Exponential Growth and Decay Models Practice

Sample questions

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1. Sample 1difficulty 2/5

   The general solution of $dy/dt = k y$ is

   • A

     $y = e^{kt} + C$

   • B

     $y = k t^2/2 + C$

   • C

     $y = kt + C$

   • D

     $y = C e^{kt}$

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   Why

   Standard exponential growth/decay solution.

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2. Sample 2difficulty 2/5

   For the logistic $dP/dt = k P(1 - P/M)$, the population approaches

   • A

     $M$ (carrying capacity)

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   • B

     $0$

   • C

     $\infty$

   • D

     $k$

   Why

   Long-term limit is $M$ (assuming $0 < P_0 < M$ initially).

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3. Sample 3difficulty 3/5

   The logistic differential equation models population with

   • A

     $dP/dt = k$

   • B

     $dP/dt = k P$

   • C

     $dP/dt = k P (1 - P/M)$ with carrying capacity $M$

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   • D

     $dP/dt = -k P^2$

   Why

   Logistic equation has both growth and self-limiting term.

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4. Sample 4difficulty 3/5

   A bacteria population doubles every $3$ hours. Its growth constant $k$ satisfies $e^{3k} = 2$, so $k =$

   • A

     $3 \ln 2$

   • B

     $\dfrac{\ln 2}{3}$

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   • C

     $\dfrac{1}{3 \ln 2}$

   • D

     $\ln 2$

   Why

   $3k = \ln 2 \Rightarrow k = (\ln 2)/3$.

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5. Sample 5difficulty 3/5

   The function $P(t)$ shown is consistent with:

   • A

     Constant

   • B

     Exponential growth

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   • C

     Quadratic decay

   • D

     Linear growth

   Why

   The increasing concave-down then leveling shape suggests exponential or logistic growth, not linear.

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