Download Ch 3 Exponential Functions Bl 3.notebook

Ch 3 Exponential Functions Bl 3.notebook
C
April 16, 2013
Chapter 3 Exponential Functions
Logistic Functions
Logarithmic Functions
`
Mar 17­9:06 PM
Exponential Functions
Equation: y = abx
a is the initial value
b is the growth factor and r is the rate
of growth (1 + r) = b
when b > 1 exponential growth (r > 0)
when 0 < b < 1 exponential decay (r < 0)
Verbal Representation
Exponential Growth
Decay
Compound Interest
Radiactive Decay
Population Growth
Half Life
Appreciation
Depreciation
Mar 17­9:12 PM
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Ch 3 Exponential Functions Bl 3.notebook
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Mar 17­9:45 PM
Explain how you know that this table of
values is exponential
Mar 17­9:46 PM
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Ch 3 Exponential Functions Bl 3.notebook
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Mar 17­9:10 PM
Mar 17­9:10 PM
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Ch 3 Exponential Functions Bl 3.notebook
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Finding the equation of an Exponential
Function
Given a starting point and the percent
of growth.
Ex: The population in Mexico grows at
a rate of 2.6% per year. In 1980 there
were 67.38 million people.
Growth Factor is b = 1 +.026
Let t=0 in 1980, then a = 67.38
P=67.38(1.026)t
Mar 29­8:23 PM
On August 2,1988 a US District Court
imposed a fine on the city of Yonkers NY
for defying a federal court order involving
housing desegregation. The fine started
out as $100 for the first day and doubled
daily until the city chose to obey the
court order.
What was the daily percent growth rate
of the fine? (200%)
Find a formula for the fine as a function
of t, the number of days since August 2,
1988. 100(2)x
If Yonkers waited 30 days before obeying
the court order, what was the fine?
100(230)=10,737,418,240
Mar 29­9:08 PM
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Ch 3 Exponential Functions Bl 3.notebook
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Class work/discussion
p270 1-10, 25 to 30, 31 to 34
p 279 1-18
Homework
p 270 # 11-14
p 279 # 21,29,31,32
Mar 17­10:28 PM
p 270 #11­12
x
f(x)
g(x)
­2
6
108
­1
3
36
0
3
36
1
3/2
12
2
3/8
4/3
13
14
(0,3) (2,6)
(0,2) (1, 2/e)
Apr 10­9:15 AM
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Ch 3 Exponential Functions Bl 3.notebook
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0
p 279
21) (0,4) (5,600.25)
29) (475,000) 3.75% each year
When 1 million
31) yr 1890 -> 6250 people growth 2.75%
yr 1915 _______ yr ? 50,000
yr 1940
32) yr 1910 -> 4200 rate2.25%
yr ? 20,000
Yr 1930
yr 1945
Apr 10­9:21 AM
Goal: Another exponential function
Discovering e
What is the number e?
Compound interest
limit of compounding
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Ch 3 Exponential Functions Bl 3.notebook
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Mar 17­9:56 PM
e = lim (1 + 1/x)x
x->
∞
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Ch 3 Exponential Functions Bl 3.notebook
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Mar 17­10:00 PM
Mar 17­10:01 PM
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Ch 3 Exponential Functions Bl 3.notebook
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The number e is a famous irrational number.
(almost as important as π)
It is found in many applications of
mathematics. The first few digits of e are
2.7182818284590452353602874713527
It is often known as Euler's number after
Leonhard Euler
Mar 29­9:19 PM
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Ch 3 Exponential Functions Bl 3.notebook
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lim(1+1/n)n = e
n->∞
Mar 29­9:33 PM
Any exponential function f(x) = abx can be
rewritten as f(x) = a ekx,
for an appropriately chosen real number
constant k.
If a>0 and k > 0, f(x) = a ekx is an exponential
growth function
If a>0 and k < 0, f(x) = a ekx is an exponential
decay function
Mar 17­10:19 PM
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Ch 3 Exponential Functions Bl 3.notebook
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The Natural Base e
Function
Family
Exponential
F(x) = ex
Domain
Continuity
all real numbers
Range (0,∞)
increasing
for all x
Decreasing
Bounded
below by
x axis
No local
extrema
End behavior
x -> ∞ y->∞
Vertical
Asymptote
None
Horzontal
Asymptotes
y=0
Symmetry
NO
x -> -∞ y-> 0
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Apr 15­10:35 AM
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Ch 3 Exponential Functions Bl 3.notebook
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Find the equation given two points
Find the equation of the exponential function
that goes through the point (-2,45/4) and
(1, 10/3)
Step 1 Write two equations using each of
the points in the form y = abx
45/4 = ab-2
10/3 = ab1
Step 2 Divide the equation so the a drops
out. and solve for b
27/8 = b-3 or 8/27 =b3
b = 2/3
Step 3 Use b and one of the points to solve
for a
10/3 = a(2/3)1
5=a
Step 4. Write equation with a and b
y = 5 (2/3)x
Mar 29­8:30 PM
Classwork
FMC Green Book
p 117 # 4-7
Precalc: p 126 # 21,25,
Homework Precalc Book
p 271 # 45-48, 55, 56
FMC p117 #13,15
Mar 29­8:41 PM
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Ch 3 Exponential Functions Bl 3.notebook
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Apr 12­11:12 AM
(-3, 3.1569)
( 2, 9.2256)
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Ch 3 Exponential Functions Bl 3.notebook
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Goal: Logistic Growth
Exponential and Logistic modeling
In many growth situations there is a limit to
the possible growth. The growth often
begins as an exponential manner, but the
growth eventually slows and levels out.
