maths integral

iii) would be asking has the trapezoid approximation converged to "n" decimal places, using part ii).
What did you get for the previous parts?

0.458658
0.575532
0.618518
0.634173
0.639825

Because part (iii) says "state", I would suggest that no further calculations are expected and that you have to use the answers/info from the previous parts of the question to answer part (iii).
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Can you pls edit your response? The aim is to help the OP arrive at an answer, rather than do it.
If you look at the difference is converging, I'd argue you could state the integral to 2 dp, but that is for the OP to analyze.

Oops! Sorry! I've just edited it.

Out of interest, can you explain to me about the difference converging thing being able to justify the integral is to 2 dp please? I get that you can calculate the difference between each of the values found in part (ii), and that it shows that the values are converging, but how would you actually write it in terms of an exam question to justify the 2dp with just these values? If this was an exam question, I would be able to justify it to 1dp, but not 2. (Feel free to DM me if you'd rather not post the answer here). Thanks!

If you calculate the successive differences between the approximations, can you see how they're going to zero roughly as a geometric sequence, what is the ratio? Use that to guestimate the next couple of approximation errors and justify that this wouldn't change 0.64.... very much.

You'd also really say that the trapezoid approximation always underestimates the true value (obvious from sketching the function in part i) and the original function is quite nice (smooth), so looking at the differences in the approximation sequence makes sense.

Thanks for that! I get a bit flustered when answering questions with text in exams and prefer to prove it with maths than try to explain it in words. So do you think it would be okay to write down the calculations of the difference between, say, the last 3 values, then say that this shows the differences are tending towards zero (is this the correct term - "tending" ? I mean the right pointing arrow) so therefore it is justified to say that the value of the integral would be 0.64 to 2 d.p.?

I was hoping you / the OP would do the maths behind the previous post, but the question asks for some justification so that means words. Its not enough to show that the differences tend to zero, but they have to go to zero fast enough. If the differernce sequence was geometric with ratio 0.99, you'd have to compute a lot of approximations. If it was geometric with ratio 0.001, you'd be there after a couple of approximations. So stating the rough ratio and what the next couple of approximation errors (sequence differences) would be, would be enough to justify the 2 dp accuracy.

So looking at the values in part (ii), the ratio is not fixed, but is roughly between 1 and 1.25 for the values given (I have worked out the ratio by dividing the value by it's previous value). Is this what you mean by stating the rough ratio? As it gets closer to the actual answer of the integral, the ratio would get closer to 1, right? So, how would I then work out the next couple ..

Ah, so what I did wrong was to work out the ratio between the values, rather than the ratio of the DIFFERENCES between the values.

So, now I've done that, the ratio is approx 0.36. Is this right? So the next couple of values in the sequence would be:

0.639825 + (0.005652 x 0.36) = 0.64185972
0.64185972 + (0.00203472 x 0.36) = 0.6425922192

Therefore, I can now justify that it would be 0.64 to 2 d.p.?

Pretty much. I'd simply note that the new value is a third of the previous which converges pretty quickly and would affect 3dps onwards.

If the original sequence is geometric (exponential) then so is the difference, so you could have analysed in terms of the original one, but having ratios close to 1 means its difficult to say anything about the rate of convergence. Analysing successive differences means its easier to say how quickly things change.

What do you mean by "the new value is a third of the previous"? (Thanks for being so patient, btw!)

The ratio is ~1/3 so the new difference is ~a third of the previous one.

@mqb2766 - please don't think I'm being rude or pedantic in anyway, but can I ask what your maths situation is? Are you a student, teacher etc? It's just that my teachers (I'm in Y13 - UK high school) have told us that when an A level question says "state" and it comes after other parts of the question where you've had to calculate things, that it's usually just asking you to extract/interpret info from the previous calculations without having to do more calculations. When I've done past papers / past questions similar to this, this is what I've done. I completely get what you've explained, and I'm certainly not questioning your maths in anyway, but this is why I thought that this question was simply a case of looking at the answers in part (ii) and then making a judgement. (At the moment, I'm revising just in case I don't get the grade I want and need to resit the maths A level in October).

This is why I asked you if you were a student, teacher, graduate etc. To me, it's clear that it's converging to 0.64, but it's not clear that the sequence differences are going down by a third each time without doing the actual calculation. Bravo if you can do all that mentally - I would have to write it down! I'm asking as an A level student, and just going on what my teachers have taught us and past papers and questions I have been doing. Personally, I think that in an exam, if they wanted the level of justification you are suggesting, they may word the question something like "show how you can justify that the value of the integral is to 2 d.p." or "prove that the value of the integral is 0.64 to 2 d.p.". We have also been advised to look at the number of marks allocated to the question as an indication to how they are expecting us to answer it. In this case, I agree that the question here is a bit vague. If it was in an exam and it was allocated 3 marks, then I would definitely start to look at the relationship between the values etc., but if it was a 1 mark question, I would say 0.6 and move on.