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.
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 =.
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
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 =
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. = 00. 0 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.
Sice 00.5% =.005, we have: A0 = 5000 A = 5000.005 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.
(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)