Notes on Exponential Growth and Decay Models | MATH 1920, Study notes of Calculus

Download Notes on Exponential Growth and Decay Models | MATH 1920 and more Study notes Calculus in PDF only on Docsity! Exponential Growth and Decay models An application of Differential Equations (Section 7.4) One model of population growth simply states that the growth rate of the population is proportional to the size of the population. If P (the dependent variable) is the population size at time t (the independent variable), this models says that dP dt  kP where k is the constant of proportionality called the growth-rate coefficient or the relative growth rate. Note that assuming Pt  0 for all t, then if k  0 then dPdt  0 (growth) while if k  0 then dP dt  0 (decay). Question #1: What is the equilibrium solution for this differential equation? Question #2: Can you guess what the nontrivial (non-equilibrium) solutions to this differential equation look like? Hint: What function’s derivative is a constant multiple of itself? The above assumptions for population growth also apply in other situations as well: in nuclear physics (radioactive decay) and in finance (continuously compounded interest) to name two. In general, if yt is a function that represents the quantity at time t of some substance whose rate of growth (or decay) is proportional to its size, then we have dy dt  ky y0  y0 where y0 is the initial quantity at time t  0. Note that this equation is separable. So we solve using the method of separation of variables (see textbook page 532) obtaining yt  y0ekt What is the significance of the relative growth rate k? If dydt  0.04y and t is measured in years, then y grows at a rate of 4% per year. Examples:  Population growth: #6 in textbook on page 532.  Radioactive decay: #8 in textbook on page 533 and radiocarbon dating (#11 on page 533).  Continuously compounded interest. The account balance, A, for an account that compounds interest n times per year at an annual rate of r for t years is A  P 1  rn nt where P is the initial investment (i.e. A0  P). What happens to this expression as we compound more and more frequently: monthly, daily, hourly, every second, every millisecond, etc... (i.e. what happens as n → )? limn→P 1  r n nt  P limn→ 1  r n nt  P limn→ 1  r n n r rt Now if we let m  nr and noting that m →  as n →  we have P limn→ 1  r n n r rt  P limm→ 1  1 m m rt  Pert This is what we call continuously compounded interest A  Pert HOMEWORK: Section 7.4 #’s 3,5,9,11,12,17