4.3: One-unit Growth/Decay Factor
4.3: One-unit Growth/Decay Factor MAT 170 EVANGELISTA - Precalculus
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This 6 page Class Notes was uploaded by Sarai843 on Thursday October 8, 2015. The Class Notes belongs to MAT 170 EVANGELISTA - Precalculus at Arizona State University taught by Knutson in Fall 2015. Since its upload, it has received 90 views. For similar materials see MAT 170: Precalculus in Mathematics (M) at Arizona State University.
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Date Created: 10/08/15
Thursday December 17 y Module 4 Exponential and Logarithmic Functions 43 Oneunit GrowthDecay Factor De nition of Exponential FuncUons Defn An exponential function is a function of the form fxabquotx ex let fx 512 x is fan exponential function yes a 5 b 12 let fx 214quotx4 is fan exponential function NO 4 let fx 24 2x is fan exponential function yes with algebra 24quot2x 24quot2quotx 216quotx aquotnquotm aAnm 4quotxquot2 aquotnaquotm aquotnm a is called the initial value b is the oneunit growth growth b gt 1 or decay b lt 1 we can always write b 1r for some number rr b1 ex b16 b0314 b1r 16115 b1r r15 O3141r r O68 r is called the oneunit percent change Thursday December 17 y Notes 1 We assume 6 gt 0 2 b is sometimes called the base We are NOT allowed to have a negative base for exponential functions with decimalsfractions as the exponent ex fx 52quotx f13 DNE f12 52quot12 f14 DNE f12 DNE Why is b called the oneunit growthdecay factor Supposed our input changes by oneunit from xto X1 The output will change from fX to fX 1 But fX ab X and fX 1 ab X1 Rules of exponents fx1 abquotx1 abquotxbquot1 fx1 bfx f4f3 b fx1fX b Why is rcalled the oneunit percent change Suppose a city s population increased by 5000 people last year Is this significant Maybe take into account the original population Thursday December 17 y Suppose the input changes by 1 from xto X1 The output will change from fx to fx1 The change in output is fx1fx The oneunit percent relative change is fX1fxfx abquotx1abquotXabquotx abquotxb1abquotx b1 r a is called the initial value because f0 abAO a Rememberl Anything to the power of Oequals 1 Thursday December 17 y Module 4 Precalculus Workbook Problems 43 Exponential and Logarithmic Functions Page 146 10 amp 12 x 1 2 3 4 gX 260 299 34385 691 605 1 O a Find the ratio of output values that correspond to increases of 1 in the input value in order to determine the growth or decay factor 1x1v1x b 1211 299260 115 b Determine the 1unit percent change by comparing the change in the output values to the function value at the beginning of a 1unit interval for X find the value of r b1r r1151 r150r15 o c Identify or determine the value of the function when X 0 find the value of a 1x abe 1x a115quotx 11 a115 260 a115 a 226057 d Use the information from parts a through c to determine a function formula for the relationship round to 1decimal place fX abAX fX 2261 1 15 X Thursday December 17 y 12 Look at P146 12 in Workbook Precalculus Pathways to Calculus A Problem Solving Approach 5th edition for the graph a Find the ratio of output values that correspond to increases of 1 in the input value in order to determine the growth or decay factor b f3lf4l 08 b Determine the 1unit percent change by comparing the change in the output values to the function value at the beginning of a 1unit interval for X 081 r O2 r c Identify or determine the value of the function when X O fX a08quotx f4 aO8quot4 12207 aO8quot4 a 1220708lA4 4999 0 Use the information from parts a through c to determine a function formula for the relationship fxl 5008le ALL CORRECT answers are underlined Thursday October 22 y More 43 Workbook Problems Module 43 Page 147 13 Let fx 34119 x a What does the number 34 represent for this function The number 34 represents the initial value or a for this formula b What does the number 119 represent for this function The number 119 represents the oneunit growth in this problems case factor c Fill in the blank Whenever X increases by 1 then new output value is o of the old output value The new output value is 119 To find this number you take the oneunit growth factor and change it to its percent form eX 119100 119 you are essentially using the growth or in some cases decay factor to find the new output value d What is the 1unit percent change and what does it tell us b1 r 1191 r r 19 or 19 It tells us the change in the oneunit orowthdecav factor
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