April 21st question


Assume that a certain network can be approximated by a particular continuous model, and its number of nodes increases at a rate given by the function \( f(t) = e^{\frac{-t}{1000}+10} \).

Given that at \( t=0 \), the number of nodes is 0, what is the approximate maximum number of nodes that this network can have? Tip: get an expression for the number of nodes \( N(t) \) and its limit when \( t \rightarrow \infty \).

  1. \( e^{\frac{-1}{3}+10} \).
  2. \( 1000\ e^{\frac{-1}{3}+10} \).
  3. \( 1000\ e^{10} \).
  4. \( e^{10} \).
  5. None of the above

Original idea by: Christian Konishi

Comentários

  1. Anônimo22 de abril de 2023 às 07:01

    Something is wrong with this question. According to the formula, when t = 0 the number of nodes is e¹⁰, not 0.

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    Respostas
    1. Anônimo22 de abril de 2023 às 11:19

      f(0) is indeed e^{10}, but f is not the number of nodes, it is the rate that they increase. The idea is that I can get an expression for the number of nodes by integrating it and applying the condition N(0) = 0

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