Ex: Population in a fixed enviroment
(fish in an aquarium)
Plant growth
Mar 17­10:33 PM
Logistic Growth Function
Let a,b,c and k be positive constants,
with b < 1.
A logistic growth function in x is a
function that can be written in the
form
c
c
f(x) =
or f(x) =
1 + a e-kx
1 + abx
where the constant c is the limit to
growth.
Mar 17­10:39 PM
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Ch 3 Exponential Functions Bl 3.notebook
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Basic Logistic Function
Function
Family
Exponential
Logistic
F(x) =
Domain
all real numbers
Range (0,1)
1
1 + e-x
Continuous
Concaved up
Concaved Down
(-∞,0)
(0,∞)
increasing
for all x
Decreasing
Bounded
below and
above
No local
extrema
Vertical
Asymptote
None
Horzontal
Asymptotes
y=0, y = 1
End behavior
x -> ∞ y->1
Symmetry
NO
x -> -∞ y-> 0
Mar 17­10:44 PM
Graph the function
f(x) = 20/(1+2e-3x)
Determine the horizontal asymptotes
Y-intercept
Mar 17­10:58 PM
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Ch 3 Exponential Functions Bl 3.notebook
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p270 #49 Logistic Function
Function
Family
Exponential
Logistic
F(x) =
Domain
Continuous
Concaved up
Concaved Down
Range
increasing
5
1 + 4e-2x
Bounded
Vertical
Asymptote
Horzontal
Asymptotes
End behavior
Symmetry
x -> ∞ y->
Decreasing
local
extrema
x -> -∞ y->
Mar 17­11:05 PM
graph
y-intercept
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Ch 3 Exponential Functions Bl 3.notebook
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Ex 1
Suppose a radioactive substance decays at a
rate of 3.5% per hour. What percent of the
substance is left after 6 hours.
Mar 17­11:11 PM
Ex 2
A new franchise of fast food
restaurants is expected to grow
at a rate of 8% per year. There
are presently 200 restaurants.
What is the expected number of
restaurants in 15 years?
Mar 17­11:14 PM
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Ch 3 Exponential Functions Bl 3.notebook
April 16, 2013
Each year the local country club sponsors a tennis tournament. Play starts with 128 participants. During each round, half of the players are eliminated. How many players remain after 5 rounds?
Mar 17­11:18 PM
In the movie the blob, a substance
doubles every 3 hours. The initial mass
of a substance is .6 grams. What is the
hourly growth rate? How large is the
substance after 24 hours? after
3days?
Mar 17­11:24 PM
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Ch 3 Exponential Functions Bl 3.notebook
April 16, 2013
Classwork p 280 #24,26,28, 34(Use e), 43
Group discussion; 35-38
Homework p280 # 23,25, 27,33, 39, 44
Mar 17­11:18 PM
Apr 16­12:48 PM
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Ch 3 Exponential Functions Bl 3.notebook
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Apr 16­1:10 PM
Solve the equations
5x+1 = 54
72x+1=73x-2
Apr 16­9:58 AM
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Ch 3 Exponential Functions Bl 3.notebook
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Solving equation without logs
Rewrite the equation in the same
base
Work with only the exponents
Example
2(X+5) = 8(2x+1)
Apr 15­11:38 AM
32x-1=27x
53x-8=252x
What about 5x=7
Can you do this the
same way?
Apr 16­9:52 AM
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Ch 3 Exponential Functions Bl 3.notebook
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What's my inverse?
What undoes x+3
x-6
5x
x/3
x2
x3
1/x
2x
ex
Mar 17­11:24 PM
The inverse of an exponential function
f(x)=bx is the logarithmic function with
the base of b.
If f(x) = bx then f-1(x) = logb(x)
If x > 0 and 0 < b≠1, then
y=logb(x) iff bY=x
Apr 6­11:07 PM
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Ch 3 Exponential Functions Bl 3.notebook
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log28 = 3 because 23=8
log3√3 =
log5(1/25)
log41
log77
Apr 6­11:14 PM
Basic Properties of Logarithms
• logb1 = 0
• logbb = 1
• logbby = y
• blogbx = x
Common log is base 10
• log 1 = 0
• log 10 = 1
• log10y = y
• 10logx = x
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Ch 3 Exponential Functions Bl 3.notebook
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Because of their special calculus properties, logarithyms with the natural base e are used in so many situations that they have a special notation for a log with base e.
THE NATURAL LOG ­ LN LN without a subscript is understood to mean logex =ln x
Basic Properties of Natural Logarithms
Let x and y be real numbers with x > y
• ln1 = 0 because e0=1
• ln e = 1 because e1=1
• lney = y because ey=ey
• eln x = x because ln x =ln x
Apr 6­11:37 PM
Classwork p291 1­30 all
Homework p 292 31­40
p 324 2,4,9,10 Apr 6­11:47 PM
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Ch 3 Exponential Functions Bl 3.notebook
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What does the graph of a logarithmic function look like. To draw a logarithmic graph first draw the function ex, now draw the feflection over y = x line
Apr 6­11:56 PM
The Natural Log (LN)
Function
Family
Inverse
function of
Exponential
ex
increasing
for all x
Decreasing
F(x) = ln x
Domain
Continuous
(0,∞)
Range
all real numbers
Bounded
on the left by
y axis
No local
extrema
End behavior
x -> ∞
y->∞
x ­> 0+
y -> -∞
Vertical
Asymptote
x=0
Horzontal
Asymptotes
none
Symmetry
NO
Apr 6­11:59 PM
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Ch 3 Exponential Functions Bl 3.notebook
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Class work p292 41­44
Apr 7­12:02 AM
Apr 7­12:02 AM
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