2) exponential growth and decay
Embed Size (px)
Transcript of 2) exponential growth and decay

Exponential functions
To explore the properties of exponential functions and their graphs.
To understand the nature and behavior of exponential growth and decay.
To solve problems leading to exponential equations

ExponentialFunctions
Inanexponentialfunctiontheunknownappearsintheexponentlikein:
f(x)=2x

Plot the graph of y=2x for 5 x 5

ExponentialFunctions
Inthisgraphasxgetslargerpositivevalues,theyvaluesgetverylarge,itapproachestoinfinity
Forlargernegativevaluesofx,theyvaluesgetsmaller,approachingtozero.Butitneverreacheszero.Wesaythatthexaxisisanasymptotetothegraph

Plot the graph of y=2x for 5 x 5
always increasing
always positive
asymptote:
yintercept:

OnGDCexplorethegraphsof:
g(x)=3x h(x)=4xf(x)=2x
y=2xy=3xy=4
x
y intercept:
asymptote:

Plotthegraphof
always decreasing
always positive
asymptote:
yintercept:
Note: can be written as

Conclusions: y=ax ,a>1
Exponential growth
Asymptote: y=0
y intercept: (0,1)
the graph is above the xaxis
asiny=2x
 Conclusions: y=ax ,0

Exponentialfunctionscanbeusedtomodelreallifesituationsas:growingofaninvestment,ofatumor,ofapopulation,etc.

Thenumberofbacteriainaculture,N,ismodelledbythefunction: N=1000x20.2t,t0wheretismeasuredindays.
Findtheinitialnumberofbacteriaintheculture.
Findthenumberofbacteriaafter5days.
Howlongdoesittakeforthenumberofbacteriatogrowto4000?
1000
2000
10days
UseyourGDCtoplotthegraphofN(t).

Mexico'spopulationisincreasingatarateof2%peryear.CalculatehowlongwillittakeforMexico'spopulationtodouble.

InNigeriathepopulationisincreasingatarateof6.2%peryear.CalculatehowlongwillittakeforNigeria'spopulationtodouble.

ThetemperatureT,indegreesCelsius,ofacoolingliquidismodelledbytheequation:
T=24+72(0.6)3t
wheretisthetimeinminutesafterthecoolingbegins.
a)Whatistheinitialtemperatureoftheliquid?b)Findthetemperatureoftheliquidafter2min.c)Howlongdoesittakefortheliquidtocoolto26oC?d)Whattemperaturedoesthemodelpredicttheliquidwilleventuallyreach?

Wehavebeenworkingwithexamplesofexponentialgrowthanddecayandweuseddifferentbases,baseslike10and2arecommonlyusedformodellingcertainapplications.Howeverthemostimportantbaseisanirrationalnumberdenotedwiththeletter"e".(e2.71828)

Doyouremembertheformulatocalculatecompoundinterest?
A:futurevalueC:initialvaluer:rateofinterestn:numberofperiods

wearegoingtostudywhathappenswhenwecompoundtheinterestcontinuously
wecanwriterasadecimal,thentheformulawillbe:

Ifwecompoundtheinteresttwiceayeartheformulawillbe:
Tosimplifythisstudy,let'smaker=1,C=1

Ifwecompoundtheinterestthreetimesayeartheformulawillbe:
Ifwecompoundtheinteresttwelvetimesayeartheformulawillbe:
wecancontinuetocompoundtheinterestdaily,minutely,secondly...

Usethemenutableinyourcalculatortocomplete:
compounding n
annualsemiannual
quaterly
monthly
daily
hourlyeveryminute
everysecond
12
4
12
365
8760
52560031536000
22.25
2.4414...
2.613035...
2.714567...
2.718126...2.71827921..2.71828247..

Nowmovetothegraphmenuandplotthegraphof
wesaythatthisfunctiontendstonumbere

Wesaythattheexpressiontendstoe:
The irrational number e ( Euler's number)
http://torus.math.uiuc.edu/eggmath/Expon/numbere.html
http://abcnews.go.com/Technology/WhosCounting/story?id=99501&page=1

Therearemanyrealsituationsofcontinuouschange,tomodelthemthemostsuitablefunctionis
Useyourcalculatortodrawthesefunctions.
continuousgrowth
continuousdecay

UseyourGDCtodrawtheexponentialfunction:
Inreallife,thegrowthofbacteriaandothernaturalphenomenasuchaspopulationgrowthfollowanexponentialmodel.

Solvetheequation:
Buteisjustanumber,soitcanbeusedasanyother
Useyourcalculatortofind:
e2 2e1 e2

Solvetheequation:

Solvetheequation:

