AP Calculus BC

posted: 02-Mar-2026 & updated: 03-Mar-2026

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Unit 6

• Given the logistic differential equation $\dfrac{dA}{dt} = A\left(20-\dfrac{A}{4}\right)$ where $A(0) = 15$, what is $\lim_{t\to\infty} A(t)$?

  Solution: $$ \begin{eqnarray*} \frac{dA}{A(20-A/4)} = dt &\Longleftrightarrow& \left( \frac{1}{A} + \frac{1}{80-A} \right) dA = 20dt \\ &\Longleftrightarrow& \log A - \log (80 - A) = 20 t + C \\ &\Longleftrightarrow& \log \left( \frac{A}{80 - A} \right) = 20 t + C \\ &\Longleftrightarrow& \log \left( \frac{1}{80/A - 1} \right) = 20 t + C \\ &\Longleftrightarrow& \log ( 80/A - 1 ) = -20 t - C \\ &\Longleftrightarrow& 80/A = e^{-20 t - C} + 1 \\ &\Longleftrightarrow& A = \frac{80}{e^{-20 t - C} + 1} \end{eqnarray*} $$