06.09.2019: One Chapter of Math – Calculus Made Easy: Chapter 3

Today’s soundtrack is Bolt Thrower: Those Once Loyal, a plodding melodic death metal album with lyrics centred around war and tanks.

This morning, I’m reading the third chapter of Thompson’s Calculus Made Easy, “On Relative Growings.”

Source: Khan Academy

Remember the quadratic formula? I’ve included it at the right. Look at its variables. Those with a fixed value (the constants) are shown by using letters from the first half of the alphabet. The value capable of varying (the variable) is shown using a letter from the second half of the alphabet. This is not an accident! Thompson (1998) tells us that in calculus, we use letters from the first half of the alphabet to denote constants; we use letters from the second half of the alphabet to denote variables.

What of a variable that depends on another variable, and that variable depends on constants? It’s sort of a reversed mathematical adverb to the adjective to the noun – but instead of influencing the former, it is influenced by it. In mathematics, though, we don’t differentiate between a variable that’s dependent on another variable versus a variable that’s dependent on a constant (or set of constants). We just call it another letter from the second half of the alphabet and move on. So if x is dependent on a and b, and there’s another value that’s dependent on x, we call this fourth variable y.

Let’s play around with x and y a little bit. As we learned in Chapter 2, dx is a little bit of x. If we add dx to x, we get x+dx. Since y is dependent on x, y becomes y+dy. Now, these two look the same! But this does not mean that dy is the same value as dx. It will grow in proportion to y‘s relation to x. Thompson (1998) explains it thus: “[T]he value of the ratio dy/dx is the same as that of the ratio y/x. And here we’ve arrived at the crux of the matter, for “right through the differential calculus we are hunting, hunting, hunting for a curious thing, a mere ratio, namely, the proportion which dy bears to dx when both of them are infinitely small” (Thompson, 1998, p. 47). This ratio is called a derivative. For this to work, of course, the two variables must be connected; also, they must be able to be expressed as functions of the other (implicit functions). Think of the explicit function d=s/t and how it can also be expressed as the implicit function s=dt. Note here that if we know that x and y are related, we can show that they are related (even if we don’t say their ratio) by expressing this with the notation “F(x, y, z) (implicit function) or by x=F(y,z), y=F(x, z) or z=F(x, y) (explicit function)” (Thompson, 1998, p. 49).

We’re now in new territory, dear reader(s). Whereas up until this point, we would algebraically search for x, we are now looking for dy/dx. This search for “the value of dy/dx is called ‘differentiating'” (Thompson, 1998, p. 49).

That’s it for today! Next time, we’ll be reading the fourth chapter.

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