Solveworksheetexponentialequations
Bookpage48,Ex2Bandpage51Ex2C
ExponentialEquationsandFunctions2014.docx
Y11 EVExponential Equations and Functions
1 NO GDC Solve the following equations
1)
5)
9)
2)
6)
10)
3)
7)
11)
4)
8)
12)
2 No GDC Solve the equations
1
1) 4x  3 (2 x) + 2=0
2) 25x  6 ( 5x) +5=0
3) 9x  10 (3x) +9 =0
4) 25 x 10 (5x)+25 =0
3 No GDC (a) Complete the table..
Function
yintercept
Increasing or Decreasing
Horizontal asymptote
Root
(b) Sketch each of the functions given in (a). Show clearly all the features found in the table.
4. No GDC Consider the function f(x) = p(0.5)x + q ,where p and q are constants. The graph of f(x) passes through the points (0, 6) and (1, 4) and is shown below.
(a)Write down two equations relating p and q.
(b)Find the value of p and of q.
(c)Write down the equation of the horizontal asymptote to the graph of f(x).
5. No GDC The following diagram shows the graph of y = 3x + 2. The curve passes through the points
(0, a) and (1, b). .
(a)Find the value of
(i)a;(ii)b.
(b)Write down the equation of the asymptote to this curve.
6.No GDC. Consider the function f(x) = 1.25 ax, where a is a positive constant and x 0.The diagram shows a sketch of the graph of f. The graph intersects the yaxis at point A and the line L is its horizontal asymptote.
(a)Find the ycoordinate of A.
The point (2, 1) lies on the graph of y = f(x)
(b)Calculate the value of a.
(c)Write down the equation of L.
7.Shiyun bought a car in 1999. The value of the car V, in USD, is depreciating according to the exponential model
V = 25 000 1.50.2t, t 0,
where t is the time, in years, that Shiyun has owned the car.
(a)Write down the value of the car when Shiyun bought it.
(b)Calculate the value of the car three years after Shiyun bought it. Give your answer correct to two decimal places.
(c)Calculate the time for the car to depreciate to half of its value since Shiyun bought it.
8.A rumour spreads through a group of teenagers according to the exponential model
N = 2 (1.81)0.7t
where N is the number of teenagers who have heard the rumour t hours after it is first started.
(a)Find the number of teenagers who started the rumour.
(b)Write down the number of teenagers who have heard the rumour five hours after it is first started.
(c)Determine the length of time it would take for 150 teenagers to have heard the rumour. Give your answer correct to the nearest minute.
Answers:
1.
2
1. x= 3
2. x= 2
3. x=2
4. x=5
5. x= 1/3
6. x=0
7. x=1/3
8. x=2/3
9. x=11/4
10. x=0, x=2
11. x=3/2
12. x=2
2. (1) x= 0, x=1 (2) x= 0, x=1 (3) x= 0, x=3 (4) x=1
4.(a)p + q = 6; 0.5p + q = 4 (b) p = 4, q = 2 (c) y = 2
5.(a)(i)a = 3 (ii) b = 2 (b) y = 2
6.(a)0.25 (b) a = 2 (c) y = 1.25
7.(a) 25 000 USD (b)19 601.32 USD (c) 8.55
8.(a)N = 2 (b) 16.0 (3 s.f.) (c) 624 minutes
SMART Notebook

Attachments
ExponentialEquationsandFunctions2014.docx
Y11 EVExponential Equations and Functions
1 NO GDC Solve the following equations
1)
5)
9)
2)
6)
10)
3)
7)
11)
4)
8)
12)
2 No GDC Solve the equations
1
1) 4x  3 (2 x) + 2=0
2) 25x  6 ( 5x) +5=0
3) 9x  10 (3x) +9 =0
4) 25 x 10 (5x)+25 =0
3 No GDC (a) Complete the table..
Function
yintercept
Increasing or Decreasing
Horizontal asymptote
Root
(b) Sketch each of the functions given in (a). Show clearly all the features found in the table.
4. No GDC Consider the function f(x) = p(0.5)x + q ,where p and q are constants. The graph of f(x) passes through the points (0, 6) and (1, 4) and is shown below.
(a)Write down two equations relating p and q.
(b)Find the value of p and of q.
(c)Write down the equation of the horizontal asymptote to the graph of f(x).
5. No GDC The following diagram shows the graph of y = 3x + 2. The curve passes through the points
(0, a) and (1, b). .
(a)Find the value of
(i)a;(ii)b.
(b)Write down the equation of the asymptote to this curve.
6.No GDC. Consider the function f(x) = 1.25 ax, where a is a positive constant and x 0.The diagram shows a sketch of the graph of f. The graph intersects the yaxis at point A and the line L is its horizontal asymptote.
(a)Find the ycoordinate of A.
The point (2, 1) lies on the graph of y = f(x)
(b)Calculate the value of a.
(c)Write down the equation of L.
7.Shiyun bought a car in 1999. The value of the car V, in USD, is depreciating according to the exponential model
V = 25 000 1.50.2t, t 0,
where t is the time, in years, that Shiyun has owned the car.
(a)Write down the value of the car when Shiyun bought it.
(b)Calculate the value of the car three years after Shiyun bought it. Give your answer correct to two decimal places.
(c)Calculate the time for the car to depreciate to half of its value since Shiyun bought it.
8.A rumour spreads through a group of teenagers according to the exponential model
N = 2 (1.81)0.7t
where N is the number of teenagers who have heard the rumour t hours after it is first started.
(a)Find the number of teenagers who started the rumour.
(b)Write down the number of teenagers who have heard the rumour five hours after it is first started.
(c)Determine the length of time it would take for 150 teenagers to have heard the rumour. Give your answer correct to the nearest minute.
Answers:
1.
2
1. x= 3
2. x= 2
3. x=2
4. x=5
5. x= 1/3
6. x=0
7. x=1/3
8. x=2/3
9. x=11/4
10. x=0, x=2
11. x=3/2
12. x=2
2. (1) x= 0, x=1 (2) x= 0, x=1 (3) x= 0, x=3 (4) x=1
4.(a)p + q = 6; 0.5p + q = 4 (b) p = 4, q = 2 (c) y = 2
5.(a)(i)a = 3 (ii) b = 2 (b) y = 2
6.(a)0.25 (b) a = 2 (c) y = 1.25
7.(a) 25 000 USD (b)19 601.32 USD (c) 8.55
8.(a)N = 2 (b) 16.0 (3 s.f.) (c) 624 minutes
SMART Notebook
Page 1Page 2Page 3Page 4Page 5Page 6Page 7Page 8Page 9Page 10Page 11Page 12Page 13Page 14Page 15Page 16Page 17Page 18Page 19Page 20Page 21Page 22Page 23Page 24Page 25Page 26Page 27Page 28Attachments Page 1