If F and G are constant-less indefinite integrals, must there exist an x such that F(x)+G(x) = F(20+G(10) ?

[Hormuz]

Let F and G be any two real-valued functions continuous in the closed interval between 10 and 20, and obtained by integration of other functions. I am wondering whether there must always be a value of x, between 10 and 20, such that F(x) + G(x) = F(20) + G(10). Is there a theorem that would have this as a consequence? And if not, can you find a counter-example? I haven't been able to. Many thanks for any thoughts you might have on this.

 

[Dr.Peterson]

Let F and G be any two real-valued functions continuous in the closed interval between 10 and 20, and obtained by integration of other functions. I am wondering whether there must always be a value of x, between 10 and 20, such that F(x) + G(x) = F(20) + G(10). Is there a theorem that would have this as a consequence? And if not, can you find a counter-example? I haven't been able to. Many thanks for any thoughts you might have on this.

First, what is a "constant-less indefinite integral"? What does that add to "continuous"?

More important, (a) why would that be relevant, and (b) why do you think this would be true? Where did the question come from? What example did you have in mind?

My first thought is, what if the maximum of F occurs at x=20, and the maximum of G occurs at x=10? What are your thoughts?

 

[Dr.Peterson]

Let F and G be any two real-valued functions continuous in the closed interval between 10 and 20, and obtained by integration of other functions. I am wondering whether there must always be a value of x, between 10 and 20, such that F(x) + G(x) = F(20) + G(10). Is there a theorem that would have this as a consequence? And if not, can you find a counter-example? I haven't been able to. Many thanks for any thoughts you might have on this.

I had another thought. Are you assuming that F and G are antiderivatives of positive functions, so that they are both increasing functions? That would make a big difference -- either in explaining why you are missing counterexamples (if the condition is not required, but you are accidentally assuming it), or why your conjecture (with that added condition) would be true.

When you clarify your own thinking for us, you will hopefully also be clarifying the conditions of your conjecture.

 

[Hormuz]

First, what is a "constant-less indefinite integral"? What does that add to "continuous"?

More important, (a) why would that be relevant, and (b) why do you think this would be true? Where did the question come from? What example did you have in mind?

My first thought is, what if the maximum of F occurs at x=20, and the maximum of G occurs at x=10? What are your thoughts?

What I meant (as you correctly surmised) is that F and G are antiderivatives, and in both cases the "constant of integration" is 0 (is there a better name for this constant when it is considered outside the context of integration?). The question arises in the context of two bodies at different temperatures, insulated from the environment, exchanging heat until they come to a thermal equilibrium. What will be their equilibrium temperature? The heat lost by the "hot" body must be equal to the heat gained by the "cold" body, and each of these heats is calculated as a definite integral of the given body's heat capacity (as a function of temperature) between the initial and the final temperatures. From this, one arrives at an equation in which the unknown is the final equilibrium temperature, and the question arises: why must this equation always have a solution? Taking 10 and 20 as attention-focusing "cold" and "hot" initial temperatures, the question reduces (I believe) to the one I posed.

I am not sure if your proposed maxima make a difference; my first impulse is to say no. But I will think of this some more.

 

[Hormuz]

I had another thought. Are you assuming that F and G are antiderivatives of positive functions, so that they are both increasing functions? That would make a big difference -- either in explaining why you are missing counterexamples (if the condition is not required, but you are accidentally assuming it), or why your conjecture (with that added condition) would be true.

When you clarify your own thinking for us, you will hopefully also be clarifying the conditions of your conjecture.

It should not be assumed that F and G are increasing functions, since heat capacities may, for some substances in some temperature regions, decrease with temperature. However, it is quite possible that in my search for counterexamples I was always unwittingly playing with increasing functions. I will try some others.

 

[Dr.Peterson]

What I meant (as you correctly surmised) is that F and G are antiderivatives, and in both cases the "constant of integration" is 0 (is there a better name for this constant when it is considered outside the context of integration?).

Technically, there is no one constant of integration; you may have run across the situation where you integrate a function two different ways, and find that they differ by a constant. All you really mean is that you chose an antiderivative, namely the one that looked most natural to you. Or you may mean the antiderivative whose value is zero at some meaningful point (such as, looking ahead, each initial temperature?). That would really be the definite integral from the specified starting point to any given x.

So this really contributes nothing to the problem! You simply have two functions that have derivatives, as far as I can see -- unless it says something about initial values.

