Exponential Growth and Decay

The model equation

Populations and radioactivity

Newton's law of cooling

Logarithmic linearization

Which differential equation expresses proportional change, the growth and decay model?

A population grows as P(t) = P_0 e^(kt). What is its doubling time T, defined by P(T) = 2 P_0?

Newton's cooling dT/dt = k(T - T_env) with k negative has what solution form?

For radioactive decay m(t) = m_0 e^(-kt) with k positive, what does the sign structure express?

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?

The pure exponential P(t) = P_0 e^(kt) eventually outruns any environment. Which model adds the carrying capacity K to correct this?

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.

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.

In y = y_0 e^(kt), what value does y take at t = 0? Answer like y with a subscript, spelled out.

To solve e^(kt) = 2 for t, one applies the natural logarithm. As a single word, what function undoes the exponential? Answer in lowercase.