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how long does vyvanse last - Rice University

📅 May 15, 2026 👤 scraface

May 15, 2026

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📌 P''(t) = 6t - 12.

📌 Evaluate at the critical points:

📌 P''(1) = 6(1) - 12 = -6 < 0 \Rightarrow \text{local maximum at } t = 1, \\

📌 P''(3) = 6(3) - 12 = 6 > 0 \Rightarrow \text{local minimum at } t = 3.

📌 Thus, the pressure reaches a local maximum at $ t = 1 $ hour (1:00 AM) and a local minimum at $ t = 3 $ hours (3:00 AM).

📌 \boxed{\text{Local maximum at } t = 1 \text{ hour, local minimum at } t = 3 \text{ hours}}

📌 Question: A zoologist observes that the number of daily interactions between two species in a reserve is modeled by the function $ I(t) = \frac{t^2 + 4}{t + 2} $, where $ t $ is time in days since the start of the study. Find the slant asymptote of $ I(t) $ as $ t \to \infty $.

📌 Solution: To find the slant asymptote of $ I(t) = \frac{t^2 + 4}{t + 2} $, we perform polynomial long division of $ t^2 + 4 $ by $ t + 2 $:

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\sum_{k=1}^{9} k^3 = \left( \frac{9 \cdot 10}{2} \right)^2 = (45)^2 =...
Now compute \( 2025 \mod 9 \). A quick way: sum the digits of 2025:
+ 0 + 2 + 5 = 9 \Rightarrow 2025 \equiv 0 \mod 9.
Alternatively, note that \( 45 \equiv 0 \mod 9 \), so \( 45^2 \equiv 0...
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We seek the smallest positive integer \( n \) such that:
n^3 \equiv 1 \pmod{1000}.
We solve \( n^3 \equiv 1 \pmod{8} \) and \( n^3 \equiv 1 \pmod{125} \)...
Step 1: Solve \( n^3 \equiv 1 \pmod{8} \)

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