Having difficulty

[littlord123]

The body temp of a female mammal varies over a foxed period. For humans the period is about 28 days. The temp, T, in Celcius, varies with, t, time in days and can be represented by the cubic function: T(t)=-0.0003t^3+0.012t^2-0.112t+36. Find the maximum and minimum temp of the mammal and what time do they occur.

 

[tkhunny]

#1 This is a TERRIBLE problem setup. A cubic function has AT MOST TWO min and max values so ANY modelling of periodic data is SEVERELY handicapped. One should CLEARLY specify the Domain to avoid errors created by the horrid model. I would STRONGLY advise that you quit looking at this model after 26 days or so or you mammal is likely to pass away due to unacceptably low body temperature.

#2 Having difficulty with what? You should have learned that the 1st Derivative is useful for finding turning points (minimum and maximum) and the 2nd derivative is useful for classifying them as a minimum or a maximum. Did you, or did you not find these two things in your course materials or classroom discussion? The answer had better be "yes" or there are limited possibilities: 1) Your teacher/professor is grossly incompetent, 2) You textbook was written by a 5th grader, or 3) You've been doing a lot of sleeping or day-dreaming when you should have been studying or otherwise paying attention.

Please show your best work on the first two derivatives and interpret the results. That much should get us most of the way to a solution.

 

[littlord123]

ya I know T'(t)=-0.0009t^2+0.024t-0.112
T"(t)=0.00018t+0.024
I guess I am just having difficulty with the algebra involved in finding the turning points from there.

 

[Dr.Peterson]

The body temp of a female mammal varies over a foxed period. For humans the period is about 28 days. The temp, T, in Celcius, varies with, t, time in days and can be represented by the cubic function: T(t)=-0.0003t^3+0.012t^2-0.112t+36. Find the maximum and minimum temp of the mammal and what time do they occur.

Please show us what particular difficulties you are having. I assume one of them isn't finding the meaning of "foxed".

One thing you should ask your teacher is, over what time interval are you to find the max and min? I suspect the intention was to look at t varying from 0 to (around) 28 days (and maybe then repeating, with a jump, unless it turns out to be 25 days); but that should have been stated.

ya I know T'(t)=-0.0009t^2+0.024t-0.112
T"(t)=0.00018t+0.024
I guess I am just having difficulty with the algebra involved in finding the turning points from there.

So, what is the first thing you do with the derivative to find a turning point? You should at least be able to write the equation.

 

[Deleted member 4993]

ya I know T'(t)=-0.0009t^2+0.024t-0.112
T"(t)=0.00018t+0.024
I guess I am just having difficulty with the algebra involved in finding the turning points from there.

At what value of "t" we get T'(t) = 0 ?

 

[tkhunny]

ya I know T'(t)=-0.0009t^2+0.024t-0.112
T"(t)=0.00018t+0.024
I guess I am just having difficulty with the algebra involved in finding the turning points from there.

Well, it is ugly. You're not going to get some nice, clean answer. Many students are pretty accustomed to things coming out with pretty numbers. It's time to give up that hope. They're just numbers. Proceed as you would if they were easier to represent.

 

[JeffM]

I think the statement of the problem is not quite so terrible. I suspect we do not have it word for word. "Foxed" and "Celcius" are clues.

Moreover a cubic NEVER has three extrema. It may have no local extrema or one local minimum and one local maximum so the domain issue is relevant only if the local extrema fall outside the domain. In any case, we can deduce from the repeated word "period "that the relevant domain for this specific problem is

[MATH]0 \le t \le 28.[/MATH]
Solving on that basis, there is one minimum and one maximum during the period.

 

[lex]

so the domain issue is relevant only if the local extrema fall outside the domain

Might the domain be relevant even if the local extrema are within the domain, if the minimum/maximum value of the function within the domain does not occur at a local minimum or maximum? E.g. the minimum might occur at the end of the interval.

