Find the approximate model (+ differentiation)

[Aminta_1900]

This is the last question in the second modulus on calculus, which so far had dealt with the differentiation of products, quotients and functions of functions. So I am guessing that it may involve these. It shows this graph of the growth of a stick insect population over time and gives four questions.

1) "Find the growth factor between the numbers of stick insects in consecutive time intervals of two weeks. Hence find the relationship between x (time in days) and y (number of insects) that can be used as an approximate model for the data given in this graph".

The given answer is: ~1.5 and y = 6.7(15^x/14)

But I'm not sure at all how this function can be deduced from the data given.

The next three questions pertain to how I am measuring x in number of weeks, even though the question says x is measured in days.

2) "Use the model to estimate the number of insects 45 days after the first recorded entry on the graph"

Here I am taking the 45 + 14 = 59 days = 8.43 weeks, so imputing x = 8.43 into y = 6.7(15^x/14) and getting 34.2 insects as the estimate, even though the given answer is 37. Is this still a correct estimate? The question said that time should be in days rather than weeks but this would give insect growth in the 100s of thousands, which cannot be right. But the next questions throw me further off.

3) " Estimate how many days after the first recorded entry the number of insects had grown to 100"

Here, using x in weeks again, I am using 6.7(15^x/14) = 100, so changing the subject x = 13.97. 13.97 - the first two weeks = 11.94 weeks = 83.6 days, but the answer is 79 days, and the approximation is again a bit off. But is this my answer valid?

4) "Find the rate at which the numbers are growing 10 weeks after the first entry"

Here I am using the derivative of 6.7(15^x/14) as

0 * 15^x/14 + 6.7 * [ {(1 * 14) - (x * 0)}/196 ] * 15^x/14

= 0.479(15^x/14)

And imputing x = 12, as per weeks = 10 + 2 = 12. My answer is 4.88 insects per week, but the actual answer is 5.5 "per day"!

What am I doing wrong in these estimates in measuring x in weeks? or perhaps have I got the derivative wrong for the last question? Thanks a lot.

 

[khansaheb]

But I'm not sure at all how this function can be deduced from the data given.

Try an "exponential" relationship.

Number = A + B * e ^(weeks)......or.......

Number = A + B * e^(weeks)

Calculate A and B using "least-square-fit" or software like MS-excel.

 

[Dr.Peterson]

1) "Find the growth factor between the numbers of stick insects in consecutive time intervals of two weeks. Hence find the relationship between x (time in days) and y (number of insects) that can be used as an approximate model for the data given in this graph".

The given answer is: ~1.5 and y = 6.7(15^x/14)

But I'm not sure at all how this function can be deduced from the data given.

Let's focus on this part, because you'll be changing everything else. Note, in particular, that if they say to take x as days, then you have to do that; taking it as weeks means you are solving a different problem.

What they are asking for here is a very approximate formula. It will not be correct for all the points; and your answer might not agree exactly with theirs!

They tell you to find the ratio between consecutive points on the graph; for example, the first ratio (from week 2 to 4) is 15/10 = 1.5. Others will be similar (do it!), so for a rough approximation, you may as well use 1.5 as the ratio.

Now, can you show how you would find a formula? It looks like, if there is a typo in your formula (namely a missing decimal point and missing parentheses), they are using a standard formula of the form [math]y=Ar^{x/n}[/math] or something like that, where r will be 1.5, and A and n can be determined from a couple points. (I can see where the number 14 probably came from; I can't be sure how they put it there without seeing their general formula.)

You will get a slightly different formula depending on which points you use. We need to see what form you have been taught, and how far you can get in using it.

I don't think you are expected to use any curve-fitting software; but if you were told to do so, we'll want to see the instructions you were given.

 

[Dr.Peterson]

2) "Use the model to estimate the number of insects 45 days after the first recorded entry on the graph"

Here I am taking the 45 + 14 = 59 days = 8.43 weeks, so imputing x = 8.43 into y = 6.7(15^x/14) and getting 34.2 insects as the estimate, even though the given answer is 37. Is this still a correct estimate? The question said that time should be in days rather than weeks but this would give insect growth in the 100s of thousands, which cannot be right. But the next questions throw me further off.

3) " Estimate how many days after the first recorded entry the number of insects had grown to 100"

Here, using x in weeks again, I am using 6.7(15^x/14) = 100, so changing the subject x = 13.97. 13.97 - the first two weeks = 11.94 weeks = 83.6 days, but the answer is 79 days, and the approximation is again a bit off. But is this my answer valid?

I think I'd better be more explicit about the error in their formula as you copied it. I hadn't looked at your work on the later parts, assuming it was all about your using weeks instead of days, but it's mostly from having the wrong formula. (If you had understood how they got it, you probably would have seen the error.)

