Friday 13 April 2018

The Calculus

Calculus is pretty much a required course in many US universities. You can teach it like a cold bath before breakfast - bracing and 'good for you' but not, for ordinary students, easy. I was rather good at Calculus when I sleep-walked through my very expensive education: I could learn the tricks by heart because I was learning a lot of things by heart and therefore was 'in training'. I think some effort was made to explain to us why differentiation [that's half of the Calculus, the other half being integration, it's inverse] might be useful. The students in my remedial maths class on the last Friday before the Easter break were distracted from the task at hand because they had a test/quiz/exam on Calculus immediately afterwards. I was singing Lockhart's Lament three years ago because Calculus, as normally taught, makes no sense. It is just a heap of random rules which you have to remember.

In that Lockhart Lament Rant I noted ". . . in nearly 40 years of teaching and research in science; writing thousands of lines of code; analysing genome sequences; [discovering novel genes] mapping genetic data across chunks of the planet; I've never, ever, had call for using any of the tricks and algorithms of The Calculus.  Why this should be a required skill taught to every science student is a mystery to Dr Lockhart . . . and to me." And occasionally I tell that to my students.  They justifiably bridle at this and respond with "Then why do you expect and require us to learn all these rules and remember them just until the next examination?". I think I agree with the Latin and Cold Baths argument that mastering the Calculus is as good a training for the mind as any. It is difficult, it is logical and it requires you to integrate <ho ho little Calculus joke there> disparate information to make sense of it. If you can make sense of Calculus your powers of reasoning with be pushed and made stronger. Call me a Protestant but you can't leave all the difficult and disagreeable things on the side of the plate because a diet of Mars Bars and ice-cream is ultimately borrrring . . . and will only make you both fat and costive.

Put on the spot that recent Friday, I tried to recall my own triumphs with the Calculus in the 1960s but could only really remember that differentiation was a technique for measuring the slope of the graph of a function. I wasn't much help to my struggling students. I should maybe have forced them to do some Excel because that would at least have distracted them from their calculus-anxiety. In fact a bit of plotting with Excel helps (me) make sense of the rules of differentiation. Because I knew that the derivative of a function [which most of us get by knowing The Rules through rote learning] tells you what the slope of the graph of that function is at any place on the graph.  If f(x) = 2x (or, put another way, if y = 2x) then the derivative (following the rules) dx/dy = 2. This is telling us that for all values of x the slope of the graph equal 2 or that it is a, fairly steep, straight line.

For f(x) = x2, the values of y start off large and positive, decrease stedaily and then rise again. The derivative (following the rules) dx/dy = 2x. This says that the slope of  f(x) = x2 varies as a function of x. For negative values of x, the slope is always negative (below the x-axis) and the slope is always positive for positive values of x. When x = 0 the slope is zero because the curve is flat.
Furthermore, as x gets further from zero, the slope of the graph gets bigger. How much bigger? Twice the value of x bigger. When x = 0 the slope is zero or flat. When x = 0.5 the slope dx/dy = 2x is 1 [in other words the angle of the curve, which they call a parabola, at that point = 45o].

For f(x) = x3, then dx/dy = 3x2. Here the graph of the function [blue dots] always trends upwards, there is no negative slope, so all values of the derivative [orange dots] are above the zero line.
But the slope [dx/dy = 3x2] of x3 starts off steep [when x = -4, dx/dy = 48], eases off to no slope at all when x = 0, and then gets progressively steeper again. And this is shown in the high values for the derivative away from zero; but, as for the previous graph of x2, the derivative = zero [flat line] when x = zero.

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