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

nycmathdad
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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.
 
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nycmathdad said:
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.
 
nycmathdad said:
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).
 
Country Boy said:
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).

Screenshot_20210402-184607_Drive.jpg
 
Beer soaked ramblings follow.
Country Boy said:
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.
 
Now that we know my error (or should I call it sin), how is 75 done?
 
Beer soaked ramblings follow.
nycmathdad said:
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.
 
In 75, you are given that [math]u(t)= (u_0- T)e^{kt}+ T[/math] AND it is specified that k< 0. As t goes to infinity [math]e^{kt}[/math] goes to 0 so u(t) goes to [math](u_0- T)(0)+ T= T[/math]
 
Country Boy said:
In 75, you are given that [math]u(t)= (u_0- T)e^{kt}+ T[/math] AND it is specified that k< 0. As t goes to infinity [math]e^{kt}[/math] goes to 0 so u(t) goes to [math](u_0- T)(0)+ T= T[/math]

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?
 
  • #10
Beer soaked ramblings follow.
nycmathdad said:
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.
 
  • #11
Thank you everyone. I will look further into this chapter.
 

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