understood that part of calculus at all! :(

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There's a very nice picture you can draw to show why $\frac{d}{dx} \int_a^x f(s) \, ds = f(x)$. Also, keep in mind that $f'(t)$ is the instantaneous rate of change of $f$ at time $t$, and $f'(t) dt$ is a tiny change in $f$ during a very short period of time, and by adding up all the tiny changes you get the total change: $\int_a^b f'(t) \, dt = f(b) - f(a)$. –  littleO Nov 9 '12 at 6:42
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The wikipedia article on Fundamental Theorem of Calculus has a nice picture explaining the inverse relationship between differentiation and integration, in terms of a graph. See the "Geometric intuition" section. –  littleO Nov 11 '12 at 23:32
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I think the entire concept behind integration is hard to grasp for students who are not familiar with analysis. They tend to think of it only as the "inverse operation of derivatives", which is quite restrisctive, in my opinion.

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By far the hardest thing for me was notation for partial derivatives. Having never been shown just how to actually interpret the symbols, I had great difficulty parsing what was actually meant by various expressions.

Eventually, I abandoned the more common notations entirely, and fell back on other notation we had been shown, like $f_1(x,y,x)$, which means "the derivative of $f$ in the first argument, evaluated at $(x,y,x)$".

These days, I have a deeper understanding of just what my problem was. Roughly speaking, $dx$ makes sense on its own, but $\partial/\partial x$ does not; the latter depends not on $x$, but on a curve $x$ is being viewed as a parameter for. (e.g. it suffices to view $x$ as a component in a coordinate system)

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I've always been fond of the notation Spivak uses for partial derivatives in Calculus on Manifolds: $D_i f(x)$ rather than $\frac{\partial f(x)}{\partial x_i}$. –  littleO Nov 9 '12 at 6:38
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Calculus was hard for me until I learned how to visualize things. Learning calculus only by writing symbols and solving problems with many $\varepsilon$'s will not make anyone understand it. If you learn to visualize all the basic concepts as limit, derivative, integration, etc. then the symbolic part is a lot easier.

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I just finished a summer Calculus course, and here is what I found most challenging,

  • I did have a difficult time with the $dx$ notation. It seemed to me that most times it worked like a "operator" as in "the derivative of...". Yet, at other times, like when solving a "Related Rates" problem, it seemed to behave as a variable. I am sure with further study I would be able to sort this out, but at the time it seemed to be a big source of confusion.

  • The other big thing I found challenging was how to properly work with "combined" rules--such as when you combine, say, the chain rule with the multiplication rule, inverse functions, etc. With simple functions, I found most of the rules of differentiation quite easy to apply, but was often unsure of the correct sequence when you have to combine rules.

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The logic behind the $\mathbb {\epsilon}$ - $\mathbb { \delta}$ definition.

This might have been mentioned in an answer above but since you are a teacher (or are instructing one) I think I must express some frustration I have had with your standard freshman year Calculus course.

My teacher - and many others I've heard - take the $\epsilon$ - $\delta$ definition of limits for granted. They just get away with repeating the statement "For any given $\epsilon \gt 0 $ $ \;\;\; \exists \delta \gt 0$ such that $|f(x) - L| \lt \epsilon $ whenever $|x - a| \lt \delta$". The students in my country (Sri Lanka) get entrance to university based on a very competitive final paper in school. Almost all of them are very capable of strong and mind-wrecking computations. But almost all of them have difficulty understanding limits in their first semester, as a result hate calculus and then despise pure mathematics in general. And I blame the teaching for this trend. The tutor fails to instill on the student the logic behind the definition. Not many know the fact that you are required to prove the existence of $\delta$ and not just an implication $|x - a| \lt \delta \implies |f(x) - L| \lt \epsilon $. It has been a couple of months since I joined this site and I repeatedly come across questions posted with similar dilemmas and all of them have flimsy logical foundation. And that is the issue.

