LETU MATH 1203 - Exponential Functions and Models

Unformatted text preview:

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 rateDefinitionAn exponential functionNote the variable is in the expon entThe base is aC is the coefficient, also considered the initial value (when x = 0)( )xf x C a= �Explore ExponentialsGiven f(1) = 3, for each unit increase in x, the output is multiplied by 1.5Determine the exponential function( )xf x C a= �x f(x)1 0.75234Explore ExponentialsGraph these exponentialsWhat 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. ExponentialSuppose you have a choice of two different jobs at graduationStart at $30,000 with a 6% per year increaseStart at $40,000 with $1200 per year raiseWhich should you choose?One is linear growthOne 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 $1200Rate 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 $55,600ExampleConsider a savings account with compounded yearly incomeYou have $100 in the accountYou 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 InterestCompleted tableCompounded InterestTable of results from calculatorSet Y= screen y1(x)=100*1.05^xChoose Table (♦ Y)Graph of resultsAssignment ALesson 5.3APage 415Exercises 1 – 57 EOOCompound InterestConsider an amount A0 of money deposited in an accountPays annual rate of interest r percentCompounded m times per yearStays in the account n yearsThen the resulting balance An01m nnrA Am�� �= +� �� �Exponential ModelingPopulation growth often modeled by exponential functionHalf life of radioactive materials modeled by exponential functionGrowth FactorRecall formulanew balance = old balance + 0.05 * old balanceAnother way of writing the formulanew balance = 1.05 * old balanceWhy equivalent?Growth factor: 1 + interest rate as a fractionDecreasing ExponentialsConsider a medicationPatient takes 100 mgOnce it is taken, body filters medication out over period of timeSuppose 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 ExponentialsCompleted chartGraphGrowth Factor = 0.85Note: when growth factor < 1, exponential is a decreasing functionSolving Exponential Equations GraphicallyFor our medication example when does the amount of medication amount to less than 5 mgGraph the functionfor 0 < t < 25Use the graph todetermine when( ) 100 0.85 5.0tM t = � <General FormulaAll exponential functions have the general format:WhereA = initial valueB = growth ratet = number of time periods( )tf t A B= �Typical Exponential GraphsWhen B > 1When B < 1( )tf t A B= �Using e As the BaseWe have used y = A * BtConsider letting B = ekThen by substitution y = A * (ek)tRecall B = (1 + r) (the growth factor)It turns out that k r�Continuous GrowthThe constant k is called the continuous percent growth rateFor Q = a bt k can be found by solving ek = bThen Q = a ek*tFor positive aif k > 0 then Q is an increasing functionif k < 0 then Q is a decreasing functionContinuous GrowthFor Q = a ek*tAssume a > 0k > 0k < 0Continuous GrowthFor the functionwhat is thecontinuous growth rate? The growth rate is the coefficient of tGrowth rate = 0.4 or 40% Graph the function (predict what it looks like)0.43tQ e= �Converting Between FormsChange to the form Q = A*BtWe know B = ekChange to the form Q = A*ek*t We know k = ln B (Why?) 0.43tQ e= �94.5(1.076)tQ =Continuous Growth RatesMay be a better mathematical model for some situationsBacteria growthDecrease of medicine in the bloodstreamPopulation growth of a large groupExampleA 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.071tBy what percent does the population increase each year (What is the yearly growth rate)?Use b = ekAssignment BLesson 5.3BPage 417Exercises 65 – 85


View Full Document

LETU MATH 1203 - Exponential Functions and Models

Download Exponential Functions and Models
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Exponential Functions and Models and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Exponential Functions and Models 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?