Small Change Algorithm...Help!!

[eldoubleu02]

Hey everyone

I am currently studying senior mathematics (year 12) in Queensland, Australia.
I take the subject Mathematics C which is the highest level math subject that you can study.
At this present time I am writing an assignment on small change and methods to set a limit to small change.

The final question of this assignment asks for us to...

Without the restriction of a simple method, develop an algorithm to set the limit of a small change. Show the development and refinement of the model. Present the final method with a clear and detailed method of its attributes.
The final goal of the task is to be able to develop an algorithm that is able to decide the upper and lower limits of the change (delta x) that keeps an approximation (approximating roots eg. √5 = √4+1) within appropriate limits.

The task so far has been based around approximating roots (as shown above) so I am assuming that this will come into play when developing the algorithm. We also had to decide on an appropriate amount of percentage error [(approx roots value - calculator value)/calculator value] and this has been decided as +- 3%.

I'm really stuck on this final question and could use some help!

Thanks so much guys...any more questions don't hesitate to ask.

El

 

[tkhunny]

There is a bit of language difficulty, here. Let's see how far we can get.

1) What does a "differential" do? Can you use the calculus?
2) What have you seen of the "Infinite Series" or "Taylor Series"?

3) Do you have ANY idea what I'm talking about or are there more language difficulties?

 

[JeffM]

In what anguage was the problem originally wtitten because it does not seem to have been English?

For example, "simple" makes no sense. Might it have been "single" and got scrambled somehow?

 

[Dr.Peterson]

I suspect it might be helpful if we saw the entire problem, not just this part. We need context! I'm not even sure what "set the limit of a small change" means.

At a couple points, what they say depends on knowing what was done earlier; and having a sense of what methods you have seen would help us know what to suggest. That could also help in interpreting the less-than-perfect English.

 

[eldoubleu02]

Hi everyone, would it help if I attached a photo of the entire task sheet?

Thank you

 

[eldoubleu02]

In what anguage was the problem originally wtitten because it does not seem to have been English?

For example, "simple" makes no sense. Might it have been "single" and got scrambled somehow?

That is exactly how the problem is written - word for word. It definitely says "simple".
This is from a school in Queensland, Australia where we speak English only. I am Australian and speak English only.

 

[eldoubleu02]

There is a bit of language difficulty, here. Let's see how far we can get.

1) What does a "differential" do? Can you use the calculus?
2) What have you seen of the "Infinite Series" or "Taylor Series"?

3) Do you have ANY idea what I'm talking about or are there more language difficulties?

Definitely no language barrier - I only speak English and this assignment is from a school in Queensland, Australia that speaks only English.
Yes, you would be able to use calculus. I have studied differential equations. I have also heard of and studied the Infinite and Taylor series.
Thanks

 

[Dr.Peterson]

Hi everyone, would it help if I attached a photo of the entire task sheet?

As long as it's readable, that would be good.

As for the language questions, parts do seem to be poor English, but I'm hoping it will make more sense once we see the context.

 

[tkhunny]

Well, there may still be cultural differences. The "small change" is not a familiar terminology. It would be VERY helpful if you would show us some fo YOUR work. Good notation doesn't care what language you speak.

Anyway, think on this: https://www.wolframalpha.com/input/?i=series+sqrt(x)+at+x=4

If you're approximating [math]\sqrt{5}[/math] as a "small change" from [math]\sqrt{4}[/math], that may be quite a challenge. 5-4 = 1 is not a "small change".

 

[eldoubleu02]

 

[eldoubleu02]

Well, there may still be cultural differences. The "small change" is not a familiar terminology. It would be VERY helpful if you would show us some fo YOUR work. Good notation doesn't care what language you speak.

Anyway, think on this: https://www.wolframalpha.com/input/?i=series+sqrt(x)+at+x=4

If you're approximating [math]\sqrt{5}[/math] as a "small change" from [math]\sqrt{4}[/math], that may be quite a challenge. 5-4 = 1 is not a "small change".

 

[tkhunny]

Perfect. If you had started with that, there would have been no confusion. We appreciate your patience while we figure out exactly what it is you are doing.

This is a linearization method or maybe a linear approximation using the method of differentials.

Notes:

1) 1/4 = 0.25 > 20% * 4

2) "Percentage Error" totally breaks down when the "Actual" value is zero (0). Then what do you do?

3) Personally, I wouldn't think 2.25 is a very "good" approximation to [math]\sqrt{5}[/math], but, quite obviously, your assignment author disagrees with me.

4) It might be important to know where the [math]f(x) + f'(x)\Delta x[/math] came from. Can you establish why this might be a good approximation using the point (4,2) and the slope of the square root function at that point? It may require the good old point-slope form of a line.

5) Can you think what might improve this linear approximation?

 

[JeffM]

It makes a huge change to provide context. Our guidelines ask that you post your exercise completely and accurately for exactly that reason.

Now, it is not YOUR fault that the writing in the exercise is below par: "decide" and "determine" for example are not perfect synonyms. Consequently, I can see why you are having trouble in figuring out how to proceed. But it is now clear by context what "simple method" means, namely methods similar to those previously given for you to explore.

