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Do Now
1.) If there are initially 100 fruit flies in a sample,
and the number of fruit flies decreases by one-half
each hour, How many fruit flies will be present
after 5 hours?
4 - 26 - 2012
2.) Sara bought 4 fish. Every month the number of
fish she has doubles. After 6 months she will have
how many fish.
4 - 27 - 2012
1.) Simplify.
2573(1  0.92)
Do Now
2.)

0.0292 
40001 

4 

3
18,211.40582
3.)
P 1  r 
t
4
4118.08
Evaluate using this formula when
P is 1219, r is 0.12, and t is 5.
12191  .12 
5
2148.29
Do Now
1.) How many half-lives would it take to have
a 700 gram sample of uranium reduce to under
3 grams of uranium ?
4 - 30 - 2012
2.) If there are initially 10 bacteria in a culture, and
the number of bacteria double each hour, find the
number of bacteria after 24 hours.
5 - 4 - 2012
Do Now
When a person takes a dosage of I milligrams of a
medicine, the amount A ( in milligrams) of
medication remaining in the person’s bloodstream
after t hours can be modeled by the equation .
A  I (0.71)
t
Using the formula, Find the amount of medication
remaining in a person’s bloodstream if the dosage was
500 mg and 2.5 hours has lapsed.
5 - 4 - 2012
Compound Interest
Do Now
A  P(1  )
r nt
n
You want to have $ 20,000 in your account
after 18 years. Find the amount your initial
deposit should be if the account pays 4.5%
annual interest compounded monthly.
Identify:
A=
P=
r=
n=
t=
5 - 4 - 2012
Compound Interest
Do Now
A  P(1  )
r nt
n
You want to have $ 20,000 in your account after 18 years.
Find the amount your initial deposit should be if the account
pays 4.5% annual interest compounded monthly.
A = 20,000
P= ?
r = .045
20000  P(1 
n = 12
.045 (12)(18)
12
)
20000  P(2.2445)
8,910  P
t = 18
5 - 8 - 2012
Do Now
y  350(1  .75)
In the equation
, which of the following is true?
10
a) There is a Growth Rate?
g) The initial amount is 350?
b) There is a Decay Rate?
h) The time period is 10 ?
c) The Decay Rate is 75% ?
I) “y” is the final amount?
d) The Decay Rate is 25% ?
j) This is an Exponential
….Growth
….Decay
e) The Decay Factor is .25 ?
f) The Decay Factor is (1 - .75) ?
Do Now
1.)
If you invested $ 2,000 at a rate of 0.6%
compounded continuously, find the
balance in the account after 5 years, use the
formula
(.006)( 5 )
rt
5 - 9 - 2012
A  Pe
A  2000e
$ 2,060.91
2. ) Simplify the Expression
25
18e
12
6e
13
3e
Do Now
5 - 10 - 2012
y  P(1  r )
t
Exponential Growth
y  P(1  r ) t
r

A  P 1  
n

a  Pe
rt
Exponential Decay
nt
Compounded Interest
ex) Compounded daily
Compounded monthly
Compounded quarterly
Continuously Compounded Interest
5 - 11 - 2012
Do Now
1.) RE-Write log 6 36  2
2.) RE-Write
10  1
3.) Evaluate
log 4 245
4.) Graph
0
y  log 4 x
in Exponential form
in Logarithmic form
p. 478
Ch 7.1 Exponential Growth
What you should learn:
Graph and use Exponential Growth
functions.
Write an Exponential Growth model
that describes the situation.
A2.5.2
7.1 Graph Exponential Growth Functions
Exponential Function
• f(x) = bx
where the base b is a positive
number other than one.
• Graph f(x) = 2x
Notice the end behavior
• As x → ∞ f(x) → ∞
• As x → -∞ f(x) → 0
• y = 0 is an asymptote
What is an Asymptote?
• A line that a graph approaches as you move
away from the origin
The graph gets closer
and closer to the line
y = 0 …….
