2Law of Exponential Change Suppose we are interested in a quantity that increases ordecreases at a rate proportional to the amount present…Can you think of any examples???If we also know the initial amount of , we can model thissituation with the following initial value problem:Differential Equation:Note: k can be eitherpositive or negativeWhat happens in eachof these instances?Initial Condition:
3Law of Exponential Change Let’s solve this differential equation:Separate variablesIntegrateExponentiateLaws of Logs/Exps
4Law of Exponential Change Let’s solve this differential equation:Def. of Abs. ValueLet A = + eC–Apply theInitial Cond.Solution:
5Law of Exponential Change If y changes at a rate proportional to the amount present(dy/dt = ky) and y = y when t = 0, thenwhere k > 0 represents growth and k < 0 representsdecay. The number k is the rate constant of theequation.
6Compounding InterestSuppose that A dollars are invested at a fixed annual interestrate r. If interest is added to the account k times a year, theamount of money present after t years isInterest can be compounded monthly (k = 12), weekly (k = 52),daily (k = 365), etc…
7Compounding Interest Solution: What if we compound interest continuously at a rate proportionalto the amount in the account?We have another initial value problem!!!Differential Equation:Initial Condition:Look familiar???Solution:Interest paid according to this formula is compoundedcontinuously. The number r is the continuous interest rate.
8RadioactivityRadioactive Decay – the process of a radioactive substanceemitting some of its mass as it changes forms.Important Point: It has been shown that the rate at which aradioactive substance decays is approximately proportional tothe number of radioactive nuclei present…So we can use ourfamiliar equation!!!Half-Life – the time required for half of the radioactive nucleipresent in a sample to decay.
9Guided Practice 1. k = – 0.5, y(0) = 200 Solution: Find the solution to the differential equation dy/dt = ky, k aconstant, that satisfies the given conditions.1. k = – 0.5, y(0) = 200Solution:
10Guided Practice 2. y(0) = 60, y(10) = 30 Solution: or Find the solution to the differential equation dy/dt = ky, k aconstant, that satisfies the given conditions.2. y(0) = 60, y(10) = 30Solution:or
11Guided Practice (a) (b) Suppose you deposit $800 in an account that pays 6.3% annualinterest. How much will you have 8 years later if the interest is(a) compounded continuously? (b) compounded quarterly?(a)(b)
12Guided Practice This is always the half-life Find the half-life of a radioactive substance with the given decayequation, and show that the half-life depends only on k.Need to solve:This is always the half-lifeof a radioactive substancewith rate constant k (k > 0)!!!
13The sample is about 866.418 years old Guided PracticeScientists who do carbon-14 dating use 5700 years for its half-life. Find the age of the sample in which 10% of the radioactivenuclei originally present have decayed.Half-Life =The sample is about years old
14There were 1250 bacteria initially Guided PracticeA colony of bacteria is increasing exponentially with time. At theend of 3 hours there are 10,000 bacteria. At the end of 5 hoursthere are 40,000 bacteria. How many bacteria were presentinitially?There were 1250 bacteria initially
15Guided PracticeThe number of radioactive atoms remaining after t days in asample of polonium-210 that starts with y radioactive atomsis(a) Find the element’s half-life.Half-life =days
16Guided PracticeThe number of radioactive atoms remaining after t days in asample of polonium-210 that starts with y radioactive atomsis(b) Your sample is no longer useful after 95% of the initialradioactive atoms have disintegrated. For about how manydays after the sample arrives will you be able to use thesample?days