Exponential and Trigonometric Functions Lesson #1
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1 Epoetial ad Trigoometric Fuctios Lesso # Itroductio To Epoetial Fuctios Cosider a populatio of 00 mice which is growig uder plague coditios. If the mouse populatio doubles each week we ca costruct a table to show the populatio umber (P) after t weeks. We ca also represet this iformatio graphically as show below: Ca we fid a relatioship betwee P ad t? Eamie the followig: So, the relatioship that coects t P ad t is P = 00. This is a epoetial relatioship ad the graph above is a epoetial graph. t Notice that we could write P(t) = 00 as P is a fuctio of t. I the table above, we have evaluated P( t) for various values of t.
2 Evaluatig Epoetial Fuctios Recall that if we are asked to fid f( a), we fid the value of the fuctio whe = a. Eample: For the fuctio f() =, fid the values of: (a.) f(0) (b.) f() (c.) f( ) (a.) 0 f(0) = (b.) f() = (c.) f(0) = f() = 4 f(0) = f() = f( ) = f( ) = 4 f( ) = 4 Graphig Simple Epoetial Fuctios We will cosider the simplest epoetial fuctio y =. A table of values ca be costructed for y =.
3 The graph of y = is show below: Horizotal Asymptote We say that the graph of y = is asymptotic to the -ais, or the -ais is a horizotal asymptote for the graph of y =. All epoetial graphs have a horizotal asymptote, but ot ecessarily the -ais. λ For the geeral epoetial fuctio y = k a + c: o a ad λ cotrol how steeply the graph icreases or decreases o c cotrols vertical positio ad y = c is the equatio of the horizotal asymptote o if k > 0, a > if k > 0, 0< a < i.e., icreasig i.e., decreasig if k < 0, a > if k < 0, 0< a < i.e., decreasig i.e., icreasig
4 Horizotal Asymptotes From our previous discussio, we oted that for the geeral λ epoetial fuctio y = k a + c, y = c is the horizotal asymptote. We ca actually obtai reasoable accurate sketch graphs of epoetial fuctios usig: o the horizotal asymptote o the y-itercept o two other poits, say = ad = Eample: Sketch the graph of y =. For y =, the horizotal asymptote is y =. Whe = 0, 0 y = y = y = So, the y-itercept is. Whe =, Whe =, y = y = 4 y = y = y = 4 y =
5 Epoetial Growth I this sectio, we will eamie situatios where quatities are icreasig (growth) or decreasig (decay) epoetially. Populatios of aimals, people, bacteria, etc. usually grow i a epoetial way while radioactive substaces ad items that depreciate usually decay epoetially. Biological Growth Cosider a populatio of 00 mice, which uder favourable coditios is icreasig by 0% each week. To icrease a quatity by 0%, we multiply it by 0% or.. So, if P is the populatio after weeks, the P0 = 00 {the origial populatio} P = P. = P = P. = 00 (.) P = P. = 00 (.) etc. ad from this patter, we see that P = 00 (.). We ca also look at the situatio i this way: This is a eample of a geometric sequece ad we could have foud the rule to geerate it. Clearly r =. ad sice P = Pr, the P = 00 (.) for = 0,,,,... Fiacial Growth 0 A further eample showig compoudig growth could be that of a ivestmet of $5000 at 6% p.a. for a period of moths, where this iterest is paid mothly. 6% p.a. meas a mothly growth of 6% 0.5% =. So, each moth our ivestmet icreases by 0.5%. This meas that our ivestmet icreases to 00.5% of what it was the previous moth.
6 Sice 00.5% =.005, we have: A0 = 5000 A = A = A.005 = 5000 (.005) A = A.005 = 5000 (.005) etc. Therefore, i geeral: A 5000 (.005) =. We ca also look at the situatio i this way: This is a eample of a geometric sequece with A0 = 5000 ad r =.005. Cosequetly, A = Ar. So, 0 = for = 0,,,,... A 5000 (.005) Note that for growth to occur, r >. Eample: The populatio size of rabbits o a farm is give, approimately, by R = 50 (.07), where is the umber of weeks after the rabbit farm was established. (a.) What was the origial rabbit populatio? (b.) How may rabbits were preset after 5 weeks? (c.) How may rabbits were preset after 0 weeks? (d.) Sketch the graph of R agaist ( 0). (e.) How log it would it take for the populatio to reach 500? R = 50 (.07), where R is the populatio size is the umber of weeks from the start. 0 (a.) Whe = 0, R = 50 (.07) R = 50 R = 50 i.e., 50 rabbits origially. 5 (b.) Whe = 5, R = 50 (.07) R 7.95 i.e., 8 rabbits. 0 (c.) Whe = 0, R = 50 (.07) R 80.6 i.e., 8 rabbits.
7 (d.) (e.) From the graph, the approimate umber of weeks to reach 500 rabbits is 4. This solutio ca also be foud usig the solver facility of a graphig calculator: Aswer: 4.0 or by fidig the itersectio of ad y = 500 (as show) y = 50 (.07)
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