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.
The hardest thing for me was to understand what is meant when someone writes $\mathrm d x$.
I still don’t know…
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.
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.
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! 🙁
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.
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)
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.
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.
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.
The area formula, namely, how could
$$\sum_{i=1}^{\infty} y_i \delta x_i=\int y dx$$
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.
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) $”.
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.”
Partial Differentiation using the third law
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.