Page:Calculus Made Easy.pdf/74

 above five minutes when “bang went saxpence.” If he were to spend money at that rate all day long, say for $$12$$ hours, he would be spending $$6$$ shillings an hour, or £$$3$$. $$12s$$. per day, or £$$21$$. $$12s$$. a week, not counting the Sawbbath.

Now try to put some of these ideas into differential notation.

Let $$y$$ in this case stand for money, and let $$t$$ stand for time.

If you are spending money, and the amount you spend in a short time $$dt$$ be called $$dy$$, the rate of spending it will be $$\dfrac{dy}{dt}$$, or rather, should be written with a minus sign, as $$-\dfrac{dy}{dt}$$, because $$dy$$ is a decrement, not an increment. But money is not a good example for the calculus, because it generally comes and goes by jumps, not by a continuous flow–you may earn £$$200$$ a year, but it does not keep running in all day long in a thin stream; it comes in only weekly, or monthly, or quarterly, in lumps: and your expenditure also goes out in sudden payments.

A more apt illustration of the idea of a rate is furnished by the speed of a moving body. From London (Euston station) to Liverpool is $$200$$ miles. If a train leaves London at $$7$$ o’clock, and reaches Liverpool at $$11$$ o’clock, you know that, since it has travelled $$200$$ miles in $$4$$ hours, its average rate must have been $$50$$ miles per hour; because $$\tfrac{200}{4}=\tfrac{50}{1}$$. Here you are really making a mental comparison between