In that Lockhart

*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) = x

^{2}, 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) = x

^{2}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 = 45

For f(x) = x

But the slope [dx/dy = 3x

^{o}].For f(x) = x

^{3}, then dx/dy = 3x^{2}. 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 = 3x

^{2}] of x^{3}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 x^{2}, the derivative = zero [flat line] when x = zero.
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