Name ____________________ AP Calculus AB Exponential Growth and Decay (6.4) Law of Exponential Change: Suppose we are told that a quantity, y, increases at a rate proportional to the amount present. If we know the amount present at t = 0 (say the amount is yo) we can find y as a function of t by solving the following initial value problem: dy ky dt Initial condition: y = yo when t = 0 Law of exponential change: If y changes at a rate proportional to the amount present ( dy ky ) and y = yo when t =0, then y y o e kt where k>0 represents growth and k<0 dx represents decay. K is called the rate constant of the equation. Name ____________________ AP Calculus AB Three examples of exponential change: Compound Interest Suppose you invest $10,000 in a bank account that pays 5% in interest a year. The interest can be given to you once at the end of the year, or at intervals throughout the year. When given at intervals throughout the year it is called compound interest. The advantage of compound interest is that you earn interest on the interest. Example: $10,000 at 5%. 1) Not compounded: After one year you’ll have $10,500 2) Compounded every three months (quarterly) .05 * 10,000 ) 1st Quarter - $10,125 ( 4 2nd Quarter - $10,125 + $126.56 = $10,251.56 3rd Quarter – $10,251.56 + $128.14 = $10,379.70 4th Quarter - $10,379.70 + $129.75 = $10,509.45 You get an extra $9.45 over not compounding Formula for compound interest is: r A(t ) Ao (1 ) kt k Where: Ao original amount invested A(t ) Amount at any time t r annual interest rate k # of times compounded per year What amount will we have after 1 year on our $10,000 (5% rate) if we compound: .05 12 ) $10,511.52 12 .05 52 Every week : A(1) 10,000(1 ) $10,512.46 52 : .05 365 Every day : A(1) 10,000(1 ) $10,512.67 365 .05 8760 Every hour : A(1) 10,000(1 ) $10,512.70 8760 Every month : A(1) 10,000(1 Name ____________________ AP Calculus AB Now let’s say that instead of being added at discreet intervals let’s say the interest is being compounded continuously, i.e. k . We have: r kt A(t ) Ao lim(1 ) k k r k t A(t ) Ao lim((1 ) ) k k k Let m r 1 mr t A(t ) Ao lim[(1 ) ] m m 1 m rt A(t ) Ao [ lim (1 ) ] m m 1 m Recall that lim (1 ) e m m A(t ) Ao e rt A(1) 10,000e.05 $10,512.71 Name ____________________ AP Calculus AB Another way to think about continuous compound interest is with the following differential equation: dA rA Where the rate of change in the amount is proportional to the amount dt and where r = interest rate, and A is the amount currently in the bank. Also, the initial condition is that at t=0 A=Ao. In this case: A(t ) Ao e rt Example: Suppose you have $1,000,000 deposited in an account that pays 10% interest annually. How much money will you have in ten years if: a) The bank compounds monthly b) The bank compounds continuously Name ____________________ AP Calculus AB Radioactive Decay – Occurs when atoms shed some of their mass and transform into a new element. The decay of a radioactive element is roughly proportional to the amount present. It can be modeled by: dy ky y y o e kt where y is the mass at any time t, yo is the initial amount of the dt element, and k is a constant. Half-life: The time required for half of a radioactive element to decay. Example: Find the half-life of a radioactive substance with decay equation: y y o e kt and show that the half-life depends only on k. y y o e kt 1 y o y o e kt 2 Name ____________________ AP Calculus AB Newton’s Law of Cooling: The rate at which an object’s temperature is changing is proportional to the difference between its temperature and the temperature of the surrounding medium. dT k (T TS ) dt T TS (T0 TS )e kt Where TO Initial temperatu re TS Surroundin g temperatu re T Temperatur e of substance at any given time t. Name ____________________ AP Calculus AB Example: A hardboiled egg at 98oC is put in a pan under running 18oC water to cool. After 5 minutes, the egg’s temperature is found to be 38oC. How much longer will it take the egg to reach 20oC? Name ____________________ AP Calculus AB Resistance proportional to velocity: The resistance (air) an object experiences is proportional to its velocity. The slower an object moves the less its forward progress is resisted by the air through which it passes. F kv V VO e k t m Where V = Velocity at any given time t Vo = Initial velocity m = Mass of object k = Constant Name ____________________ AP Calculus AB Example: For a 50kg ice skater the k is about 2.5 kg/s a) How long will it take for the skater to coast from 7 m/s to 1 m/s? b) How far will the skater coast until s/he comes to a complete stop? Problem Set: Day 1: p338 1-10, Day 2: p338 11, 13, 15, 19, 23 + handout