Finding the Double Derivative from a table

[Karim]

Hi! In this question we're asked to find the first and second derivative at x = 0.5. I was able to find the first and second derivatives but according to the mark scheme, my answer for the second derivative was wrong. For the first derivative at 0.5, I found the average of the derivatives between x=0.5 and x=0.75, and x=0.25 and x=0.5. I tried doing the same to get the second derivative, but it was wrong.


The mark scheme uses this technique to find the second derivative, could someone please explain how they were able to get this equation?

 

[Dr.Peterson]

View attachment 34599
Hi! In this question we're asked to find the first and second derivative at x = 0.5. I was able to find the first and second derivatives but according to the mark scheme, my answer for the second derivative was wrong. For the first derivative at 0.5, I found the average of the derivatives between x=0.5 and x=0.75, and x=0.25 and x=0.5. I tried doing the same to get the second derivative, but it was wrong.

View attachment 34600
The mark scheme uses this technique to find the second derivative, could someone please explain how they were able to get this equation?

Please show your work for both derivatives; it may be valid even if you got a different answer, and there may be more worth discussing.

The trouble with estimates is that they are ultimately guesses about how the function really behaves; there can be more than one way to estimate the same value. If they expect you to do it a certain way, then they should have taught it. (I take it you haven't been taught that formula?)

The trouble with this problem is that the function is not at all well-behaved, so that different ways of estimating can be expected to produce very different results. Do you see how f(x) is bouncing up and down? The graph doesn't look at all smooth:

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So in real life, I'd ask for better data!

But taking what you are given, I would estimate the derivative at each point, either by your method, or by the symmetric (central) difference quotient (which is equivalent to yours); and then do the same again to estimate the derivative of that derivative. This is done in Example 1.90 here.

Doing that for the given values of x, which is what I first did (and likely what you did), won't produce the result (or formula) that you are shown. That formula comes from estimating the first derivatives not at the given points, but between them (e.g. using data for 0.25 and 0.5 to estimate f'(0.375)), and then using the same method to estimate the derivative at 0.5.

 

[Karim]

I got it, thank you!

I was just confused because the mark-scheme used different techniques in finding the first and second derivatives.

 

[Karim]

Wouldn't this also be a correct way of going about the question? You mentioned that this won't get us the same answer, why is that so? Even if it's a correct way to do it, it's not even close to the original answer so why is that?

 

[Dr.Peterson]

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I got it, thank you!

I was just confused because the mark-scheme used different techniques in finding the first and second derivatives.

Their formula is fairly standard; I had trouble finding a simple statement of it, but it is what Wikipedia shows in a general form here. But you're right that it uses a different spread of points, in each case doing the best that can be done with the given points, rather than being consistent from one derivative to the other in the deltas.

View attachment 34603
Wouldn't this also be a correct way of going about the question? You mentioned that this won't get us the same answer, why is that so? Even if it's a correct way to do it, it's not even close to the original answer so why is that?

Yes, this is what I did, in trying to do what you would be likely to do if you expected consistency. (Except that your answer should be -10.6.) In a sense, it may be considered a more accurate estimate.

I explained the reason for the difference: the irregularity in the function, which causes the estimate to vary significantly with different deltas. Here I've fit parabolas to the three points used in "their" work (red, f" = -28.8), and to the three points used in "ours" (green, f" = -10.6):

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(Yes, those numbers are the actual second derivatives of the two parabolas, and agree with the estimates.)

If the data had f(0.5) = 0.25, say, the results would have been more consistent, because the data would better fit a smooth curve (f" = -4.6):

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