But NEVER reaches it
2 raised to any power
Will NEVER be zero!!
y=0
Example 1
• Graph
y  3
1
2
x
• Plot (0, ½) and (1, 3/2)
• Then, from left to right, draw a
curve that begins just above the xaxis, passes thru the 2 points, and
moves up to the right
What do you think the Asymptote is?
y=0
Example 2
• Graph y = - (3/2)x
•
•
•
•
•
•
•
y=0
Plot (0, -1) and (1, -1.5)
Connect with a curve
Mark asymptote
D = ??
All reals
R = ???
All reals < 0
Example 3
Graph y = 3·2x-1 - 4
• Lightly sketch y = 3·2x
• Passes thru (0,3) & (1,6)
• h = 1, k = -4
• Move your 2 points to the right 1
and down 4
• AND your asymptote k units (4
units down in this case)
Now…you try one!
Example 4
• Graph y = 2·3x-2 +1
• State the Domain and
Range!
• D = all reals
• R = all reals >1
y=1
When a real-life quantity increases by a
fixed percent each year, the amount y of
the quantity after t years can be modeled
by the equation
t
y  a(1  r )
where
•
•
•
•
a - Initial principal
r – percent increase expressed as a decimal
t – number of years
y – amount in account after t years
Notice that the quantity (1 + r) is called the Growth Factor
Example
The amount of money, A, accrued at the end of n years when a certain
amount, P, is invested at a compound annual rate, r, is given by
A  P(1  r )
t
If a person invests $310 in an account that pays 8% interest
compounded annually, find the approximant balance after 5 years.
A  310(1  .08)
A = $455.49
5
Compound Interest
Consider an initial principal P deposited in an
account that pays interest at an annual rate, r,
compounded n times per year.
A  P(1  )
r nt
n
• P - Initial principal
• r – annual rate expressed as a
decimal
• n – compounded n times a year
• t – number of years
• A – amount in account after t
years
Compound Interest example
A  P(1  )
r nt
n
• You deposit $1000 in an account that pays 8% annual
interest.
• Find the balance after 1 year if the interest is compounded
with the given frequency.
• a) annually
b) quarterly
c) daily
A=1000(1+ .08/1)1x1
=
1000(1.08)1
≈ $1080
A=1000(1+.08/4)4x1 A=1000(1+.08/365)365x1
=1000(1.02)4
≈1000(1.000219)365
≈ $1082.43
≈ $1083.28
p. 486
Ch 7.2 Exponential Decay
What you should learn:
Goal 1 Graph and use Exponential Decay
functions.
Goal 2
Write an Exponential Decay model
that describes the situation.
A2.5.2
7.2 Graph Exponential decay Functions
Discovery Education – Example 3: Exponential Decay-Bloodstream
P. 486
7.2 Exponential Decay
Exponential Decay
• Has the same form as growth functions
f(x) = a(b)x
• Where
• BUT:
a>0
0<
b<1
(a fraction between 0 & 1)
Recognizing growth and decay functions
• State whether f(x) is an exponential
growth or DECAY function
• f(x) = 5(2/3)x
• b = 2/3, 0 < b < 1 it is a decay function.
• f(x) = 8(3/2)x
• b = 3/2, b > 1 it is a growth function.
• f(x) = 10(3)-x
• rewrite as f(x)= 10(1/3)x
so it is decay
Recall from 7.1:
• The graph of y= abx
• Passes thru the point (0,a) (the y intercept
is a)
• The x-axis is the asymptote of the graph
• a tells you up or down
• D is all reals (the Domain)
• R is y>0 if a>0 and y<0 if a<0
• (the Range)
Graph:
• y = 3(1/4)x
• Plot (0,3) and
(1,3/4)
• Draw & label
asymptote
• Connect the dots
using the
asymptote
y=0
Domain = all reals
Range = reals>0
Graph
• y = -5(2/3)x
• Plot (0,-5) and (1,10/3)
• Draw & label
asymptote
• Connect the dots
using the
asymptote
y=0
Domain : all reals
Range : y < 0
Now remember: To graph a
general Exponential Function:
•
•
•
•
y = a bx-h + k
Sketch y = a bx
h= ??? k= ???
