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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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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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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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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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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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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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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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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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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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 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."
The area formula, namely, how could
$$\sum_{i=1}^{\infty} y_i \delta x_i=\int y dx$$