Limit of Newton's Law of Cooling....1

[nycmathdad]

Given u(t) = (u_0 - T)e^(kt) + T, find the limit of u(t) as t tends to positive infinity.

The answer is T. How is the answer found? Seeking a hint or two.

 

[jonah1]

Beer soaked ramblings follow.

Given u(t) = (u_0 - T)e^(kt) + T, find the limit of u(t) as t tends to positive infinity.

The answer is T. How is the answer found? Seeking a hint or two.

Problem 1.5.75.a.
Some details left out.
Suggest you post a screenshot of the entire problem instead of making helpers guess the condition for k.

 

[HOI]

Given u(t) = (u_0 - T)e^(kt) + T, find the limit of u(t) as t tends to positive infinity.

The answer is T. How is the answer found? Seeking a hint or two.

That is simply NOT true! as t goes to positive infinity, e^{kt} goes positive infinity. u(t) would go to negative infinity, not T. It is true if t tended to negative infinity or if the exponential were e^(kt).

 

[nycmathdad]

That is simply NOT true! as t goes to positive infinity, e^{kt} goes positive infinity. u(t) would go to negative infinity, not T. It is true if t tended to negative infinity or if the exponential were e^(kt).

I posted the answer given in the textbook. Let me look screen shot the question.

Look at 75 parts (a) and (b).

 

[jonah1]

Beer soaked ramblings follow.

That is simply NOT true! as t goes to positive infinity, e^{kt} goes positive infinity. u(t) would go to negative infinity, not T. It is true if t tended to negative infinity or if the exponential were e^(kt).

As I intimated, nycmathdad omitted the detail that k<0.

 

[nycmathdad]

Now that we know my error (or should I call it sin), how is 75 done?

 

[jonah1]

Beer soaked ramblings follow.

Now that we know my error (or should I call it sin), how is 75 done?

If you have done your reading, it should be clear to you what $\lim_{t \to \infty} e^{kt}$ if k<0.
If not, I suggest you whip out your calculator and try out values for increasing values of t for a specific constant value of k<0 like -1/2.

 

[HOI]

In 75, you are given that \(\displaystyle u(t)= (u_0- T)e^{kt}+ T\) AND it is specified that k< 0. As t goes to infinity \(\displaystyle e^{kt}\) goes to 0 so u(t) goes to \(\displaystyle (u_0- T)(0)+ T= T\)

 

[nycmathdad]

In 75, you are given that \(\displaystyle u(t)= (u_0- T)e^{kt}+ T\) AND it is specified that k< 0. As t goes to infinity \(\displaystyle e^{kt}\) goes to 0 so u(t) goes to \(\displaystyle (u_0- T)(0)+ T= T\)

I don't understand why e^(kt) goes to 0 as t goes to positive infinity. In other words, why does e^(kt) become zero?

 

[jonah1]

Beer soaked ramblings follow.

I don't understand why e^(kt) goes to 0 as t goes to positive infinity. In other words, why does e^(kt) become zero?

For every action, there is an equal and opposite reaction.
If you didn't read your book, then you you won't understand why.

 

[nycmathdad]

Thank you everyone. I will look further into this chapter.