Diff EQs - Central Dauphin School District

What do all of these have to do with Calculus?!?!? Application of Differential Equations AP Calculus Mrs. Mongold Common Problems: Exponential Growth and Decay Compound Interest Radiation and half-life Newtons law of cooling Other fun topics Suppose we are interested in a quantity that increases at a rate proportional to the amount present. If we also know the amount at a certain time, we can find y as a function of time by solving the differential equation:

dy ky dx dy ky dx y e kt e C dy kdx y y Ce kt dy y kdx ln y kt C y e kt C When t = o, y = y0.

Solving for C, C = y0 y y0 e kt Initial Amount The rate of change of a population of rabbits is proportional to the number of rabbits present at any given time. If 10 rabbits are present initially, and 195 rabbits are present in 6 months, how many rabbits will there be in 2 years? dr kr dt r Ce r (0) 10 r (6) 195 kt r 10e kt 195 10e r (24) ?

k ( 6) k (6) 19.5 e ln 19.5 6k k .4951 r 10e.4951t .4951( 24 ) r (24) 10e r (24) 1,446,974 Suppose you deposit $800 in an account that pays 6.3% annual interest. How much will you have 8 years later if interest is: A. compounded continuously B. compounded quarterly dA .063 A dt dA .063dt A

dA A .063dt ln A .063t C A e.063t C A Ce.063t A 800e .063( 8 ) r A P 1 k kt A $1324.26 .063 A 800 1 4

A $1319.07 ( 4 )(8 ) The rate at which the population of a group of organisms grows is directly proportional to the number of organisms in the population. If the population at time zero is 3,500 and the population after one year is 5,250, what will the population be after 3 years? dy ky dt y y0 e kt 5250 3500e k (1) 3 ln k 2 y 3500e t ln

3 2 y (3) 3500e ( 3) ln 3 2 y (3) 11,812.5 Half-Life y y0 e kt After t years, half of the original amount is left 1 y0 y0 e kt 2 1 e kt

2 1 ln kt 2 ln 1 ln 2 k t ln 2 k t t is the half-life. K does not depend on the initial amount! Example life of carbon-14 is 5700 years. Find the age of a sample in which 10% of the original radioactivity is depleted. 90% is still present

.9 e .9 y 0 y 0 e ln 2 t 5700 ln 2 ln .9 t 5700 866 t ln 2 t 5700 ln 2 ln 2 k t 5700

Newtons Law of Cooling dT k T Ts dt Rate of change of objects temp with respect to time is Proportional to Difference between temp of surroundings and temp of object When a murder is committed, the body, originally at 37C, cools according to Newtons Law of Cooling. Suppose on the day of a very difficult calc quiz, Mrs. Mongolds body is found at 2pm with a body temperature of 35C in a room with a constant temperature of 20C. 2 hours later the body temperature is 30C. Being the bright students you are, you should be able to tell Mr. Miller when the heinous crime was committed. dT

k T Ts dt dT T Ts kdt T 20 15e kt 30 20 15e k ( 2 ) ln T Ts kt C 10 15e k ( 2 ) 2 ln 2k 3 T Ts Ce kt k .203 T 20 Ce kt 35 20 Ce k ( 0 ) 15 C

37 20 15e ( .203)t T 20 15e ( .203) t 17 15e ( .203) t 17 ln .203t 15 .62 t min .62 60 37 min hr The crime was committed at 1:23 pm