Quiz 7-1:Quiz 7-1:

1. Where does the graph cross the y-axis?1. Where does the graph cross the y-axis?

2. f(1) = ?2. f(1) = ?

3. Horizontal asymptote = ?3. Horizontal asymptote = ?

4. How was the function transformed4. How was the function transformed to get f(x) above? to get f(x) above?

3)2(5)( 2 xxf

dabxf cx ))(1()1()(

5. Domain = ?5. Domain = ?

6. range = ?6. range = ?

xxg )2(5)(

7-2, and 7-37-2, and 7-3Exponential Decay Exponential Decay

And And

The growth/decay factor (base) “e”The growth/decay factor (base) “e”

Putting it all together:Putting it all together:

dabxf cx ))(1()1()(If negative:If negative:

Reflect across x-axisReflect across x-axis

2)4.0(10)( xxf

Initial value:Initial value:Crosses y-axis hereCrosses y-axis here

Growth factor:Growth factor:

If negative:If negative:Reflect across y-axisReflect across y-axis

Horizontal shiftHorizontal shift

vertical shiftvertical shift

2)5(3)( xxf

xxf 2)7.0(4)(

5)1.1(6)( 2 xxf

Population GrowthPopulation Growth

changePPP 01

We can rewrite the change in the populationWe can rewrite the change in the population as some percentage of the original population.as some percentage of the original population.

The population at the end of one period of time equals theThe population at the end of one period of time equals the initial population plus the growth/decay of the population.initial population plus the growth/decay of the population.

change %001 PPP If “r” is the % change in population (decimal equivalent), If “r” is the % change in population (decimal equivalent), then we can rewrite that as: then we can rewrite that as:

rPPP 001 Factoring out the common factor results in: Factoring out the common factor results in:

)1(01 rPP

Population Growth Population Growth over “t” time periods:over “t” time periods:

trPtP )1()( 0 PopulationPopulation (as a (as a function of time)function of time)

InitialInitial populationpopulation

GrowthGrowth raterate

time time

It’s just a formula!!!It’s just a formula!!!

The initial population of a colony of bacteriaThe initial population of a colony of bacteria is 1000. The population increases by 50% is 1000. The population increases by 50% every hour. What is the population after 5 hours?every hour. What is the population after 5 hours?

5)50.01(1000)5( P 7593)5( P

Percent rate of changePercent rate of change (in decimal form)(in decimal form)

5)5.1(1000)5( P

Interest (savings account)Interest (savings account)kt

krPtA )1()(

AmountAmount (as a (as a function of time)function of time)

Initial amountInitial amount (“principle”)(“principle”)

Interest Interest raterate

time time

In problems like this, the words in the problem will give In problems like this, the words in the problem will give you all but one of the quantities in the equation. The you all but one of the quantities in the equation. The quantities are:quantities are: A(t) A(t) PP rr kk tt

Number of times theNumber of times the interest is paid per yearinterest is paid per year(compounded)(compounded)

It will be up to you to solve for the missing quantity.It will be up to you to solve for the missing quantity.

kt

krPtA )1()(

A bank account pays 3.5% interest per year compounded A bank account pays 3.5% interest per year compounded monthly. If you initially invest $200, monthly. If you initially invest $200, how much moneyhow much money will you have after 5 yearswill you have after 5 years? ?

5*12)12035.01(200$)5( A 19.238$)5( A

What is the unknown quantity? What is the unknown quantity? A(t) A(t) PP rr kk tt

A(5) = ?A(5) = ?

kt

krPtA )1()(

A bank account pays quarterly compounded interest. If A bank account pays quarterly compounded interest. If you initially invested $1000 and after 10 years had $2200 you initially invested $1000 and after 10 years had $2200 in your account, what was the annual interest rate? in your account, what was the annual interest rate?

10*4)41(10002200 r

0796.0r

What is the unknown quantity? What is the unknown quantity?

