Dear Yoni,

[MyndFyre]

I have forgotten calculus.

I remember limits, polynomial derivatives, and polynomial integration.

I need a refresher!  Please help!

Love and affections,

--MyndFyre

[Yoni]

That's way too general.

Ok, so you have derivatives and integrals and stuff.

[Rule]

Robert Adams wrote the best introductory book to calculus that I've seen.  Cornell uses it for its first year honours course:
Adams, Robert A., Calculus Single Variable (paperback) (Edition: 5) Pearson Education Canada, 2003 (ISBN: 0-201-79805-0).

Greetings, and welcome to the first edition of Rule's "Learn Calculus better than an Engineering Graduate in 3 days":

I quickly wrote out some tasks that should help you gain a more fundamental understanding of calculus.  The tasks are self contained for the most part, so you should only refer to supplementary info in completing them when suggested.  For example, since I haven't defined what a "riemann sum" is you should briefly acquaint yourself with them, and perhaps be familiar with basic left, right and midpoint rectangular riemann sums.  The tasks (1-8) shouldn't take you longer than three days, and once you are finished, any particular application of single variable calculus (e.g. integrating polynomials, integrating by parts, determining volumes of revolution, partial fractions, chain rule calculus, series expansions, solutions to ordinary differential equations, related rates, optimization, etc..) should become trivial to you.  This is a different approach to learning calculus than you'll get in any college/university, and since I wrote it out, I imagine it's the best one .  On completion of tasks 1-8 you should have a deeper understanding of calculus than most 3rd year engineering students, or most engineering graduates for that matter.  On completion of task 100, perhaps only about 1000 people in the world will have a deeper understanding of calculus than you .

Note that about 1/4 of the tasks can be easily done in about 5 minutes, while others will require varying levels of reasonably deep thinking and concentration.  The latter may still be completed in a matter of minutes or hours, although it's possible and reasonable to expect that you may need days to do them.

Task 1) Read on limits, understand when they exist, how to evaluate them in a variety of situations, etc.

Task 2)  A derivative is supposed to represent an instantaneous rate of change; for example, the derivative of f(x) at x = a, is the slope of the tangent line at x =a, and tells you how f(x) is changing with x at x=a.  Derivatives are often closely associated with tangent line approximations, and in my opinion, one should consider a function to be differentiable at a point a when the tangent lines to f(x) converge to one line (or plane in higher dimensions) as you approach a from any direction.

From this information alone, mathematically derive a definition for the derivative of a single variable function, f(x).   [Scroll down for task 3 once you finish task 2 ]




























































Task 3) Understand and explain how these definitions are equivalent.
d/dx f(x) = lim h --> 0 [f(x+h) - f(x)]/h    =  lim x-->a   [f(x)-f(a)] / (x-a),
if the respective limits exist.

Task 4) Derive a convenient method for determining the derivative of a polynomial.  Hint: see binomial expansion thm. 

Task 5) Derive a convenient method for determining the derivative
of h(x) = f(x)*g(x).  Using this definition, derive a convenient method for determining the derivative of a function q(x) = r(x)/m(x).  Afterwards, look up and read about the chain rule.



Task 6) Derive (e.g. do not look up a proof for) the fundamental theorem of calculus using the so-called "riemann sums" and the definition of the derivative you derived (or understood) in task 1. Hint: Draw a curve, g(x)=d/dx f(x) in two dimensions, with a starting point x = a, and a terminal point x = b.

Task 7) Many functions have a power series expansion.  They can be written as
f(x) = Sum[a(n) * (x-b)^(n)] where n goes from 0 to infinity.  Assuming f(x) is infinitely differentiable, derive this formula for the coefficients a(n)
a(n) =  the (n-1)th derivative of f(x) at x=b divided by n!
[Note a(n) means a is a function of n. For example a(n) might = n^2, in which case a(0) = 0, a(1) = 1, a(5) = 25, etc...]
Hint: Write out the terms like this:  f(x) = a₀ + a₁*(x-b)
+ a₂*(x-b)² + ... + a_n*(x-b)ⁿ.  From there the proof shouldn't be very difficult.

When b = 0, the series expansion is known as a Maclaurin series, which is a special case of the Taylor series given above.

The Maclaurin series of e^x, for example =
1 + x + x^2/(2!) + x^3/(3!) + ... + x^(n)/n!
here a(n) = 1/(n!)

Taylor series expansions are incredibly powerful (e.g. they are often used to make approximations in physics, they are used to solve differential equations, Yoni suggested it as a solution for approximating some power in a recent thread here, etc).  I also just think it's pretty cool how so many functions can be written as an infinite sum of polynomials.  Try   graphing e^x on your calculator (or mathematica), and then graph 1 + x + x^2/2 + x^3/6 + x^4/64.

Using taylor series and integration we can come up with better and better approximations of pi.

For example, the power series expansion of 1/[1-x] = sum(x^n)
so 1/[1-(-x^2)] = 1/[1+x^2] = sum( (-1)^n * x^2n).
Integrating both sides wrt. x we get
arctan(x) = sum ( -1)^n *x^(2n+1)/(2n+1) ) .
tan(pi/4) = 1
so arctan(1) = pi/4

write some loops in a computer program that evaluates say 100000 terms of that series at x=1 , multiply by 4, and you've got a decent approximation to pi .

Here's a question I once thought about when I first started learning calculus (but resolved): 
The fundamental theorem of algebra says that a polynomial of degree n will have
n roots in the complex plane.  e^x has no roots, yet a more and more accurate maclaurin series approximation of e^x is a polynomial of an higher and higher degree, and hence has more and more roots.  How can this "paradox" be resolved?
...
Task 8: (Ok, I imagine by now you're on day 3 or so)
Read about evaluating integrals using substitution.  Then derive integration by parts using the product rule that you derived in task 5.

...
/* Learn classical vector calculus.  Don't waste too much time doing this, because in my opinion it's mostly an exercise in familiarizing yourself with higher dimensions and allowing you to think about things in such a way that modern differential geometry (task 100) will at least be approachable.  If/when you do learn classical vector calculus, be sure to understand what the derivative matrix is, what a jacobian is, what the definition of a derivative of a multivariable function is, what the definition fo the directional derivative is and how to do change of variables in multiple dimensions (not substitution, but creating your own coordinate systems to make integration easier).  Also understand the implicit and inverse function theorems.  All of these topics are ordinarily *very* poorly covered (if covered at all), yet they are the most important topics in higher level mathematics in physics.  Avoid the Stewart's multivariable calculus textbook; the "chain rule" is one of the only important concepts in classical multivariable calculus which receives adequate attention in standard textbooks.  Colley's "Vector Calculus" is all right, but I'm sure there are much better.  */
...
Task 100: Understand and prove the Generalized Stokes' theorem,
integral[dw] over a manifold = integral[w] over the boundary of the manifold.  Show how this one theorem can be used to prove every single fundamental theorem in n dimensional calculus (including, for example, the fundamental theorem of calculus in 1 variable), as well as identities such as div(curlF) = 0.

Now reteach yourself calculus properly using tensors, differential forms, and the above version of Stokes' theorem.  Hopefully at this point (I am talking to the future Myndfyre of about 3 years from now), you haven't wasted too much time learning calculus the classical/engineering way .