Exponential Growth and Decay Dr. Dillon Calculus II SPSU Fall 1999 Today’s Goals Identify growth and decay problems Learn to solve growth and decay problems Which is which? “Growth’’ refers to exponential growth f ( x) k a x , where a 1 “Decay” refers to exponential decay f ( x) k a x , where 0 a 1 Recognize the Problem Key words – population – radioactive decay – Newton’s Law of Cooling One differential equation says it all dy k y, k a constant dx A differential equation? Information about the derivative of a function. To solve a differential equation, find the function. We want y, a function of x, which satisfies dy k y, k a constant dx Technicalities The TI-89 solves differential equations. This one is easy to solve by hand. One diff. eq. underlies every growth & decay problem. We only have to solve it once. The Solution dy dy ky k dx dx y dy y k dx ln ( y) kx C ye kxC y kx eC kx C ' , where C' C e e e Thus y C ' e kx Are we there yet? No. So far, we just have an outline of the type of problem we want to solve. Example A population of bacteria doubles in twenty minutes. How long will it take to triple in size? Where is the diff. eq.? If P is the size of any population at time t then P grows at a rate proportional to itself, i.e. dP k P, for some constant k dt Thus we know... P(t ) Ce kt where P(t) is the size of the population at time t It’s always true... that C is the value of the function (in this case P) when the variable (in this case t) is zero that k is a feature of the situation at hand (in this case, the bacteria in your petri dish) Showing that C=P(0) P(t ) C e k t P(0) C k 0 C 0 C e e which gives us… t k e)0(P ) t(P A Note about Notation dy If k y, then y y0e kx , where y0 is the value of y when x 0. dx dP If k P, then P P0e kt , where P0 is the value of P when t 0. dt If f ' ( x) k f ( x), then f ( x) f (0)e kx. dP(t ) If k t , then P(t ) P(0)e kt . dt Always ask... What do we know? What are we looking for? Find k k 20 P 2P(0) P(0)e P(0)e 20k 2e 20k ln( 2) 20k k ln( 2) / 20 thus, for this population the model is ln( 2) ln(2 ) t P P(0)e 20 P(0) exp t 20 Finally... Find t when P=3P(0) as follows ln( 2) ln( 2) 3P(0) P(0) exp t 3 exp t 20 20 ln( 2) 20 ln( 3) ln( 3) t t 20 ln( 2) The Moral A diff eq underlies every problem The solution is always of the form P(t ) P(0)e kt k is different in every problem Work with what you know to find what you seek.