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Class Note for MATH 1314 at UH


Class Note for MATH 1314 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 25 views.

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Date Created: 02/06/15
Math 1314 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 mount 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 infinity mt lim P 1 L mgtoo m This is a fairly complicated limit to evaluate so we will omit the details mt lim P 1 L Pequot mgtoo 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 Pe Present value P A6 All values are as defined above Example 1 Find the accumulated amount when 5000 is invested for 8 years in an account that pays 25 annual interest compounded continuously Example 2 Suppose your current annual salary is 50000 per year Suppose in ation remains at a at rate of 3 per year for the next 12 years What will your salary need to be in order to have the same purchasing power you have now Assume that in ation compounds continuously Exponential Functions Recall the graph of an exponential function such as f x 3 79724757574444 1234557291 This is an exponential growth function The function increases rapidly each time you increase the x value by 1 unit you multiply the preceeding y value by 3 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 We ll look at a function Qt Qoek for exponential growth problems and a different function Qt Qoe39k 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 tand 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 Qt kQOek and the decay rate can be found using Qt oneikt Exponential Growth Example 3 A biologist wants to study the growth of a certain strain of bacteria She starts with a culture containing 35000 bacteria After two hours the number of bacteria has grown to 53000 How many bacterial will be present in the culture 7 hours after she started her study What will be the rate of growth 7 hours after she started her study Assume the population grows exponentially Example 4 A think tank began a study of population growth in a small country 3 years ago At the beginning of the study the population was 4200000 Two years later it was 5100000 What will the population be 4 years form now What will the growth rate be in 4 years Assume the population grows exponentially Exponential Decay Example 5 At the beginning of a study there are 150 grams of a substance present After 23 days there are 1112 grams remaining How much of the substance will be present after 65 days What will be the rate of decay on day 65 of the study Assume the substance decays exponentially Example 6 A certain drug has a half life of4 hours Suppose you take a dose of 1000 milligrams of the drug How much of it is left in your bloodstream 45 hours later Example 7 The halflife of Carbon 14 is 5770 years Bones found from an archeological dig were found to have 18 of the amount of Carbon 14 that living bones have Find the approximate age of the bones 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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