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# College Algebra MTH 103

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This 9 page Class Notes was uploaded by Donny Graham on Saturday September 19, 2015. The Class Notes belongs to MTH 103 at Michigan State University taught by Jennifer Powers in Fall. Since its upload, it has received 46 views. For similar materials see /class/207306/mth-103-michigan-state-university in Mathematics (M) at Michigan State University.

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Date Created: 09/19/15

41 Exponential Functions and Their Graphs In this section you will learn to evaluate exponential functions graph exponential functions use transformations to graph exponential functions use compound interest formulas An exponential function f with base I is defined by fx bx or y V where b gt 0 b at 1 and x is any real number Note Any transformation of y bx is also an exponential function Example 1 Determine which functions are exponential functions For those that are not explain why they are not exponential functions a fx2x7 Yes No b gxx2 Yes No c hx1x Yes No d fxxx Yes No e hx310 x Yes No t fx 3 15 Yes No g 8X3 15 Yes No h hx 2x 1 Yes No Example 2 Graph each of the following and find the domain and range for each function a f x 2X domain range 1 X b gx domain range Page 1 Section 41 Characteristics of Exponential Functions f x 7quot bgt1 0ltblt1 Domain Range Transformations of gng bx C gt 01 Order of transformations is H S R V orizontal gx 17 graph moves 0 units left gx bH graph moves 0 units right tretchghrink gx cbx graph stretches if c gt 1 Vertical graph shrinks if 0 lt c lt 1 StretchShrink gx b graph shrinks if c gt 1 Horizontal graph stretches if 0 lt c lt 1 Re ection gx bx graph re ects over the x axis gx b graph re ects over the y axis Xertical gx bx c graph moves up 0 units gx bx c graph moves down 0 units Page 2 Section 41 Example 3 Use f x 2X to obtain the graph gx 2 3 1 Domain of g Range of g Equation of any asymptotes of g f x ex is called the natural exponential function where the irrational number 6 approximately 2718282 is called the natural base The number 6 is defined as the value that 11 approaches as n gets larger and larger n Example 4 Graph fx ex gx 6H and hx eX on the same set of axes Page 3 Section 41 Periodic Interest Formula Continuous Interest Formula m r A P1 Azpen A 2 balance in the account Amount after t years P Elincipal beginning amount in the account r 2 annual interest rate as a decimal n number of times interest is compounded per year t time in years Example 5 Find the accumulated value of a 5000 investment which is invested for 8 years at an interest rate of 12 compounded a annually b semi annually c quaiterly d monthly e continuously Page 4 Section 41 41 Homework Problems 1 Use a calculator to find each value to four decimal places a 5 b 7 C 2 53 d 62 e 6 2 f e 25 g 7171 2 Simplify each expression without using a calculator Recall 17 17 17 and hwy If r f F l l a 6565 b 35 2 c 125 8 d 59 3 e 4242 1 bmb For Problems 3 14 graph each exponential function State the domain and range for each along with the equation of any asymptotes Check your graph using a graphing calculator 3 fx3x 4 fx3X 5 fx3 x 6 fx 7 fx2X 3 8 fx2x 3 9 fx2 5 5 10 fx 2 X 11 fx 2 31 12 fxk 4 13 fxe x2 14 fx e 2 15 10000 is invested for 5 years at an interest rate of 55 Find the accumulated value if the money is a compounded semiannually b compounded quarterly c compounded monthly d compounded continuously 16 Sam won 150000 in the Michigan lottery and decides to invest the money for retirement in 20 years Find the accumulated value for Sam s retirement for each of his options a a certificate of deposit paying 54 compounded yearly b a money market certificate paying 535 compounded semiannually c a bank account paying 525 compounded quarterly d a bond issue paying 52 compounded daily e a saving account paying 519 compounded continuously 41 Homework Answers 1 a 162425 b 4518079 c 0254 d 73891 e 1353 f 12840 g 3183 2 a 36 5 b 9 c 174 d 125 e 4 f 73 3 Domain ooooRange 000 y 0 4 Domain oo ooRange oo 0 y0 5 Domain oo ooRange 0 00 y0 6 Domain oo ooRange 0 00 y 0 7 Domain oo ooRange 3 00 y 