Move your 2 points h units left or right
…and k units up or down
• Then sketch the graph with the 2 new
points.
Example graph y=-3(1/2)x+2+1
• Lightly sketch y=3·(1/2)x
• Passes thru (0,-3)
& (1,-3/2)
• h=-2, k=1
• Move your 2 points
to the left 2 and up
1
• AND your
asymptote k units
(1 unit up in this
case)
y=1
Domain : all reals
Range : y<1
Using Exponential Decay
Models
• When a real life quantity decreases by
fixed percent each year (or other time
period), the amount y of the quantity after t
years can be modeled by:
• y = a(1-r)t
• Where a is the initial amount and r is the
percent decrease expressed as a decimal.
• The quantity 1-r is called the
decay factor
Discovery Ed - Using functions to Gauge Filter Eff
Ex: Buying a car!
• You buy a new car for $24,000.
• The value y of this car decreases by
16% each year.
• Write an exponential decay model for
the value of the car.
• Use the model to estimate the value
after 2 years.
• Graph the model.
• Use the graph to estimate when the
car will have a value of $12,000.
• Let t be the number of years since
you bought the car.
• The model is: y = a(1-r)t
•
= 24,000(1-.16)t
•
= 24,000(.84)t
•
Note: .84 is the decay factor
• When t = 2 the value is
y=24,000(.84)2 ≈ $16,934
Now Graph
The car will have a
value of $12,000 in 4 years!!!
p. 492
7.3 Use Functions Involving e
What you should learn:
Goal 1 Will study functions involving the
Natural base e
Goal 2 Simplify and Evaluate expressions
involving e
Goal 3 Graph functions with e
A3.2.2
7.3 Use Functions Involving e
The Natural base e
• Much of the history of mathematics is marked
by the discovery of special types of numbers
like counting numbers, zero, negative
numbers, Л, and imaginary numbers.
7.3 Use Functions Involving e
Natural Base e
• Like Л and ‘i’, ‘e’ denotes a number.
• Called The Euler Number after Leonhard
Euler (1707-1783)
• It can be defined by:
e= 1 + 1 + 1 + 1 + 1 + 1 +…
0! 1! 2! 3! 4! 5!
= 1 + 1 + ½ + 1/6 + 1/24 + 1/120+...
≈ 2.718281828459….
7.3 Use Functions Involving e
• The number e is irrational – its’
decimal representation does not
terminate or follow a repeating
pattern.
• The previous sequence of e can also
be represented:
n
(1+1/n)
• As n gets larger (n→∞),
gets closer and closer to
2.71828…..
• Which is the value of e.
7.3 Use Functions Involving e
Examples
3
e
7
e
·
4
e
3
10e
5e2
(3e-4x)2
(-4x)2
9e
2e3-2
9e-8x
2e
9
8x
e
7.3 Use Functions Involving e
More Examples!
8
24e
5
8e
3e3
-5x
-2
(2e )
-2
10x
2 e
10x
e
4
7.3 Use Functions Involving e
Using a calculator
7.389
2
• Evaluate e using
a graphing
calculator
x
• Locate the e
button
• you need to use
the second button
7.3 Use Functions Involving e
Evaluate e-.06
with a calculator
7.3 Use Functions Involving e
Graphing
• f(x) =
rx
ae is a natural base
exponential function
• If a > 0 & r > 0 it is a growth function
• If a > 0 & r < 0 it is a decay function
7.3 Use Functions Involving e
Graphing examples
• Graph y = ex
• Remember
the rules for
graphing
exponential
functions!
• The graph
goes thru
(0,a) and (1,e)
(1,2.7)
(0,1)
7.3 Use Functions Involving e
Graphing cont.