40)41(2.2 r

Divide left/right by 1000Divide left/right by 1000

4040thth root left/right root left/right

41019906.1 r Subtract 1, multiply by 4Subtract 1, multiply by 4

%96.7r

Your turn:Your turn: kt

krPtA )1()(

A bank account pays 14% interest per year A bank account pays 14% interest per year compounded quarterly. If you initially invest $2500, compounded quarterly. If you initially invest $2500, how much money will you have after 7 years? how much money will you have after 7 years?

1. 1.

2. 2. Five years after you made a single deposit in anFive years after you made a single deposit in an earning 3% compounded monthly, it contains $580.81.earning 3% compounded monthly, it contains $580.81. What was your initial deposit? What was your initial deposit?

Exponential DecayExponential Decay dabxf x )(

‘‘a’ is the a’ is the initial valueinitial value f(0) = ‘a’ f(0) = ‘a’ ‘‘b’ is called the b’ is called the decay factordecay factor

0 < ‘b’ < 10 < ‘b’ < 1

xxf )5.0(4)( Table of valuesTable of values

xx f(x)f(x)x)5.0(4

000)5.0(4 44

111)5.0(4 22

0

2

4

6

8

10

12

14

16

18

20

-3 -2 -1 0 1 2 3 4 5

x (input value)

f(x) (

outp

ut v

alue

s)

222)5.0(4 11

333)5.0(4 0.50.5

44 4)5.0(4 0.250.25

½½

½½

½½

½½-1-1 1)5.0(4

88-2-2

2)5.0(4 1616

‘‘d’ shifts everything up or downd’ shifts everything up or down

Exponential Growth and Exponential Growth and DecayDecay

xabxf )(

exponential growthexponential growth: growth factor > 1: growth factor > 1

exponential decayexponential decay: growth factor 0 < b < 1 : growth factor 0 < b < 1

Your turn:Your turn: dabxf x )(

For each of the following what is the:For each of the following what is the: a. “initial value”?a. “initial value”? b. “decay factor”?b. “decay factor”? c. “horizontal asymptote”c. “horizontal asymptote” d. Any reflections (across x-axis or y-axis)d. Any reflections (across x-axis or y-axis)

xxf )3.0(2)( 3.3.

4.4.xxg )5(10)(

5.5. 45.0)( xxf

Identifying the Parts of the Identifying the Parts of the function:function:

dabxf x )(‘‘aa’ is the ’ is the initial valueinitial value f(0) = ‘a’ f(0) = ‘a’

‘‘bb’ is called the ’ is called the decay factordecay factor (if 0 < b < 1) (if 0 < b < 1)

2)3.0(10)( xxf

Initial valueInitial value: : 1010 y-intercepty-intercept: f(0) = 10 + 2 = 12: f(0) = 10 + 2 = 12

decay factordecay factor: : 0.30.3

‘‘dd’ shifts graph up/down ’ shifts graph up/down andand is the horizontal asymptote is the horizontal asymptote

Horizontal asymptoteHorizontal asymptote: y = 2: y = 2

Graphing Exponential DecayGraphing Exponential DecayUse the “power of the calculator” or: Use the “power of the calculator” or:

0

1

2

3

4

5

6

7

8

9

10

-5 -4 -3 -2 -1 0 1 2 3 4 5

x (input value)

f(x)

(o

utp

ut

valu

es)

xxf )5.0(6)(

f(1) = ?f(1) = ?

3. Horizontal 3. Horizontal asymptoteasymptote

1. f(0) = ?1. f(0) = ?

2. Some other point2. Some other point

f(0) = 6f(0) = 6

f(1) = 3f(1) = 3

y = 0y = 0

Domain = ?Domain = ? Range = ?Range = ?All real #’sAll real #’s y > 0 y > 0

Where does the number ‘e’ come Where does the number ‘e’ come from?from?

Named after the Swiss mathematician Named after the Swiss mathematician Leonard Euler (1707 – 1783).Leonard Euler (1707 – 1783).

