What is optimisation?

Watch this to learn the skills of optimisation for Higher Maths (and put another shrimp on the BBQ)

Last Christmas I was lucky enough to go and visit some friends in Australia. When I was packing I realised I’d bought them far too many bulky and heavy Christmas presents and that I’d have to leave some behind. I decided to pack the combination of presents that would be highest in value to my friends, but that didn't exceed the weight limit imposed by the airline.

But how could I decide which ones to pack? As I’m a mathematician I thought about trying all possible combinations, calculating their value and weight, and picking the combination that was below the weight limit but maximized the value. But there were simply too many combinations to try out and I was worried I’d take so long I’d miss the plane – so in the end I just paid for an extra bag of luggage!.

This problem is a lot like the optimization questions in the Higher Maths course, but there are no easy fixes like paying for more baggage allowance. Let me show you what I mean…

This is thinkfour!

Usually the higher course optimisation questions are to do with length, area or volume, but they can have other contexts too.
Often you get questions about boxes or trays and you want to either maximise the volume or minimise the surface area.

Optimisation questions tend to be split into parts (a) and (b) and many students avoid the part (a), losing up to three marks.

To master this part of the question you need to have a clear strategy. Come up with two expressions, perhaps for area and volume, like in this example. Next you rearrange one of the expressions and finally substitute it into the other one. There is usually a fair bit of algebraic manipulation involved so be careful. You should practice these part (a) questions and try not to avoid them.

In part (b) you are just solving a stationary points question, but it is in a context, so often the notation is different from using y and x. You will differentiate, equate to zero and solve.

However, because this is an optimisation question and you are asked to find a maximum or minimum, it is important you prove the nature of your stationary point with a nature table or second derivative test.

It is also important to relate the answers back to the context of the question. Does it need units? And do you need to reject a negative solution as it makes no sense in the context of the question? Your answers for optimisation questions should be as full as possible, with words relating your answers back to the question context.

Optimisation questions can be worth a lot of marks, and tend to follow a similar structure. Try practising three of four in a row and have a look at where the marks are allocated using the marking instructions. This will help you to ensure you are including everything you need to.

And, when it comes to delivering presents to the other side of the world? My advice is simple: don’t be so generous and remember to buy smaller and lighter gifts!

This was thinkfour, thanks for watching.