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温度与种群的建模
参数 $a$、$k$ 和 $c$ 在函数 $F(t) = c + ae^{-kt}$ 描述物体在周围环境中冷却或升温的过程中各自起什么作用？
1. Introduction
Introduction
2. Newton's Law of Cooling revisited
Newton's Law of Cooling revisited
3. A more realistic model for population growth
A more realistic model for population growth
4. Summary
Summary
5. Finding the cooling constant
An object warms as F(t) = 275 - 203 e^(-kt) and F(20) = 101. What is k?
6. Exact time to a target temperature
For F(t) = 34.6 e^(-0.0154t) + 37.7, how do you find when F = 42.4, and what does the average rate on [25, 30] mean?
7. Sensitivity of the cooling constant
If the same drink started at 65 degrees instead of 72.3, how would k change and why?
8. Logistic graph behavior
For P(t) = A/(1 + M e^(-kt)), where does the population grow fastest, and what does A represent?
9. Logistic parameters from three conditions
A logistic model has P(0) = 2 and approaches 9 as t grows. What is M?
10. Average rates along a logistic curve
A logistic population's average rates on [0,2], [2,4], [4,6], [6,8] are 0.650, 0.796, 0.836 and 0.747 thousand per year. What does the pattern show?
11. Logistic epidemic modeling
A cruise ship with capacity 5000 starts with 5 infected and reaches 20 after one day. In the logistic model, what is A and what is M?
12. A cooling model approaches the surrounding temperature
For positive a and b, log(ab) equals log(a) + log(b).