The number ‘e’ can be found on your calculatorThe number ‘e’ can be found on your calculator by “2nd” + “ln” + “1” = 2.718by “2nd” + “ln” + “1” = 2.718 281828459045 …281828459045 …

‘‘e’ is an e’ is an irrational numberirrational number like pi or the square like pi or the square root of a prime number. The numbers after theroot of a prime number. The numbers after the decimal point go on decimal point go on foreverforever without any repetition without any repetition of number patterns.of number patterns.

The slope of the The slope of the tangenttangent line at any point on the line at any point on the curve is curve is

The number ‘e’ The number ‘e’ (Named after Leonard Euler, a Swiss (Named after Leonard Euler, a Swiss mathematician)mathematician)

‘‘e’ is a very unique numbere’ is a very unique number

xexf )( xeThe slope of the The slope of the tangenttangent line at x = 0 is line at x = 0 is 10 eThe slope of the The slope of the tangenttangent line at x = 1 is line at x = 1 is ......59045235367182818284.21 e

Exponential Functions and Exponential Functions and the number ‘e’the number ‘e’

xabxf )(

kxaexf )(

Any exponential function of the form: Any exponential function of the form:

Can be written in the form:Can be written in the form:

Exponential Growth: k > 0Exponential Growth: k > 0

Exponential Decay: k < 0Exponential Decay: k < 0

Exponential Functions and Exponential Functions and ‘‘e’e’

What processes does ‘e’ What processes does ‘e’ have to do with?have to do with?

Think of a bacteria cell. OverThink of a bacteria cell. Over a certain period of time it splitsa certain period of time it splits from one cell into two cells. The from one cell into two cells. The number of bacteria doubles.number of bacteria doubles.

This type of growth occurs in “spurts”. This type of growth occurs in “spurts”. At one instant of time it is a single bacteria.At one instant of time it is a single bacteria.The next instant it is two bacteria. The next instant it is two bacteria. The number of bacteria doubles.The number of bacteria doubles.

A(t) = 2 M A(t) = 2 M In “t” time periods it In “t” time periods it doublesdoubles ‘t’ times. ‘t’ times.

t = 1t = 1

What processes does ‘e’ have to do What processes does ‘e’ have to do with?with?

The number e (2.718…) represents the maximum compound rate of The number e (2.718…) represents the maximum compound rate of growth from a process that grows at 100% for one time period. Sure, growth from a process that grows at 100% for one time period. Sure, you start out expecting to grow from 1 to 2. But with each tiny step you start out expecting to grow from 1 to 2. But with each tiny step forward you create a little “dividend” that starts growing on its own. forward you create a little “dividend” that starts growing on its own. When all is said and done, you end up with e (2.718…) at the end of When all is said and done, you end up with e (2.718…) at the end of 1 time period, not 2.1 time period, not 2.

Instead of in “spurts,” this type of growth occurs Instead of in “spurts,” this type of growth occurs continuouslycontinuously

growth of money in an account that pays interest continuouslygrowth of money in an account that pays interest continuously the decay of radioactive material which occurs continuouslythe decay of radioactive material which occurs continuously

t = 1t = 1

e= 2.718…e= 2.718…

The “natural” number ‘e’ The “natural” number ‘e’ worksworks perfectly with natural perfectly with natural processesprocessesExponential growth of populationsExponential growth of populations

Exponential decay of radioactive materialExponential decay of radioactive material

Your turn:Your turn:2eex

7. 7. simplifysimplify

6. 6. simplify simplify

53 42 ee x

?)3)(5( 2 eex 215 xe

Exponential Function (base Exponential Function (base ‘e’)‘e’)

What is the “initial value” ?What is the “initial value” ? xexf 23)(

Describe the transformation:Describe the transformation: 43)( 2 xexg

What is the horizontal asymptote ?What is the horizontal asymptote ?

Is it growth or decay?Is it growth or decay?

Where does Where does g(x)g(x) cross the y-axis ? cross the y-axis ?