 

[JeffM]

Might the domain be relevant even if the local extrema are within the domain, if the minimum/maximum value of the function within the domain does not occur at a local minimum or maximum? E.g. the minimum might occur at the end of the interval.

@lex

I think what I said was correct, both technically and informally. (Of course I am always a bit weak on technical aspects.)

The implication is that the underlying function is periodic (plausible with respect to a function modeling biological processes and reinforced by repeated use of the word "period"). It is also implied that the function is differentiable. We can assume that the underlying function has a domain that is greater than one period. (Human women live longer than 28 days.)

On those assumptions, which I admit I did not specify because they seemed obvious to me, we reach the following conclusion. The values of the function are bounded, and every local minimum value and every local maximum value that the function takes on will be found at some point in any closed interval that is wholly within the function's domain and has a length equal to the period.

We are told that the function is a cubic. That of course cannot be literally true. A cubic us not periodic, and its values are not bounded. A cubic may, however, equal our periodic function in a given closed interval with the length of one period if that cubic has both a local minimum and a local maximum at points not outside the endpoints of that closed interval.

Probably I could have said what I meant better than I did, but I do not think "outside" means I was excluding endpoints. In fact, I chose "outside" because I had thought about end points.

First, I'd appreciate an explanation of my error if there was one.

Second, the language here is tricky. By definition, local extrema cannot occur at end points of an interval. But local extrema of a function that is defined in a broader domain than just that interval can have local extrema at the end points of that interval. Probably, I should have made up a phrase like "sub-domain" to make my thought clearer.

 

[lex]

 

[JeffM]

Got it.

My point was that a cubic could not be a good model if the distance between two extrema exceeded the length of the period. Of course, it is not a very good model if there are discontinuities at the end points. Never checked that.

 

[Dr.Peterson]

Some of what is being discussed is what was behind my comments:

One thing you should ask your teacher is, over what time interval are you to find the max and min? I suspect the intention was to look at t varying from 0 to (around) 28 days (and maybe then repeating, with a jump, unless it turns out to be 25 days); but that should have been stated.

I had graphed this:



I saw that if the period was 28, there would be a jump; but the problem doesn't actually say anything about what the period is, except that it is "fixed", and that for humans it is 28.

If it were 25, the periodic function would be more reasonable:



But the problem doesn't even say to start at 0. That, to me, is what makes this a poorly written problem! Far too much is left to guessing. Of course, we can be reasonably sure that the local max and min are the "right" answers, but it doesn't say what needs to be said.

Of course, all that's going on here is an attempt to fit a simple problem (find the local max and min of a polynomial) into a "real-life" problem that isn't a good fit for it.

 

[JeffM]

Some of what is being discussed is what was behind my comments:

I had graphed this:

View attachment 25963

I saw that if the period was 28, there would be a jump; but the problem doesn't actually say anything about what the period is, except that it is "fixed", and that for humans it is 28.

If it were 25, the periodic function would be more reasonable:

View attachment 25964

But the problem doesn't even say to start at 0. That, to me, is what makes this a poorly written problem! Far too much is left to guessing. Of course, we can be reasonably sure that the local max and min are the "right" answers, but it doesn't say what needs to be said.

Of course, all that's going on here is an attempt to fit a simple problem (find the local max and min of a polynomial) into a "real-life" problem that isn't a good fit for it.

Although we have no clue what the problem itself said

 

[Dr.Peterson]

Although we have no clue what the problem itself said

I'd say we do have a clue -- it reads to me like it was copied by hand, with a couple typos and maybe slightly paraphrased. Mostly, we just don't know how complete it is.

At any rate, all the OP needs in order to answer the question as presumably intended is to solve a quadratic equation and check the signs of a second derivative.

 

[lex]

Yes. The missing info is what is the interval that is to be repeated. But, true, the question is probably just about straightforward calculus. I seem to have derailed the discussion from the essential.
Back to posts #3 to #7!

 

[tkhunny]

The Domain is ALWAYS relevant. I do hope the author of the problem understands that. If the OP simply copied it badly, or didn't realize the domain information was important, that's less worse.