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Note 15 for MATH 1314 with Professor Gross at UH


Note 15 for MATH 1314 with Professor Gross at UH

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This 6 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 15 views.

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Date Created: 02/06/15
M1314 lesson 15 1 Math 1314 Lesson 15 Exponential Functions as Mathematical Models In this lesson we will look at a few applications involving exponential functions We ll first consider some word problems having to do with money Next we ll consider exponential growth and decay problems Interest Problems From previous course work you may have encountered the compound interest formula mt A P 1LJ m P principal amount invested A accumulated amount r interest rate m number of times interest is compounded per year t time in years Now suppose we let the number of compounding periods increase that is we ll take the limit of this function as m goes to in nity mt lim P 1 L mam m This is a fairly complicated limit to evaluate so we will omit the details mt limP 1L Pequot mgtw m You may also have seen this formula before This is the interest formula to use when interest is compounded continuously We ll be interested in two kinds of problems those that ask for an accumulated amount and those that ask for present value We ll use two formulas Accumulated amount A Pequot Present value P Ae quot All values are as defined above Example 1 a Find the accumulated amount when 3000 is invested for 5 years in an account that pays 3 annual interest compounded continuously M1314 lesson 15 2 b Suppose you want to have 3000 in your savings account in 5 years The bank will pay 3 annual interest compounded continuously How much money should you deposit today so that you will have 3000 in 5 years Exponential Functions Recall the graph of an exponential function such as f x 3x 79724757574444 1234557291 This is an exponential growth function The function increases rapidly This kind of growth will occur for any exponential function where b gt 1 including f x 8 If f x we ll have the re ection of this graph about the y axis y 79724757574444 1234557291 This is an exponential decay function This kind of decay will occur for any exponential function where 0 lt b lt l M1314 lesson 15 3 We ll look at a function Qt Qoek for exponential growth problems and a different function Qt Qoe k for exponential decay problems In these formulas Q0 is the original amount of the substance or population under study Qt is the amount of the substance or population at time t and k is the growth constant or 7k is the decay constant depending on whether your problem is a growth problem or a decay problem We can find the rate of growth or rate of decay by finding the derivative of the growth or decay functions Thus the growth rate can be found using Q t kQOek found using Q t kQOe k and the decay rate can be Exponential Growth Example 2 A biologist wants to study the growth of a certain strain of bacteria She starts with a culture containing 25000 bacteria After three hours the number of bacteria has grown to 63000 How many bacterial will be present in the culture 6 hours after she started her study What will be the rate of growth 6 hours after she started her study Assume the population grows exponentially M1314 lesson 15 4 Example 3 A think tank began a study of population growth in a small country 5 years ago At the beginning of the study the population was 4500000 Three years later it was 6200000 What will the population be in 2 years What will the growth rate be in 2 years Assume the population grows exponentially Exponential Decay Example 4 At the beginning of a study there are 50 grams of a substance present After 17 days there are 387 grams remaining How much of the substance will be present after 40 days What will be the rate of decay on day 40 of the study Assume the substance decays exponentially M1314 lesson 15 5 Example 5 A certain drug has a halflife of4 hours Suppose you take a dose of 1000 milligrams of the drug How much of it is left in your bloodstream 28 hours later Example 6 The halflife of Carbon 14 is 5770 years Bones found from an archeological dig were found to have 22 of the amount of Carbon 14 that living bones have Find the approximate age of the bones M1314 lesson 15 6 From this lesson you should be able to Solve problems involving continuously compounded interest including problems that ask for accumulated amount and present value Solve problems involving exponential growth Solve problems involving exponential decay Find a rate of growth or decay


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