Putting it all together:Putting it all together:

daexf kx )1()(

If negative:If negative:Reflect across x-axisReflect across x-axis

210)( xexf

Initial value:Initial value:Crosses y-axis hereCrosses y-axis here

Growth factor:Growth factor:

If negative:If negative:decaydecay

Horizontal asymptoteHorizontal asymptote

vertical shiftvertical shift

xexf 2.03)(

xexf 24)(

Your turn:Your turn:daexf

xk )(For each of the following what is the:For each of the following what is the: a. “initial value”?a. “initial value”? b. Growth or decay?b. Growth or decay? c. “horizontal asymptote”c. “horizontal asymptote” d. Any reflections (across x-axis)d. Any reflections (across x-axis)

32)( xexf8.8.

9.9.xexf 45.010)(

10.10. 4)( xexf

Your turn:Your turn: trPtP )1()( 0

11.11. The population of Detroit The population of Detroit decreasesdecreases by by 3% by by 3% every every

year. In 2000 the population was 1.5 million.year. In 2000 the population was 1.5 million.What was the population in 2009?What was the population in 2009?

12. 12. The population of a small town can be modeled by: The population of a small town can be modeled by:

What is the % change in population for every timeWhat is the % change in population for every time period ‘t’ ?period ‘t’ ?

tPtP )05.1()( 0

Value of a depreciating Value of a depreciating asset.asset. trPtA )1()(

According to tax law, the value of a piece of equipment can beAccording to tax law, the value of a piece of equipment can be “ “depreciated” and the depreciation can be used as a “businessdepreciated” and the depreciation can be used as a “business expense” to reduce the amount of taxes that you pay. expense” to reduce the amount of taxes that you pay.

2)20.01(000,20$)5( A

800,12$)2( A2)8.0(000,20$)2( A

A company buys a car for $20,000. It can depreciate A company buys a car for $20,000. It can depreciate the value of the car by 20% per year. What is the the value of the car by 20% per year. What is the value of the car after 2 years? value of the car after 2 years?

Your turn:Your turn: trPtA )1()(

A company is in debt for $300,000. It is A company is in debt for $300,000. It is reducingreducing its its debt at the annual rate of 15%. What is the company’sdebt at the annual rate of 15%. What is the company’s debt after 10 years? debt after 10 years?

13.13.

$59,062.32$59,062.32

continuous compoundingcontinuous compoundingRemember that base “e” is used for things that grow (or Remember that base “e” is used for things that grow (or

decrease) continuously instead of in “spurts.” decrease) continuously instead of in “spurts.”

The more frequently the interest is paid in a savings accountThe more frequently the interest is paid in a savings account the faster the money will grow for a given interest rate.the faster the money will grow for a given interest rate.

The fastest possible rate of growth for a given interest rateThe fastest possible rate of growth for a given interest rate occurs with occurs with continuous compoundingcontinuous compounding. .

Periodic compoundingPeriodic compounding

kt

krPtA )1()( 0

continuous compoundingcontinuous compounding

rtePtA 0)(

PERTPERT

Your turn:Your turn: rtePtA 0)( You put $500 into an account earning 5% interestYou put $500 into an account earning 5% interest which is compounded continuously. How muchwhich is compounded continuously. How much money will be in the account after 6 years?money will be in the account after 6 years?

14.14.

$675.93$675.93

15.15. Radioactive material “decays” Radioactive material “decays” continuouslycontinuously (to some other element). The “decay constant” of (to some other element). The “decay constant” of Carbon-14 is k = -0.0001258. If 50 grams of C-14Carbon-14 is k = -0.0001258. If 50 grams of C-14 decays for 2000 years, how much carbon-14 willdecays for 2000 years, how much carbon-14 will remain?remain?

38.9 gms38.9 gms

HOMEWORKHOMEWORK

Section 7-1 (page 482)Section 7-1 (page 482) 2, 4, 16, 18, 20, 24, 282, 4, 16, 18, 20, 24, 28 Section 7-2 (page 489)Section 7-2 (page 489) 4, 6, 8, 16, 18, 30a4, 6, 8, 16, 18, 30a

Section 7-3 (page 495)Section 7-3 (page 495) 6, 8, 20, 22, 32, 38, 40, 56 6, 8, 20, 22, 32, 38, 40, 56

(21 total problems)(21 total problems)