Chapter 9: Modeling Our World Lecture notes Math 1030 Section CSection C.1: The Exponential FunctionExponential functionA exponential function grows or decay by the same relative amount per unit time. For any quantity Qgrowing exponentially with a fractional growth rate rQ = Q0× (1 + r)twhereQ = value of the exponentially growing quantity at time tQ0= initial value of the quantity (at t = 0)r = fractional growth rate (which may be positive or negative) for the quantityt = timeNegative values of r correspond to exponential decay. Note that the units of time used for t and r mustbe the same. For example, if the fractional growth rate is 0.05 per month, then t must also be measured inmonths.RemarkYou may notice that the exponential function is identical to the compound interest formula if we identifyQ as the accumulate balance A, Q0as the starting principal P , r as the interest rate, and t as the number oftimes interest is paid. In other words, compound interest is a form of exponential growth.Ex.1 U.S. population growth.The 2000 census found a U.S. population of about 281 million with an estimated growth rate of 0.7% peryear. Write an equation for the U.S. population that assumes exponential growth at this rate. Use theequation to predict the U.S. population in 2100.1Chapter 9: Modeling Our World Lecture notes Math 1030 Section CEx.2 Declining population.China’s one-child policy was originally implemented with the goal of reducing China’s population to 700million by 2050. China’s population was about 1.2 billion. Suppose China’s 2000 population declines ata rate of 0.5% per year. Write an equation for the exponential decay of the population. Will this rate ofdecline be sufficient to meet the original goal?2Chapter 9: Modeling Our World Lecture notes Math 1030 Section CGraphing Exponential FunctionsHow to graph exponential functionsThe easiest way to graph an exponential function is to use points corresponding to several doubling times(or half-lives in the case of decay), that is the points(0, Q0), (Tdouble, 2Q0), (2Tdouble, 4Q0) · · ·in the case of exponential growth, or(0, Q0), (Thalf,Q02), (2Thalf,Q04) · · ·in the case of exponential decay.Ex.3The growth rate of the U.S. population has varied substantially during the past century. It depends onthe immigration rate, as well as birth and death rates. Starting from a 2000 population of 281 million,project the population in 2100 using growth rates that are just 0.2 percentage point lower and higherthan the 0.7% used in Example 1. Make a graph showing the population through 2100 for each growthrate.3Chapter 9: Modeling Our World Lecture notes Math 1030 Section CAlternative Form of Exponential FunctionsForms of the exponential functionLet Q represent the quantity at any time t and Q0represent the initial value.• If given the growth or decay rate r, use the exponential function in the formQ = Q0× (1 + r)tRemember that r is positive for growth and negative for decay.• If given the doubling time Tdouble, use the exponential function in the formQ = Q0× 2tTdouble• If given the half-life Thalf, use the exponential function in the formQ = Q0×12tThalfSection C.2: Selected ApplicationsInflationInflationBecause the prices tend to change with time, price comparisons from one time to another are meaningfulonly if the prices are adjusted for the effects of inflation. We can model the effects of inflation with anexponential function in which r represents the rate of inflation.Ex.4 Monthly and annual inflation rates.The U.S. government reports the rate of inflation (as measured by the Consumer Price Index) both monthlyand annually. Suppose that, for a particular month, the monthly rate of inflation is reported as 0.8%. Whatannual rate of inflation does this imply? Is the annual rate 12 times the monthly rate? Explain.4Chapter 9: Modeling Our World Lecture notes Math 1030 Section CPhysiological ProcessesPhysiological processes are exponentialMany physiological processes are exponential. For example, the concentration of many drugs in the blood-stream decays exponentially. Alcohol is an exception since its concentration decays linearly.Ex.5 Drug concentration.Consider an antibiotic that has half-life in the bloodstream of 12 hours. A 10-milligram injection of theantibiotic is given at 1:00 p.m. How much antibiotic remains in the blood at 9:00 p.m.? Draw a graph thatshows the amount of antibiotic remaning as the drug is eliminated by the