I’m kind of taking it slowly this week, in preparation for the M257 revision marathon that I’m planning on starting on Friday, but I’ve been dipping my toes back into calculus a bit by making a start on Chapter C1 of MS221, which is about differentiation. Now, calculus is something I had trouble with in MST121, so I’ve not really been looking forward to Block C; on the other hand, I feel like I’ve got unfinished business with calculus since I didn’t get a good grasp of it the first time I encountered it, so despite my uneasiness about the topic, I’m determined to do a good job of it this time!

Although I’m finding differentiation the least interesting of all the topics we’ve studied so far in MS221, I did enjoy the bit in Section 1 of C1 about higher derivatives; there’s something fascinating to me about the way that different functions behave when repeatedly differentiated. The whole f’(ex) = ex thing is pretty impressive, but what really piqued my interest is what happens when you repeatedly differentiate sinx or cosx, i.e.:

f’(sinx) = cosx
f”(sinx) = f’(cosx)=-sinx
f(3)(sinx) =f’(-sinx) =-cosx
f(4)(sinx) =f’(-cosx) = -(-sinx) = sinx

So as far as I can tell, the higher derivatives just keep cycling through the same four functions, infinitely. I love patterns like that, so I spent a little while yesterday messing about with higher derivatives of some of the other basic functions mentioned in C1, and I the other most interesting pattern I noticed is to do with power functions; from what I can tell, a function xn has an (n+1)th derivative f(n+1) = 0, as long as n is an integer. E.g.,

f’(x4) = 4x3
f”(4x3) = 4 × 3x2 = 12x2
f(3)(12x2) = 12 × 2x1 = 24x1
f(4)(24x1) = 24 × 1x0 = 24
f(5)(24) = 0

Very interesting! (Though probably not a particularly useful result, unless there’s a question in the exam along the lines of “What is the 21st derivative of 3x20“!)

As much as I dislike sketching graphs and diagrams, I’ve been trying to get a good visual understanding of derivatives by having a look at various functions using a couple of online graph plotting tools.

  • The Derivative Plotter from Flash And Math is great for watching the relationship between the tangent and the derivative at each point on a function.
  • Matthew Crumley’s JavaScript Function Plotter is really useful for higher derivatives, since it allows nested derivative commands – so you can instruct the plotter to draw the 4th derivative of x7, for example, by inputting d(d(d(d(x^7)))).

I’ve also been reading Calculus Made Easy after seeing it recommended on God Plays Dice, and I’m finding it very helpful. It probably contains exactly the same basic material as the MS221 course unit, but I think it’s useful to have a different perspective on the topic. I’ll certainly be recommending the book to Alex when he eventually encounters calculus in his astronomy/physics studies!