3 8 Domain oo ooRange 0 00 y 0 9 Domain oo ooRange 5 00 y 5 10 Domain oo ooRange oo 0 y0 11 Domain oo ooRange oo 1 y1 12 Domain ooooRange 4oo y 4 13 Domain ooooRange 2 00 y2 14 Domain ooooRange oo0 y0 15 a 1311651 b 1314067 c 1315704 d 1316531 16 a 42944097 b 43120096 c 42572959 d 42435112 e 42353464 Page 5 Section 41 42 Applications of Exponential Functions In this section you will learn to 0 find exponential equations using graphs 0 solve exponential growth and decay problems 0 use logistic growth models Example 1 The graph of g is the transformation of f x 2X Find the equation of the graph of g HINTS 1 There are no stretches 0r shrinks 2 Look at the general graph and asymptote to determine any re ections and0r vertical shifts 3 Follow the point 0 1 on f through the transformations to help determine any vertical and0r horizontal shifts Example 2 The graph of g is the transformation of f x ex Find the equation of the graph of g Example 3 In 1969 the world population was approximately 36 billion with a growth rate of 17 per year The function f x 366001 describes the world population f x in billions x years after 1969 Use this function to estimate the world population in 1969 2000 2012 Page 1 Section 42 Example 4 The exponential function f x 8451012X models the population of Mexico f x in millions x years after 1986 a Without using a calculator substitute 0 for x and find Mexico s population in 1986 b Estimate Mexico s population to the nearest million in the year 2000 c Estimate Mexico s population to the nearest million this year Example 5 College students study a large volume of information Unfortunately people do not retain information for very long The function f x 8060 20 describes the percentage of information f x that a particular person remembers x weeks after learning the information without repetition a Substitute 0 for x and find the percentage of information remembered at the moment it is first learned b What percentage of information is retained after 1 week 4 weeks 1 year Radioactive Decay Formula I The amountA of radioactive material present at time t is given by A 1402 h where A0 is the amount that was present initially at t 0 and his the material s half life Example 6 The half life of radioactive carbon 14 is 5700 years How much of an initial sample will remain after 3000 years Example 7 The half life of Arsenic 74 is 175 days If 4 grams of Arsenic 74 are present in a body initially how many grams are presents 90 days later Page 2 Section 42 Logistic Growth Model Logistic growth models situations when there are factors that limit the ability to grow or spread From population growth to the spread of disease nothing on earth can exhibit exponential growth indefinitely Eventually this growth levels off and approaches a maximum level which can be represented by a horizontal asymptote Logistic growth models are used in the study of conservation biology learning curves spread of an epidemic or disease carrying capacity etc The mathematical model for limited logistic growth is given 6 c by f I b 0 b where a b and c are constants c gt 0 and b gt 0 1 we 1 516 As time increases t gt co the expression ae b a 7 and A gt Therefore y c is a horizontal asymptote for the graph of the function Thus 0 represents the limiting size 200000 1 1999600 become ill with in uenza t weeks after its initial outbreak in a town with 200000 inhabitants Example 8 The function f t describes the number of people f t who have a How many people became ill with the u when the epidemic began b How many people were ill by the end of the 4th week c What is the limiting size of f t the population that becomes ill d What is the horizontal asymptote for this function Example 9 The function f t 1L is a model for describing the proportion of correct responses 702 6 f t after t learning trials a Find the proportion of correct responses prior to learning trials taking place b Find the proportion of correct responses after 10 learning trials c What is the limiting size of f t as continued trials take place d What is the horizontal asymptote for this function e Sketch a graph of this function Page 3 Section 42

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