Calculus Volume 2 · 免费预览
指数增长与衰减
微分方程 y 的导数等于 k y 具有指数解 y 等于 y0 e 的 kt 次方，可用于刻画种群增长、放射性、冷却和利息等过程，其中倍增常数与半衰期常数概括了相应的动力学行为。
1. The model equation
The model equation
2. Populations and radioactivity
Populations and radioactivity
3. Newton's law of cooling
Newton's law of cooling
4. Logarithmic linearization
Logarithmic linearization
5. Model equation
Which differential equation expresses proportional change, the growth and decay model?
6. Doubling time
A population grows as P(t) = P_0 e^(kt). What is its doubling time T, defined by P(T) = 2 P_0?
7. Cooling law solution
Newton's cooling dT/dt = k(T - T_env) with k negative has what solution form?
8. Sign of decay constant
For radioactive decay m(t) = m_0 e^(-kt) with k positive, what does the sign structure express?
9. Rule of 70
The rule of 70 estimates a continuously compounding account's doubling time as 70 over r when r is in percent. Where does the 70 come from?
10. Unlimited versus limited growth
The pure exponential P(t) = P_0 e^(kt) eventually outruns any environment. Which model adds the carrying capacity K to correct this?
11. Half-life computation
Carbon-14 has k approximately 1.21 times 10 to the -4 per year. Its half-life is ln 2 over k, about how many years as a round number? Answer with a number.
12. Cooling equilibrium
As t approaches infinity, e^(kt) with k negative approaches zero, so T(t) approaches what temperature? Answer like T with a subscript, spelled out.
13. Initial value reading
In y = y_0 e^(kt), what value does y take at t = 0? Answer like y with a subscript, spelled out.
14. Logarithm conversion
To solve e^(kt) = 2 for t, one applies the natural logarithm. As a single word, what function undoes the exponential? Answer in lowercase.