Posted 28 April 2006 - 08:02 PM
I was wondering if anyone could give me some help on this subject. I am really good with maths and find it easy to follow set procedures etc when it comes to getting answers to question. However, when it comes to optimization with differentiation, i seem to lack the abstract thought needed for it! Maybe i'm over dramatising it with the word "abstract" but when i read through examples of things like finding the dimensions of 400m of fencing that willl provice the largest area in an open space, i understand what is happening at each stage...but i couldn't imagine myself thinking in such ways during an exam.
On such example gives you a cylinder and tells you that it is being designed to hold 400 litres and asks you to find the dimensions of the radius and height that would give you that volume with but using the smallest surface area. The processes that the example go through are very weird...making up your own formulae to suit the question based on various other types of equations and eventually differentiating one of the terms to find the stationary points and then find the max/min and use that to determine the smallest surface area etc.
I'm guess that there must be some sort of procedure to answering such a question. Or at least some guidelines or things you should aim for within your working to be able to answer. So can anyone tell me how to go about thinking about a question like this?
Posted 28 April 2006 - 08:21 PM
1) Part A of the question normally tends to ask you to prove a formula. You will notice that if you look at the given diagram, there will be 2 terms which represent the lengths.
A simple example:
A cylinder: height = 2h+2x
diammeter = 2x
Part A would then say something along the lines of "Prove that the volume = 2x^3-3x (not correct to the question, just an example)
Firstly, notice how there is no "h" in the formula, as there is in the diagram, and there are only x's.
You therefore have to get 'h' in terms of 'x':
height = 2h+2x
Now you can manipulate the formula to find the volume, and substitute h=-x for the height.
2) For part B - always attempt this section if you cannot do part a, as you tend to use the formula they give you in 'a'.
Normally you are asked to state the maximum/minimum, for example, volume/cost/area etc.
This is basically a differenciation question, wherby you must prove that the volume/cost/area is maximum or minimum and then state the value.
From your knowledge of differenciating you should know that:
1) Prepare the function to be differenciated, ie expand brackets/bring x's up to the top, separate fractions etc
2) differenciate and let the equation = 0 in order to determine the values
3) Factorise to find the values and then use a nature table
The nature table indicates which value of x to sub into the given formula, so it is vital that you include this step. Some people use another method than the nature table, but it basically determines where the values are maximum and minimum by subbing them into the differenciated equation.
If any of this is unclear, post an example from the past papers and i'll take you through the steps.
Posted 07 May 2006 - 06:24 PM
Thanks alot for the post though, good tips!
Posted 07 May 2006 - 06:28 PM
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