Exponential Functions and ModelsContrastDefinitionExplore ExponentialsSlide 5Contrast Linear vs. ExponentialWhich Job?ExampleCompounded InterestSlide 10Assignment ACompound InterestExponential ModelingGrowth FactorDecreasing ExponentialsSlide 16Solving Exponential Equations GraphicallyGeneral FormulaTypical Exponential GraphsUsing e As the BaseContinuous GrowthSlide 22Slide 23Converting Between FormsContinuous Growth RatesSlide 26Assignment BExponential Functions and ModelsLesson 5.3ContrastLinearFunctions Change at a constant rate Rate of change (slope) is a constantExponentialFunctions Change at a changing rate Change at a constant percent rateDefinitionAn exponential functionNote the variable is in the expon entThe base is aC is the coefficient, also considered the initial value (when x = 0)( )xf x C a= �Explore ExponentialsGiven f(1) = 3, for each unit increase in x, the output is multiplied by 1.5Determine the exponential function( )xf x C a= �x f(x)1 0.75234Explore ExponentialsGraph these exponentialsWhat do you think the coefficient C and the base a do to the appearance of the graphs?( )( )( ) 2 1.1( ) 4 0.9xxf xg x= �= �Contrast Linear vs. ExponentialSuppose you have a choice of two different jobs at graduationStart at $30,000 with a 6% per year increaseStart at $40,000 with $1200 per year raiseWhich should you choose?One is linear growthOne is exponential growthWhich Job?How do we get each nextvalue for Option A?When is Option A better?When is Option B better?Rate of increase a constant $1200Rate of increase changing Percent of increase is a constant Ratio of successive years is 1.06Year Option A Option B1 $30,000 $40,0002 $31,800 $41,2003 $33,708 $42,4004 $35,730 $43,6005 $37,874 $44,8006 $40,147 $46,0007 $42,556 $47,2008 $45,109 $48,4009 $47,815 $49,60010 $50,684 $50,80011 $53,725 $52,00012 $56,949 $53,20013 $60,366 $54,40014 $63,988 $55,600ExampleConsider a savings account with compounded yearly incomeYou have $100 in the accountYou receive 5% annual interestAt end of yearAmount of interest earnedNew balance in account1 100 * 0.05 = $5.00 $105.00 2 105 * 0.05 = $5.25 $110.25 3 110.25 * 0.05 = $5.51 $115.76 4 5 View completed tableCompounded InterestCompleted tableCompounded InterestTable of results from calculatorSet Y= screen y1(x)=100*1.05^xChoose Table (♦ Y)Graph of resultsAssignment ALesson 5.3APage 415Exercises 1 – 57 EOOCompound InterestConsider an amount A0 of money deposited in an accountPays annual rate of interest r percentCompounded m times per yearStays in the account n yearsThen the resulting balance An01m nnrA Am�� �= +� �� �Exponential ModelingPopulation growth often modeled by exponential functionHalf life of radioactive materials modeled by exponential functionGrowth FactorRecall formulanew balance = old balance + 0.05 * old balanceAnother way of writing the formulanew balance = 1.05 * old balanceWhy equivalent?Growth factor: 1 + interest rate as a fractionDecreasing ExponentialsConsider a medicationPatient takes 100 mgOnce it is taken, body filters medication out over period of timeSuppose it removes 15% of what is present in the blood stream every hourAt end of hour Amount remaining1 100 – 0.15 * 100 = 852 85 – 0.15 * 85 = 72.25345Fill in the rest of the tableWhat is the growth factor?Decreasing ExponentialsCompleted chartGraphGrowth Factor = 0.85Note: when growth factor < 1, exponential is a decreasing functionSolving Exponential Equations GraphicallyFor our medication example when does the amount of medication amount to less than 5 mgGraph the functionfor 0 < t < 25Use the graph todetermine when( ) 100 0.85 5.0tM t = � <General FormulaAll exponential functions have the general format:WhereA = initial valueB = growth ratet = number of time periods( )tf t A B= �Typical Exponential GraphsWhen B > 1When B < 1( )tf t A B= �Using e As the BaseWe have used y = A * BtConsider letting B = ekThen by substitution y = A * (ek)tRecall B = (1 + r) (the growth factor)It turns out that k r�Continuous GrowthThe constant k is called the continuous percent growth rateFor Q = a bt k can be found by solving ek = bThen Q = a ek*tFor positive aif k > 0 then Q is an increasing functionif k < 0 then Q is a decreasing functionContinuous GrowthFor Q = a ek*tAssume a > 0k > 0k < 0Continuous GrowthFor the functionwhat is thecontinuous growth rate? The growth rate is the coefficient of tGrowth rate = 0.4 or 40% Graph the function (predict what it looks like)0.43tQ e= �Converting Between FormsChange to the form Q = A*BtWe know B = ekChange to the form Q = A*ek*t We know k = ln B (Why?) 0.43tQ e= �94.5(1.076)tQ =Continuous Growth RatesMay be a better mathematical model for some situationsBacteria growthDecrease of medicine in the bloodstreamPopulation growth of a large groupExampleA population grows from its initial level of 22,000 people and grows at a continuous growth rate of 7.1% per year.What is the formula P(t), the population in year t?P(t) = 22000*e.071tBy what percent does the population increase each year (What is the yearly growth rate)?Use b = ekAssignment BLesson 5.3BPage 417Exercises 65 – 85
View Full Document