If you have not already done so, I suggest that you read

Approximation error - Wikipedia

en.m.wikipedia.org

with respect to relative and absolute error.

Does this affect your conclusions in response to question 2? In any case, what did you decide in question 2?

Because the problem permits the use of a computer, you might, again only if you have not already done so, look at

Iterative method - Wikipedia

en.m.wikipedia.org

Iterative methods are easy to implement on a computer. You did not mention them so it is possible that you are not aware of them.

 

[eldoubleu02]

Perfect. If you had started with that, there would have been no confusion. We appreciate your patience while we figure out exactly what it is you are doing.

This is a linearization method or maybe a linear approximation using the method of differentials.

Notes:

1) 1/4 = 0.25 > 20% * 4

2) "Percentage Error" totally breaks down when the "Actual" value is zero (0). Then what do you do?

3) Personally, I wouldn't think 2.25 is a very "good" approximation to [math]\sqrt{5}[/math], but, quite obviously, your assignment author disagrees with me.

4) It might be important to know where the [math]f(x) + f'(x)\Delta x[/math] came from. Can you establish why this might be a good approximation using the point (4,2) and the slope of the square root function at that point? It may require the good old point-slope form of a line.

5) Can you think what might improve this linear approximation?

Okay, thank you.

We have done differentials but I'm not too sure how I would apply them to this situation??


Thanks again

 

[eldoubleu02]

It makes a huge change to provide context. Our guidelines ask that you post your exercise completely and accurately for exactly that reason.

Now, it is not YOUR fault that the writing in the exercise is below par: "decide" and "determine" for example are not perfect synonyms. Consequently, I can see why you are having trouble in figuring out how to proceed. But it is now clear by context what "simple method" means, namely methods similar to those previously given for you to explore.

If you have not already done so, I suggest that you read

Approximation error - Wikipedia

en.m.wikipedia.org

with respect to relative and absolute error.

Does this affect your conclusions in response to question 2? In any case, what did you decide in question 2?

Because the problem permits the use of a computer, you might, again only if you have not already done so, look at

Iterative method - Wikipedia

en.m.wikipedia.org

Iterative methods are easy to implement on a computer. You did not mention them so it is possible that you are not aware of them.

Hi

That link to the approximation error is great - thank you.

What I decided in question 2 is written in the box (shown in the attached photo above). This was done in class time and was signed off by the teacher. The teacher essentially gave examples of what to do for this question, so really these methods are just a modified version of the teachers examples. Question two was really just coming up with 3 simple methods (that we knew wouldn't do that good of a job) and not over complicating it.

Thanks for your help - especially that link.

 

[JeffM]

one of the things I noticed about this exercise is that it gave little helpful guidance about how to determine what is a feasible and useful bound on allowable error.

It suggests looking at standard criteria for hypothesis texting so you should certainly do that. I must admit that I do not find much merit in those criteria: oh, we should not use this medicine because we cannot be 95% confident that it cures cancer even though our experiment showed that it did so 85% of the time. I do not suggest that you indulge in expessing doubt about the arbitrary nature of traditional hypothesis testing in this exercise; it will not help your grade. But you might keep in mind for your personal benefit that ultimately "good enough" is never an objective criterion. It always comes down to good enough for what purpose and what alternatives are available, facts that are extra-mathematical.

To explore further in this topic, you might look at articles on potential propagation of error. The main point to take from that is that any method of estimation can never get any better than the potential error in the data.

Because this is a kind of thought experiment for you, I shall not get highly specific, but I do suggest that you read up on the bisection method as a very easily understood iterative method that is easy to implement on a computer. If I recollect correctly from a course that I took over 50 years ago, it is not considered an "efficient" method of approximation and is not even feasible for problems that are not particularly complex, but you can implement it on a hand calculator when it applies.

 

[eldoubleu02]

one of the things I noticed about this exercise is that it gave little helpful guidance about how to determine what is a feasible and useful bound on allowable error.

It suggests looking at standard criteria for hypothesis texting so you should certainly do that. I must admit that I do not find much merit in those criteria: oh, we should not use this medicine because we cannot be 95% confident that it cures cancer even though our experiment showed that it did so 85% of the time. I do not suggest that you indulge in expessing doubt about the arbitrary nature of traditional hypothesis testing in this exercise; it will not help your grade. But you might keep in mind for your personal benefit that ultimately "good enough" is never an objective criterion. It always comes down to good enough for what purpose and what alternatives are available, facts that are extra-mathematical.

To explore further in this topic, you might look at articles on potential propagation of error. The main point to take from that is that any method of estimation can never get any better than the potential error in the data.

Because this is a kind of thought experiment for you, I shall not get highly specific, but I do suggest that you read up on the bisection method as a very easily understood iterative method that is easy to implement on a computer. If I recollect correctly from a course that I took over 50 years ago, it is not considered an "efficient" method of approximation and is not even feasible for problems that are not particularly complex, but you can implement it on a hand calculator when it applies.

Okay great. That's really helpful information that I will 100% take into account. Thank you so very much for being so helpful!

El