• Graph y = e-x
(1,.368)
(0,1)
7.3 Use Functions Involving e
Graphing Example
• Graph
y = 2e0.75x
• State the
Domain &
Range
(1,4.23)
(0,2)
• Because a=2 is
positive and r=0.75,
the function is
exponential growth.
• Plot (0,2)&(1,4.23)
and draw the curve.
7.3 Use Functions Involving e
Using e in real life.
• In 8.1 we learned the formula for
compounding interest n times a year.
• In that equation, as n approaches
infinity, the compound interest formula
approaches the formula for
continuously compounded interest:
A=
rt
Pe
7.3 Use Functions Involving e
Example of
Continuously compounded interest
You deposit $1000.00 into an account that pays
8% annual interest compounded continuously.
What is the balance after 1 year?
P = 1000, r = .08, and
A=
rt
Pe
t=1
= 1000e.08*1 ≈ $1083.29
7.3 Use Functions Involving e
7.4 Logarithms
What you should learn:
mathbook
p. 499
Goal 1 Evaluate logarithms
Goal 2
Graph logarithmic functions
A3.2.2
7.4 Evaluate Logarithms and Graph Logarithmic Functions
Evaluating Log Expressions
• We know 22 = 4 and 23 = 8
• But for what value of y does 2y = 6 ?
• Because 22 < 6 < 23 you would expect
the answer to be between 2 & 3.
• To answer this question exactly,
mathematicians defined logarithms.
Definition of Logarithm to base a
• Let a & x be positive numbers & a ≠ 1.
• The logarithm of x with base a is
denoted by logax and is defined:
logax = y iff
y
a
=x
• This expression is read “log base a of x”
• The function f(x) = logax is the
logarithmic function with base a.
• The definition tells you that the
equations logax = y and ay = x are
equivilant.
• Rewriting forms:
• To evaluate log3 9 = x ask yourself…
• “Self… 3 to what power is 9?”
• 32 = 9 so……
log39 = 2
Log form
• log216 = 4
• log1010 = 1
• log31 = 0
• log10 .1 = -1
• log2 6 ≈ 2.585
Exp. form
• 24 = 16
1
• 10 = 10
0
•3 = 1
• 10-1 = .1
• 22.585 = 6
Evaluate without a calculator
• log381 = 4
• Log5125 = 3
• Log4256 = 4
• Log2(1/32) =
-5
x
•3
= 81
x
• 5 = 125
• 4x = 256
x
• 2 = (1/32)
Evaluating logarithms now you try
some!
• Log 4 16 = 2
• Log 5 1 = 0
• Log 4 2 = ½ (because 41/2 = 2)
• Log 3 (-1) = undefined
• (Think of the graph of y=3x)
You should learn the following
general forms!!!
0
a
• Log a 1 = 0 because = 1
1
• Log a a = 1 because a = a
• Log a ax = x because ax = ax
Natural logarithms
•log e x = ln x
• ln means log base e
Common logarithms
•log 10 x = log x
• Understood base 10 if
nothing is there.
Common logs and natural logs with
a calculator
log10 button
ln button
• g(x) = log
• f(x) = bx
b
x is the inverse of
• f(g(x)) = x and g(f(x)) = x
• Exponential and log functions
are inverses and “undo” each
other
• So: g(f(x)) = logb
•
x
b
=x
log
x
f(g(x)) = b b = x
• 10log2 = 2
x
• Log39 = Log3(32)x =Log332x=2x
logx
• 10
= x
x
• Log5125 = 3x
Finding Inverses
• Find the inverse of:
• y = log3x
• By definition of logarithm, the inverse is
x
y=3
• OR write it in exponential form and
switch the x & y!
3y = x
3x = y
Finding Inverses cont.
• Find the inverse of :
• Y = ln (x +1)
• X = ln (y + 1)
• ex = y + 1
• ex – 1 = y
Switch the x & y
Write in exp form
solve for y