The question arises in the context of two bodies at different temperatures, insulated from the environment, exchanging heat until they come to a thermal equilibrium. What will be their equilibrium temperature? The heat lost by the "hot" body must be equal to the heat gained by the "cold" body, and each of these heats is calculated as a definite integral of the given body's heat capacity (as a function of temperature) between the initial and the final temperatures. From this, one arrives at an equation in which the unknown is the final equilibrium temperature, and the question arises: why must this equation always have a solution? Taking 10 and 20 as attention-focusing "cold" and "hot" initial temperatures, the question reduces (I believe) to the one I posed.

It sounds like one temperature is increasing while the other is decreasing, so they have to meet in the middle; but I doubt that has anything to do with what you are asking about. I'm not at all sure I'm clear about what you are actually integrating, or what you are solving.

Evidently x is not time, but perhaps temperature? And F and G represent the heat content of each object at a given temperature? So you want to find a common temperature at which the total heat content would be the same as it was initially?

Perhaps you could provide a link to an example of what you are asking about, including graphs?

And could you show us an example or two that you tried?

My guess is that the problem as you stated it does not quite match your real problem.

It should not be assumed that F and G are increasing functions, since heat capacities may, for some substances in some temperature regions, decrease with temperature. However, it is quite possible that in my search for counterexamples I was always unwittingly playing with increasing functions. I will try some others.

If F and G are heat content (not capacity) as a function of temperature, I can't picture how that could ever decrease. Their derivatives might, but that's a different issue; perhaps you are mixing them up. But I haven't seen ideas like enthalpy (is that the word?) or heat capacity in a looooong time.

 

[Hormuz]

Technically, there is no one constant of integration; you may have run across the situation where you integrate a function two different ways, and find that they differ by a constant. All you really mean is that you chose an antiderivative, namely the one that looked most natural to you. Or you may mean the antiderivative whose value is zero at some meaningful point (such as, looking ahead, each initial temperature?). That would really be the definite integral from the specified starting point to any given x.

So this really contributes nothing to the problem! You simply have two functions that have derivatives, as far as I can see -- unless it says something about initial values.

It sounds like one temperature is increasing while the other is decreasing, so they have to meet in the middle; but I doubt that has anything to do with what you are asking about. I'm not at all sure I'm clear about what you are actually integrating, or what you are solving.

Evidently x is not time, but perhaps temperature? And F and G represent the heat content of each object at a given temperature? So you want to find a common temperature at which the total heat content would be the same as it was initially?

Perhaps you could provide a link to an example of what you are asking about, including graphs?

And could you show us an example or two that you tried?

My guess is that the problem as you stated it does not quite match your real problem.


If F and G are heat content (not capacity) as a function of temperature, I can't picture how that could ever decrease. Their derivatives might, but that's a different issue; perhaps you are mixing them up. But I haven't seen ideas like enthalpy (is that the word?) or heat capacity in a looooong time.

I am sorry, but this is taking us further and further away from my original problem - which, I think, has by now been stated clearly and simply enough, and which does match - in fact, is - my "real" problem. You asked for background, so I provided it, but what I am interested in is some insight into my problem as stated. I gather that you don't know of any theorem in calculus that would be applicable.

 

[Dr.Peterson]

No, the real question is, are the functions F and G (not their derivatives) both increasing, or not; and have you tried examples in which one is increasing and the other is decreasing? You've left that hanging.

If they are both increasing, then the answer will be yes. If they are not, then it is easy to find a counterexample, if I am understanding it correctly.

I asked for some examples you tried, in order to get a better idea of your interpretation of the problem, to be sure I am not answering the wrong question. Seeing a couple graphs will make it a lot clearer.

 

[blamocur]

Let F and G be any two real-valued functions continuous in the closed interval between 10 and 20, and obtained by integration of other functions. I am wondering whether there must always be a value of x, between 10 and 20, such that F(x) + G(x) = F(20) + G(10). Is there a theorem that would have this as a consequence? And if not, can you find a counter-example? I haven't been able to. Many thanks for any thoughts you might have on this.

Cannot help thinking that I might be missing something here, but here is a simple counterexample:
[math]G(x) = x - 10[/math][math]F(x) = 30-x[/math][math]F(x)+G(x) = 20[/math][math]F(20) + G(10) = 10 + 0 = 10[/math]

 

[Hormuz]

Cannot help thinking that I might be missing something here, but here is a simple counterexample:
[math]G(x) = x - 10[/math][math]F(x) = 30-x[/math][math]F(x)+G(x) = 20[/math][math]F(20) + G(10) = 10 + 0 = 10[/math]

Both of these functions involve additive constants, which is what I was trying to rule out by saying "constant-less" or "constant of integration is 0". (There must be a correct way of conveying the idea, but I guess I haven't hit upon it.) However, one can simplify your counterexample even further by taking F to be any function and G to be -F. So let's restrict ourselves to increasing functions. Dr. Peterson says that if F and G are both increasing then the answer is yes. Why is that, please?

