Quality of teaching might have something to do with it.
But, also, calculus is much harder to understand at a rigorous, formal level than at an informal level.
On one level you can try to understand what the main concepts are about, be able to compute derivatives and integrals, solve optimization and related rates problems, and so on. I'd recommend Silvanus Thompson's Calculus Made Easy over any mainstream calculus book for this. In my opinion, the book succeeds amazingly at fulfilling the promise of its title.
But suppose you really try to read any mainstream calculus book, and understand everything. For example:
- Why are limits defined the way they are (with epsilons and deltas)?
- The book will probably touch lightly upon the Mean Value Theorem -- why is this important? What's the point?
- Why is the chain rule true? It reads dy/dx = (dy/du) (du/dx). Yay! This is just cancelling fractions, right? Any "respectable" calculus book will insist that it's not, but most students will cheerfully ignore this, still get correct answers to the homework problems, and sleep fine at night.
- Consider the function e^x. How is it defined? The informal way is to say e = 2.71828... and we define exponents "as usual". Most students are perfectly happy with this. But does this really make sense if x is irrational? Your calculus book might bend over backwards to define everything properly (e^x is the inverse to ln(x), which is defined as a definite integral), and it takes a lot of work to appreciate why.
In my experience, these sorts of issues mostly don't pop up in linear algebra, where the proofs tend to parallel the handwavy heuristics. I wonder if this had anything to do with your experience?
One difficulty students had that I encountered as a TA was some garbled prerequisites. All of the epsilon-delta definitions are written in propositional logic, and are often '2nd order statements', (that is they have nested quantifiers). This is an entirely new formal language, and its usage is very different than english. It needs to be carefully explained, but standard texts like Stewart just dress it up to look kind of like english and carry on.
In fact the mathematics curriculum DOES acknowledge that you have to teach most students this if you want them to understand it: at my university it lived in the discrete math course, which used Rosen. He devotes an entire chapter on propositional logic, and spends literally 60 pages gradually building up the complexity to arbitrary nested quantifiers; the definition of the limit appears at the end of this.
Unfortunately, discrete math also makes heavy use of sequences and series, so that Calc 2 is a prerequisite of the course... thus my program's student-victims would spend a year taking calculus and not understanding much of the formalism before they were even allowed to take the course that explained the language the basic definitions of calculus were written in! Ugh.
I think Stewart and Rosen are pretty mainstream textbooks, so i suspect this problem is very common. Perhaps you could point it out at your next faculty meeting and shuffle some prerequisites around; we'll start a math-revolution! :)
> - Why are limits defined the way they are (with epsilons and deltas)?
> - The book will probably touch lightly upon the Mean Value Theorem -- why is this important? What's the point?
> - Why is the chain rule true? It reads dy/dx = (dy/du) (du/dx). Yay! This is just cancelling fractions, right? Any "respectable" calculus book will insist that it's not, but most students will cheerfully ignore this, still get correct answers to the homework problems, and sleep fine at night.
Which "respectable" book(s) would you recommend for those who want to dive into this details? Is Tom Apostol's Mathematical Analysis? good for learning these kind of details? (They say this book is "respectable", but I would like to hear your thoughts about it. Thanks).
I don't know anything about Apostol's Mathematical Analysis. My guess would be that it demands a fairly sophisticated background of the reader, and does an excellent job of covering calculus from an extremely rigorous point of view.
I have heard that Apostol's Calculus is an excellent choice, probably somewhat more accessible to beginners, but still offering a rigorous, highbrow perspective. I've also heard the same of Spivak. I'd probably opt for one or both of these.
That's another thing I observed teaching calculus: a lot of students who have problems, are having their difficulties with the algebra they supposedly know, and not so much with the "new" material. In theory, we expect students to already be fluent in this sort of algebraic manipulation; in reality, we recognize that a calculus class provides our students an opportunity to improve at this.
I'm not sure I have any suggestion other than lots of practice. And know that you're not alone, this is a completely natural difficulty.
That's another thing I observed teaching calculus: a lot of students who have problems, are having their difficulties with the algebra they supposedly know
A. You're not alone. In the first video of his Calc I series, Professor Leonard cracks that "Calculus is the class you take to finally fail Algebra".
B. This is hardly unexpected, especially if there's any gap at all between taking Algebra and taking Calc. The simple truth is, you forget material you don't use. And most people don't use a lot of algebra in their daily lives. If even a semester or two has passed since you took algebra, you're almost certainly going to have forgotten a lot of it, unless you made a very pointed effort to keep practicing that stuff and focus on retention.
> Somehow I found linear algebra easier than calculus, but I don't know why.
I suspect the answer is that your calculus course was a lot heavier on crank-grinding: having to readily apply integration and differentiation on a wide panoply of functions, some of them you're not really familiar with (such as arccos). If you're weak on trigonometry or some algebraic manipulations, that's going to shut out the ability to do a lot of the crank-grinding without really impacting your ability to understand the concepts.
By contrast, the crank-grinding in linear algebra is a lot less involved. The most complex algebra is going to be solving polynomial equations to find the eigenvalues of a matrix, but those are generally going to mostly be quadratic equations since asking anyone to solve more complex equations by hand is going to ask for trouble. Otherwise, it's largely plug-and-chug numbers into stock formula. Gram-Schmidt orthonormalization? Pick a vector, normalize it, project the other vectors and cancel them out, and repeat until you've done all of them.
Linear algebra should be easier than calculus shouldn't it? The whole program of differential calculus is basically that we already know how to solve problems in linear algebra, so let's solve other problems by reducing them to questions of linear algebra in the tangent space.
The fundamental strategy of calculus is to replace a nonlinear function with a tangent line approximation to that function. This greatly simplifies calculations, and the approximation is often accurate enough to be useful.
I did both at the same time in university, but failed calculus 3 times and aced linear algebra at the first try.
I'd expect being either good or bad at math, not both at the same time