6.1 Exponential Growth and Decay
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Transcript of 6.1 Exponential Growth and Decay
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6.1 Exponential 6.1 Exponential Growth and DecayGrowth and Decay
6.1 Exponential 6.1 Exponential Growth and DecayGrowth and Decay
With ApplicationsWith Applications
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Exponential Expression• An expression
where the exponent is the variable and the base is a fixed number
nb
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Multiplier• The base of an
exponential expression
2n
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Growth vs. Decay• When b>1, f(x) = bx represents
GROWTH
• When 0<b<1, f(x) = bx represents DECAY
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Applications• Exponential Growth and Decay
can be found in many applications• Ex: population growth, stocks,
science studies, compound interest, and effective yield
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Basic Growth/Decay Applications:
• When dealing with most growth and decay apps, you have an equation such as:
( )( )ny initial base
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• Base is your multiplier
• Growth: multiplier = 100% + rate
• Decay: multiplier = 100% - rate
( )( )ny initial base
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Growth/Decay App. WS Problem 1:
• The population of the United States was 248,718,301 in 1990 and was projected to grow at a rate of about 8% per decade. Predict the population, to the nearest hundred thousand, for the years 2010 and 2025.
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Growth/Decay App. WS
Problem 1:• Growth Application• Initial Population = 248,718,301• Multiplier = 100% + 8% = 108%
= 1.08• Expression to model the problem:248,718,301(1.08)n
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Growth/Decay App. WS
Problem 1:• 2010: 2 decades after 1990• n = 2
2
248,718,301(1.08)
248,718,301(1.08)
290,105,026.3
nPop
Pop
Pop
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Growth/Decay App. WS
Problem 1:• Round to the nearest hundred
thousand:• 290,100,000 = Population in 2010
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Growth/Decay App. WS
Problem 1:• 2025: 3.5 decades after 1990• n = 3.5
3.5
248,718,301(1.08)
248,718,301(1.08)
325,604,866
nPop
Pop
Pop
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Growth/Decay App. WS
Problem 1:• Round to the nearest hundred
thousand:• 325,600,000 = Population in 2025
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Population Formula WS• This is for homework, but it is for
you to practice writing population formulas.
• YOU DO NOT HAVE TO SOLVE ANYTHING….JUST WRITE THE FORMULAS
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Compound Interest Formula
• Another application of exponential growth
• The total amount of an investment A, earning compound interest is:
( ) (1 )ntr
A t Pn
P = Principal, r = annual interest rate, n = # of times interest is compounded per year, t = time in years
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Example• Find the final amount of a $500
investment after 8 years, at 7% interest compounded annually, quarterly, monthly, daily.
( ) (1 )ntr
A t Pn
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Example (cont)• P = $500• r = 7% = .07• t = 8 years
• Annually, n = 1
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Example (cont):Annually
1(8)
( ) (1 )
.07(8) 500(1 )
1(8) $859.09
ntrA t P
n
A
A
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Example (cont):Quarterly
• n = 4
4(8)
( ) (1 )
.07(8) 500(1 )
4(8) $871.11
ntrA t P
n
A
A
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Example (cont):Monthly
• n = 12
12(8)
( ) (1 )
.07(8) 500(1 )
12(8) $873.91
ntrA t P
n
A
A
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Example (cont):Daily
• n = 365
365(8)
( ) (1 )
.07(8) 500(1 )
365(8) $875.29
ntrA t P
n
A
A
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Effective Yield• The annually compounded interest
rate that yields the final amount of an investment.
• Determine the effective yield by fitting an exponential regression equation to 2 points.
• Effective Yield = b - 1
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Example• A collector buys a painting for
$100,000 at the beginning of 1995 and sells it for $150,000 at the beginning of 2000. Write an equation to model this situation and then find the effective yield.
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Example (Cont)• When modeling the situation, you
use the compounded interest formula, and you let n = 1 for compounded annually.
• A(t) = ending value = $150,000• P = initial = $100,000• n = 1 and t = 5 years
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Example (Cont)
1(5)
5
150,000 100,000(1 )1
150,000 100,000(1 )
r
r
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Example (Cont)• Now we need to find the effective
yield
• First we need 2 points that would model the data:
(0, 100000) and (5, 150000)
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Example (Cont)• Plug these points into your calc• STAT, EDIT
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Example (Cont)• Then generate the Exponential
Regression: STAT, CALC, 0:ExpReg• y = abx
• a = 100000• b = 1.084• Effective yield = 1.084 – 1 = .084
= 8.4% annual interest rate
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Homework:• Finish BOTH WS
• Pg 358 #15, 18, 21, 37, 42, 47, 48
• Pg. 367 #17-23odd, 29-33odd, 47, 49