 

[Hormuz]

Evidently x is not time, but perhaps temperature? And F and G represent the heat content of each object at a given temperature? So you want to find a common temperature at which the total heat content would be the same as it was initially?

I apologise: I should have paid greater attention to your proddings in this post. Yes, x is temperature in Kelvin, and F(x) and G(x) represent the heat contents of, respectively, the "hot" body and the "cold" body at temperature x. So both functions are indeed increasing, and both are positive-valued. And the question is, if the initial temperatures of the two bodies are 20 and 10 (or a and b), what ensures that there will always exist a temperature (the equilibration temperature) such that the joint heat contents of the two bodies at that temperature are equal to their initial joint heat contents, at the two different temperatures?

 

[Dr.Peterson]

I apologise: I should have paid greater attention to your proddings in this post. Yes, x is temperature in Kelvin, and F(x) and G(x) represent the heat contents of, respectively, the "hot" body and the "cold" body at temperature x. So both functions are indeed increasing, and both are positive-valued. And the question is, if the initial temperatures of the two bodies are 20 and 10 (or a and b), what ensures that there will always exist a temperature (the equilibration temperature) such that the joint heat contents of the two bodies at that temperature are equal to their initial joint heat contents, at the two different temperatures?

Good. (It doesn't really matter that F and G are positive, but that can make it easier to think about.)

Both of these functions involve additive constants, which is what I was trying to rule out by saying "constant-less" or "constant of integration is 0". (There must be a correct way of conveying the idea, but I guess I haven't hit upon it.)

I'm not sure your thought about additive constants is meaningful, which would explain why you are having trouble expressing it. Adding a constant to either function won't affect the fact you are asking about, as it would add that constant to both sums.

(As an example of what I said before about the constant of integration, if you had the functions tan^2(x) and sec^2(x), neither has a visible constant, but they differ by a constant: sec^2(x) = tan^2(x) + 1. Would you say both are constant-less? Does that matter?)

So let's restrict ourselves to increasing functions. Dr. Peterson says that if F and G are both increasing then the answer is yes. Why is that, please?

We want to show that there is always some x such that F(x) + G(x) = F(20) + G(10).

We're now assuming that F and G are both increasing continuous functions, so F(x) < F(20) and G(x) > G(10) for all 10 < x < 20.

Suppose we define new functions f(x) = F(x) - F(20) and g(x) = G(x) - G(10). Further, define h(x) = f(x) + g(x) = F(x) - F(20) + G(x) - G(10).

Therefore

h(10) = F(10) - F(20) + G(10) - G(10) = F(10) - F(20) < 0,​

​

while

h(20) = F(20) - F(20) + G(20) - G(10) = G(20) - G(10) > 0​

Thus h is a continuous function that is negative at x=10 and positive at x=20. By the Intermediate Value Theorem, there must be a value of x between 10 and 20 such that h(x) = 0, so that F(x) + G(x) = F(20) + G(10).

You can see that this depends heavily on having increasing functions.

 

[Hormuz]

Good. (It doesn't really matter that F and G are positive, but that can make it easier to think about.)


I'm not sure your thought about additive constants is meaningful, which would explain why you are having trouble expressing it. Adding a constant to either function won't affect the fact you are asking about, as it would add that constant to both sums.

(As an example of what I said before about the constant of integration, if you had the functions tan^2(x) and sec^2(x), neither has a visible constant, but they differ by a constant: sec^2(x) = tan^2(x) + 1. Would you say both are constant-less? Does that matter?)


We want to show that there is always some x such that F(x) + G(x) = F(20) + G(10).

We're now assuming that F and G are both increasing continuous functions, so F(x) < F(20) and G(x) > G(10) for all 10 < x < 20.

Suppose we define new functions f(x) = F(x) - F(20) and g(x) = G(x) - G(10). Further, define h(x) = f(x) + g(x) = F(x) - F(20) + G(x) - G(10).

Therefore

h(10) = F(10) - F(20) + G(10) - G(10) = F(10) - F(20) < 0,​

​

while

h(20) = F(20) - F(20) + G(20) - G(10) = G(20) - G(10) > 0​

Thus h is a continuous function that is negative at x=10 and positive at x=20. By the Intermediate Value Theorem, there must be a value of x between 10 and 20 such that h(x) = 0, so that F(x) + G(x) = F(20) + G(10).

You can see that this depends heavily on having increasing functions.

Beautiful. Thank you ever so much, and my apologies again for being dense.