The main typo is that you show 15 instead of 1.5, the growth factor they used. The other is that you omitted parentheses, but your work shows that you are handling that part as if they were there. The formula they intend is y = 6.7(1.5^(x/14)).

In your work for (2), take x = 59; don't convert to weeks as if they hadn't told you x is in weeks. With the correct units and the correct formula, your work will lead to the right answer.

The same is true for (3). Your work would have been correct if you had used 1.5, except that the value you get for x will be the correct number of days, not the number of weeks.

I think you changed to weeks because that gave you better results than using days with the wrong formula.

I presume these two corrections will make the rest of your work give the right answers, too.

By the way, you should observe that the formula does not give the correct y for larger x; for example, for the last point (16 weeks, x = 112) it gives about 172, not 125. It appears that they just used the first two points to get their formula (and rounded 6 2/3 to 6.7), ignoring the other points. Clearly they did not do any curve fitting!

 

[Aminta_1900]

Thanks for the long reply. Yes, looking at the answer in the back of the textbook it is definitely written as 6.7(15^(x/2)) (written here with my corrected parentheses).

My initial assumption when it asked me to make an approximate model was to produce a linear function based on the average daily or weekly growth of insects.

So I had started, assuming day 0 was about near 0 insects, with (10 + 5 + 6 + 10 +19 + 8 + 24 + 43)/8 = 15.625 insects per 2 weeks, or 1.11 per day. So perhaps y = (10/9)x, but of course this still did not fit the answers given when substituting for x.

But I can see how the following answers would work if the function had been written correctly. Only I still have no idea how the question assumed that I could have deduced this exponential function, not having done anything similar to this previously.

I'm guessing from your correction that because the questions asks to begin from the first recorded point, that 1.5 becomes the principle of the exponential function. Other than that it is still a tough question to answer. But thanks for the insight.

 

[Dr.Peterson]

My initial assumption when it asked me to make an approximate model was to produce a linear function based on the average daily or weekly growth of insects.

So I had started, assuming day 0 was about near 0 insects, with (10 + 5 + 6 + 10 +19 + 8 + 24 + 43)/8 = 15.625 insects per 2 weeks, or 1.11 per day. So perhaps y = (10/9)x, but of course this still did not fit the answers given when substituting for x.

Only I still have no idea how the question assumed that I could have deduced this exponential function, not having done anything similar to this previously.

The key to the problem is in the first few words: "Find the growth factor between the numbers of stick insects in consecutive time intervals of two weeks." I would hope that something has been said about exponential growth and what they mean by "growth factor". This should (in principle) prevent you from assuming a linear function (along with the fact that the graph clearly looks more exponential than linear).

Has there been anything said about exponential growth? I'd still like to see what general equation they use for it, because books vary tremendously on this topic. Here is an example of a teacher defining the growth factor and such a general equation:

​

Here r is the rate (percent increase per time interval) of growth, and b is the growth factor; their "original form" is the one that is relevant here.

Note that since the time interval in your graph is 2 weeks, the x here has to be replaced with x/14, which gives the number of two-week intervals in x days.

Here is a graph I made yesterday comparing the data (blue dots), the book's estimated formula (orange), and a more accurate formula (blue curve) that takes all the data into account:

​

If you used the first and last data points rather than the first two, you would get something close to the blue curve. But you are not being taught that level of precision. I am not at all happy about how they chose to do it, when the question implies that some sort of average growth rate, rather than one single ratio, would be used.

 

[Aminta_1900]

OK thanks again for the help. I have been taught previously on exponential functions, only I was not familiar with the term "growth factor," at least not as definition for the b in the equation. Since most of the questions I have encountered with these functions have been in context of investing money I have only been familiar with

a = principle

r = interest rate + 1

x = number of periods

But maybe I had not being paying enough attention, or not having the ingenuity! I was not comfortable trying to find this type of function because, for one thing, I noticed the curve in the graph has what I'm sure are inflection points.

Also I remember getting a few questions wrong when that x was given with a denominator, as per the disparagement between days and weeks, since this was not clearly shown in examples I had been given. But I can see more clearly now where this 14 = one two-week interval on the graph. I certainly need more practice there.

I'll study these graphs closely to get a better picture of how this estimate was taken. All the best.

 

[Dr.Peterson]

It sounds like they didn't prepare you quite as well for this problem as I expected, not knowing the context.

I was not comfortable trying to find this type of function because, for one thing, I noticed the curve in the graph has what I'm sure are inflection points.

Probably what you're noticing is primarily that one point (70, 50) that's significantly higher than expected. That's just a wiggle in the data, which you will find in real life. We don't expect anything real to follow an exact exponential curve! On the other hand, if you were an entomologist, you might look at that anomaly and look for something that changed at that time, which might make you consider a different curve, perhaps two separate exponentials.

They really should have communicated more of these ideas to you; that's why I'm telling you a little more than I ordinarily might.