So my suggestion here is a better introduction to the logic behind the introductory calculus courses. One professor I know starts off by asking students to negate statements like "Every girl is pretty" and "Some parents are kind" which I think is an excellent approach.

I found it very difficult to grasp the sense behind the limit statement and took me long hours and several books to get a hold of it. And I think this can and should be avoided. Yes a student should work out problems on his own and do lot of work on his own. But the foundation should be laid. Solidly. And I think the teacher is responsible for that.

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The area formula, namely, how could

$$\sum_{i=1}^{\infty} y_i \delta x_i=\int y dx$$

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Being more algebraically minded, I found it incredibly hard to follow all the techniques of integration, because rather soon those tend to become either very hand-wavy or very technical. I think I finished my degree without ever "calculating" a concrete non-trivial integral, such as doing integration by partial fractions or

I was just too scared about the "intuitive" notion with which various "dx", which at that point are merely a meaningless symbol, suddenly get replaced by some dy dx/dy - at that time, I never was able to assure myself that that I could make that rigorous.

If this is not clear, I found a quick example on wikipedia of something that I find scary even today; I quote:

"Integrating by this substitution: $cos(x) dx = d sin(x) $".

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I took differential calculus twice in two different schools. Not until years later did I realize that I had not known what a function is and that differential calculus is the study of one particular index of a point property of a function that produces another function of the same independent variable, and what the property is, and what index is used, and why, and that the derivative of a function is the result of an operation on a function called differentiation, and that the role of the limit is simply to carry out the operation and has nothing whatever to do with the basic idea. Defining the derivative as a limit completely obfuscated what it was really about. The insistence that mathematics is abstract and axiomatic buries all of the simple intriguing ideas. No wonder John von Neumann said "In mathematics you don't understand things. You just get used to them."

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Partial Differentiation using the third law

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Hardest aspect for me was when I took an advanced mathematics course at high school when I was 15 and was introduced to the concept of $ i=\sqrt{-1}$. I felt pretty cool having become comfortable with basic calculus at a relatively young age (compared with everyone I knew) only to become absolutely befuddled with imaginary numbers. To me it was a notion so detached from reality that it wasn't even slightly tangible.

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For me,the hardest part of elementary calculus was infinite series and the idea of convergence. I learned it in an accelerated summer course taught by Elliott Mendelson and I remember going insane trying to absorb all the basic tests in one feverish night on vacation with my family in the Catskills.

The main reason infinite series was so difficult was because you can't really understand how they work-indeed, the very concepts involved-without a rigorous formulation of both real numbers and limits. Queens College was-and still sadly is,from what I hear-determined to use a pencil-pushing course with Stewart as the text.Of COURSE infinite series and sequences are going to be a garbled mess if you try to use hand waving to explain it!

In retrospect,I feel bad for Elliott-he became visibly frustrated at times trying to teach it to us via "handwaving" and endless sample calculations.I didn't understand at the time why he was frustrated.Of course,I realize now how difficult what he was trying to do was-especially in a full semester course that was crammed into 5 weeks in the dead of summer!

This is why unless I can teach it with some rigor, I may just skip it entirely when I teach calculus the first time.

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Uh-why was this answer removed?The question itself is rather subjective and personal,so I gave a personal and subjective answer. I didn't think it was any more so then any of the other responses.This site-like MO-can be very capricious sometimes. –  Mathemagician1234 Mar 25 '12 at 0:15
    
I see now it hasn't been removed-just downvoted to death. Why is still a complete mystery to me-unless my fan club's been working overtime to spite me like children again. –  Mathemagician1234 Mar 25 '12 at 3:10
    
Infinite series is part of the second semester of calculus at all of the (five) universities where I have studied. Arguably, Calc II is the hardest math class that a student will have had by that point. –  The Chaz 2.0 Apr 9 '12 at 16:40
    
(And this question is about Calc I, if I read it correctly...) –  The Chaz 2.0 Apr 9 '12 at 16:40
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