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Home » Yes or No? The Oracle Says: You Don’t Want to Believe This!, - Abbey Badges

Yes or No? The Oracle Says: You Don’t Want to Believe This!, - Abbey Badges

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 I(t) = t - 2 + \frac{8}{t + 2}.

📌 As $ t \to \infty $, the term $ \frac{8}{t + 2} \to 0 $, so the slant asymptote is:

📌 \boxed{y = t - 2}

📌 Question: A herpetologist models the population of a rare frog species in Madagascar with the function $ F(t) = \frac{100t}{t^2 + 4} $, where $ t $ is time in years. What is the maximum population growth rate?

📌 Solution: To find the maximum of $ F(t) = \frac{100t}{t^2 + 4} $, we compute its derivative using the quotient rule:

📌 F'(t) = \frac{(100)(t^2 + 4) - 100t(2t)}{(t^2 + 4)^2} = \frac{100(t^2 + 4 - 2t^2)}{(t^2 + 4)^2} = \frac{100(4 - t^2)}{(t^2 + 4)^2}.

📌 Set $ F'(t) = 0 $:

📌 - t^2 = 0 \Rightarrow t^2 = 4 \Rightarrow t = \pm 2.

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