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by: Marquise Graham


Marquise Graham
GPA 3.8


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Class Notes
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This 10 page Class Notes was uploaded by Marquise Graham on Saturday September 19, 2015. The Class Notes belongs to MAC 1105 at University of Florida taught by Staff in Fall. Since its upload, it has received 19 views. For similar materials see /class/207049/mac-1105-university-of-florida in Calculus and Pre Calculus at University of Florida.

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Date Created: 09/19/15
L22 Exponential Functions Statement If a and x are real numbers with a gt O and a 1 then y ax is a uniquely defined real number Laws of Exponents If s t a and b are real numbers with a gt O b gt 0 then The exponential function with the base a is a function of the form f X ax where 01 gt0 and 01721 The domain of f is the set of all real numbers 239 Example For fx cax a gt O a 721 c 72 O evaluate fQD f X Example For f x ax b a 2 O evaluate fX1fx Comparing Exponential and Linear Models For an exponential model f x cax a gt O a 72 1 c i O for unit increases in the input the output changes by a factor of a base fxlfxa For a linear model f x ax b a 72 O for unit increases in the input the output changes by an additive a slope fxlfxa 240 Example Find the appropriate model for each of the problem a Wages of an Assembly Line Worker A manufacturer pays its assembly line workers 1150 per hour In addition workers receive a piecework rate of 075 per unit produced Write the equation for the hourly wages Win terms of the number of units x produced per hour 241 b Size of a Population Under ideal conditions a certain bacteria population is known to double every hour Suppose that there are initially lOO bacteria Write the equation for the size of the population N after t hours 242 Graphing Exponential Functions x fx2 fltxgtlj 2quot 2 g 9 B E 7 7 6 6 5 5 3 3 239 2 139 1 S 74 73 2 1 n 1 2 3 A 5 75 7A 73 72 r n 1 2 3 A S 1 x 1 x Note Use these graphs as templates for graphing fxax with agt1 onthe left and 0lta lt1 on the right Graphing Exponential Functions Using Transformations Example Use the graph of f x 2x to graph the following Give the domain and range gx 2 2 639 hx 2x 2 j iffy m 2x 539 244 Properties of the Function f x ax a gt O a 1 l Domain Range 2 Points 41 01 and 101 are on the graph a 3 The line y O is the horizontal asymptote ifagt1then ax gtO as x gt oo if0ltalt1then ax gtO asx gtoo 4 fx ax is increasing if 01 gt1 decreasing if 0 lt a lt l 5 f x ax is a onetoone function Recall f x is a onetoone function means that f f v ltgt u v Therefore an equivalent form of the property 5 is a 2 av ltgt u v This property is useful for solVing some exponential equations 245 Example Solve the equations ETC 9x 3 246 The base 6 Consider the expression 1 for n 123 n The values for some natural n are listed below 3 n1 11 2 n n10 11 2593742 n n100 11 22704813 n n1oooo 11 22718145 n n1000000 11 22718280 n n1ooooooooo 11 22718281 n Note 11 gt271828 asn gt00 n We denote 271828 2 e and write lim1 e 71 00 n 247 Graghing y 8 Example Graph the function Find its domain and range x1 y e 3 Z48


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