Calculus: For Emily

A graph showing how one quantity varies when another quantity changes. Calculus allows scientists and mathematicians to study the rate of change - or the slope - of such graphs.

A graph showing how one quantity varies when another quantity changes. Calculus allows scientists and mathematicians to study the rate of change - or the slope - of such graphs.

I was chatting with an engineer colleague the other day when he asked for advice about his daughter, Emily. He explained that she was doing great at school, in the top set for everything, but that she was having trouble with calculus. She had asked him “Dad: What is the point of calculus?“. And he had not known how to answer.

Well, my first thought was how great it would be to have a daughter who was confiding such a question! And then I thought about my own education where I was just told things and had to learn them. And I thought maybe it’s possible she is just being told stuff, and that someone has avoided a few short paragraphs of explanation. So I thought, I would imagine a conversation in which I would brilliantly explain things:

The conversation

Michael:Calculus is the mathematics of change. It is essential for the study of any situation where things change.

Emily: Thanks Michael, that’s brilliantly explained.You really are wonderful. But everything changes all the time – so what’s so special about calculus?

Michael: That’s the point, calculus is ubiquitous. Indeed its hard to imagine what science would be like without calculus. Before Newton people couldn’t really understand the motion of any objects – whether in the heavens or on Earth. Newton had to  invent calculus so that he describe the motion of objects – and we have been using his mathematical techniques  ever since

Emily: Newton invented calculus?

Michael: Yes. Leibnitz came up with the idea at roughly the same time, but Newton certainly invented it independently.

Newton: I was first!

Michael: He thought (correctly), that if there was no friction, the rate of change of the speed of an object would be zero. He hypothesised that there existed things he called ‘forces’ whose direct effect was to affect the rate of change of the speed of an object. He didn’t equate the force on object to a simply measurable property of the object (such as its speed) – but to the rate of change of a property of the object (its acceleration). Nowadays we would call that a differential equation – but Newton had to invent the mathematics to be able to write down the equation!

Emily: OK, but what about other areas?

Michael: Here’s three examples: In ecology the population of one animal or plant affects the rate of change of the population of another animal or plant ; in chemistry  the rate at which reactions occur depends on concentrations of chemicals; and most importantly, in economics – where things change all the time – you get to pick your own favourite reason for what caused what to change.

Emily: Well thanks for taking the time to explain that, but I am still unconvinced. In fact I think I am actually more confused than I was before.

Michael: Sorry. Well stick with it. If you can master calculus – you can master anything!

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