Exponential Functions and ModelsLesson 2By Carmen GoguBastrop High SchoolMath Models

Contrast

Linear Functions Change at a constant rate Rate of change (slope) is a constantExponential Functions Change at a changing rate Change at a constant percent rate

DefinitionAn exponential function

Note the variable is in the exponentThe base is aC is the coefficient, also considered the initial value (when x = 0)

Explore ExponentialsGiven f(1) = 3, for each unit increase in x, the output is multiplied by 1.5Determine the exponential function

xf(x)10.75234

Explore ExponentialsGraph these exponentials

What do you think the coefficient C and the base a do to the appearance of the graphs?

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 growth

Which Job?How do we get each next value 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 1.06

YearOption AOption B0$30,000$40,0001$31,800$41,2002$33,708$42,4003$35,730$43,6004$37,874$44,8005$40,147$46,0006$42,556$47,2007$45,109$48,4008$47,815$49,6009$50,684$50,80010$53,725$52,00011$56,949$53,20012$60,366$54,40013$63,988$55,600

ExampleConsider a savings account with compounded yearly incomeYou have $100 in the accountYou receive 5% annual interestView completed table

At end of yearAmount of interest earnedNew balance in account1100 * 0.05 = $5.00$105.00 2105 * 0.05 = $5.25$110.25 3110.25 * 0.05 = $5.51$115.76 45

Compounded Interest

Completed table

Sheet1

YearOption AOption BAt end of yearAmount of interest earnedNew balance in account

1$30,000$40,0001100 * 0.05 = $5.00$105.00

2$31,800$41,2002105 * 0.05 = $5.25$110.25

3$33,708$42,4003110.25 * 0.05 = $5.51$115.76

4$35,730$43,6004

5$37,874$44,8005

6$40,147$46,000

7$42,556$47,200

8$45,109$48,400

9$47,815$49,600

10$50,684$50,800

11$53,725$52,000

12$56,949$53,200

13$60,366$54,400

14$63,988$55,600

Sheet2

At end of yearAmount of interest earnedNew balance in account

00$100.00

1$5.00$105.00

2$5.25$110.25

3$5.51$115.76

4$5.79$121.55

5$6.08$127.63

6$6.38$134.01

7$6.70$140.71

8$7.04$147.75

9$7.39$155.13

10$7.76$162.89

Sheet3

Compounded InterestTable of results from calculatorSet Y= screen y1(x)=100*1.05^xChoose Table ( Y)

Graph of results

Compound 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 An

Simple Interest Simple interest is a type of fee that is charged (or paid) only on the amount borrowed (or invested), and not on past interest. Simple interest is generally used only on short-term notes often on duration less than one year. The amount invested (borrowed) is called the principal (P).

Simple Interest The interest (fee) is usually computed as a percentage of the principal (called the interest rate(r)) over a given period of time (unless otherwise stated, an annual rate). Formula : I=PrtI=interestP=principalr=annual simple interest rate ( in decimal form)t=time in years

Simple Interest When solving financial mathematics problems, ALWAYS specify all variables and their values. To buy furniture for a new apartment, Megan borrowed $4000 at 8% simple interest for 12 months. How much interest will she pay? P=$4,000r= 8%= 0.08t= 12 months= 1 year I=4,000*(0.08)*1= 320$

Future or Maturity Value for Simple Interest. If a principal P is borrowed at a rate r, then after t years the borrower will owe the lender an amount A that will include the principal P plus the interest I. Since P is the amount borrowed now and A is the amount that must be paid back in the future, P is often referred to as the present value and A as the future value. When loans are involved, the future value is often called the maturity value of the loan. A= P + Prt= P*(1+rt)

Simple Interest-Future valueFind the maturity value for a loan of $2000 to be repaid in 6 months with interest of 9.4%. P=$2,000r= 9.4%=0.094t= 6 months = 6/12=1/6 yearsA=P*(1+rt)A=2,000*(1+(0.094*1/6))=$2,031

Simple Interest-Present Value of an InvestmentIf you want to earn an annual rate of 15% on your investments, how much (to the nearest cent) should you pay for a note that will be worth $6,000 in 8 months?r= 15%=0.15P=?A= $6,000t= 8/12=2/3 years P=A/(1+rt)=6,000/(1+(0.15*2/3))=5,454.5$

Exponential ModelingPopulation growth often modeled by exponential function

Half life of radioactive materials modeled by exponential function

Growth FactorRecall formula new balance = old balance + 0.05 * old balanceAnother way of writing the formula new balance = 1.05 * old balanceWhy equivalent?

Growth factor: 1 + interest rate as a fraction

Decreasing 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 hourFill in the rest of the tableWhat is the growth factor?

At end of hourAmount remaining1100 0.15 * 100 = 85285 0.15 * 85 = 72.25345

Decreasing ExponentialsCompleted chart

GraphGrowth Factor = 0.85Note: when growth factor < 1, exponential is a decreasing function

Sheet1

At end of hourAmount Remaining

185.00

272.25

361.41

452.20

544.37

637.71

732.06

Sheet2

Sheet3

Chart1

85

72.25

61.4125

52.200625

44.37053125

37.7149515625

32.0577088281

Amount Remaining

At End of Hour

Mg remaining

Sheet1

At end of hourAmount Remaining

185.00

272.25

361.41

452.20

544.37

637.71

732.06

Sheet1

0

0

0

0

0

0

0

Amount Remaining

Sheet2

Sheet3

Solving Exponential Equations GraphicallyFor our medication example when does the amount of medication amount to less than 5 mgGraph the function for 0 < t < 25Use the graph to determine when

General FormulaAll exponential functions have the general format:

WhereA = initial valueB = growth ratet = number of time periods

Typical Exponential GraphsWhen B > 1

When B < 1

Using e As the BaseWe have used y = A * BtConsider letting B = ek

Then by substitution y = A * (ek)t

Recall B = (1 + r) (the growth factor)

It turns out that

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*t

For positive aif k > 0 then Q is an increasing functionif k < 0 then Q is a decreasing function

Continuous GrowthFor Q = a ek*tAssume a > 0

k > 0

k < 0

Continuous GrowthFor the function what is the continuous growth rate? The growth rate is the coefficient of tGrowth rate = 0.4 or 40% Graph the function (predict what it looks like)

Converting Between FormsChange to the form Q = A*Bt

We know B = ekChange to the form Q = A*ek*t

We know k = ln B (Why?)

Continuous Growth RatesMay be a better mathematical model for some situations

Bacteria growthDecrease of medicine in the bloodstream

Population growth of a large group

ExampleA 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 = ek

Assignment B

Lesson 5.3BPage 417Exercises 65 85 odd