1.) if there are initially 100 fruit flies in a sample, and the number of fruit flies decreases by...
TRANSCRIPT
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?
Do Now4 - 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.
Do Now4 - 27 - 2012
Simplify.3)92.01(2573
1.) 4
4
0292.014000
2.)
trP 13.) Evaluate using this formula when
P is 1219, r is 0.12, and t is 5.
40582.211,18 08.4118
29.2148 512.11219
Do Now4 - 30 - 2012
1.) How many half-liveshalf-lives would it take to have a 700 gram sample of uranium reduce to under 3 grams of uranium ?
2.) If there are initially 10 bacteria in a culture, and the number of bacteria doubledouble each hour, find the number of bacteria after 24 hours.
Do Now5 - 4 - 2012
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 .
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.
tIA )71.0(
Compound Interest
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.
ntnrPA )1( Do Now5 - 4 - 2012
Identify:
A = P = r = n = t =
Compound Interest
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.
ntnrPA )1( Do Now5 - 4 - 2012
)18)(12(12045. )1(20000 P
A = 20,000 P = ? r = .045 n = 12 t = 18
)2445.2(20000 PP910,8
Do Now5 - 8 - 2012
In the equation
, which of the following is true?
a) There is a Growth Rate?
b) There is a Decay Rate?
10)75.1(350 y
c) The Decay Rate is 75% ?
d) The Decay Rate is 25% ?
e) The Decay Factor is .25 ?
f) The Decay Factor is (1 - .75) ?
g) The initial amount is 350? h) The time period is 10 ?
I) “y” is the final amount?
j) This is an Exponential ….Growth ….Decay
Do Now5 - 9 - 2012
1.) If you invested $ 2,000 at a rate of 0.6% compounded continuouslycontinuously, find the balance in the account after 5 years, use the formula rtPeA
2. ) Simplify the Expression
12
25
6
18
e
e
)5)(006(.2000eA $ 2,060.91
133e
trPy )1( trPy )1(
trPy )1( trPy )1(
nt
n
rPA
1
nt
n
rPA
1
rtPea rtPea
Compounded Interestex) Compounded daily Compounded monthly Compounded quarterly
Compounded Interestex) Compounded daily Compounded monthly Compounded quarterly
Continuously Compounded InterestContinuously Compounded Interest
Exponential GrowthExponential Growth
Exponential DecayExponential Decay
Do Now5 - 10 - 2012
Do Now5 - 11 - 2012
1.) RE-Write in Exponential form236log6
2.) RE-Write in Logarithmic form1100
3.) Evaluate 245log4
4.) Graph xy 4log
p. 478
What you should learn:
GraphGraph and use Exponential Growth functions.
Write an Exponential Growth model that describes the situation.
7.1 Graph Exponential Growth Functions7.1 Graph Exponential Growth Functions
A2.5.2
Ch 7.1 Exponential Growth
Exponential Function• f(x) = bx where the base bb 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
y = 0
2 raised to any powerWill NEVER be zero!!
Example 1
• Graph
• Plot (0, ½) and (1, 3/2)
• Then, from left to right, draw a curve that begins just above the x-axis, passes thru the 2 points, and moves up to the right
xy 321
What do you think the Asymptote is? y = 0
Example 2
• Graph y = - (3/2)x
• Plot (0, -1) and (1, -1.5)• Connect with a curve• Mark asymptote• D = ??• All reals• R = ???• All reals < 0
y = 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!
• Graph y = 2·3x-2 +1
• State the Domain and Range!
• D = all reals• R = all reals >1
y=1
Example 4
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
where
• a - Initial principal
• r – percent increase expressed as a decimal
• t – number of years
• y – amount in account after t years
tray )1(
Notice 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
trPA )1( If a person invests $310 in an account that pays 8% interest compounded annually, find the approximant balance after 5 years.
5)08.1(310 A
A = $455.49
Compound Interest Consider an initial principal P deposited in an account that pays interest at an annual rate, r,
compounded n times per year.
• 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
ntnrPA )1(
Compound Interest example
• 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
=1000(1.02)4
≈ $1082.43
A=1000(1+.08/365)365x1
≈1000(1.000219)365
≈ $1083.28
ntnrPA )1(
Ch 7.2 Exponential Decayp. 486
What you should learn:
GoalGoal 11
GoalGoal 22
GraphGraph and use Exponential Decay functions.
Write an Exponential Decay model that describes the situation.
7.2 Graph Exponential decay Functions7.2 Graph Exponential decay Functions
A2.5.2
7.2 Exponential DecayP. 486
Discovery Education – Example 3: Exponential Decay-Bloodstream
Exponential Decay
• Has the same form as growth functions
f(x) = a(b)x
• Where a > 0
• BUT:
0 < b < 1 (a fractionfraction between 0 & 1)
Recognizing growth and decay Recognizing growth and decay functionsfunctions
• 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
Domain = all reals Range = reals>0
y=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 realsRange : 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 realsRange : 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 aa is the initial amount and rr 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!!!
7.3 Use Functions Involving ep. 492
What you should learn:GoalGoal 11
GoalGoal 22
Will study functions involving the Natural base e
Simplify and Evaluate expressions involving e
7.3 Use Functions Involving 7.3 Use Functions Involving ee
A3.2.2
GoalGoal 33 Graph functions with 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 7.3 Use Functions Involving ee
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 7.3 Use Functions Involving ee
• 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:
• As n gets larger (n→∞), (1+1/n)n gets closer and closer to 2.71828…..
• Which is the value of e.
7.3 Use Functions Involving 7.3 Use Functions Involving ee
Examples
e3 · e4
e7
10e3
5e2
2e3-2
2e
(3e-4x)2
9e(-4x)2
9e-8x
9 e8x
7.3 Use Functions Involving 7.3 Use Functions Involving ee
More Examples!
24e8
8e5
3e3
(2e-5x)-2
2-2e10x
e10x
4
7.3 Use Functions Involving 7.3 Use Functions Involving ee
Using a calculator• Evaluate e2 using
a graphing calculator
• Locate the ex button
• you need to use the second button
7.389
7.3 Use Functions Involving 7.3 Use Functions Involving ee
Evaluate e-.06 with a calculator
7.3 Use Functions Involving 7.3 Use Functions Involving ee
Graphing
• f(x) = aerx 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 7.3 Use Functions Involving ee
Graphing examples• Graph y = ex
• Remember the rules for graphing exponential functions!
• The graph goes thru (0,a) and (1,e)
(0,1)
(1,2.7)
7.3 Use Functions Involving 7.3 Use Functions Involving ee
Graphing cont.• Graph y = e-x
(0,1)(1,.368)
7.3 Use Functions Involving 7.3 Use Functions Involving ee
Graphing Example• Graph
y = 2e0.75x • State the
Domain & Range
• Because a=2 is positive and r=0.75, the function is exponential growth.
• Plot (0,2)&(1,4.23) and draw the curve.
(0,2)
(1,4.23)
7.3 Use Functions Involving 7.3 Use Functions Involving ee
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 = Pert
7.3 Use Functions Involving 7.3 Use Functions Involving ee
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?
7.3 Use Functions Involving 7.3 Use Functions Involving ee
P = 1000, r = .08, and t = 1
A = Pert = 1000e.08*1 ≈ $1083.29
7.4 Logarithms
p. 499What you should learn:GoalGoal 11
GoalGoal 22
Evaluate logarithms
Graph logarithmic functions
7.4 Evaluate Logarithms and Graph Logarithmic Functions7.4 Evaluate Logarithms and Graph Logarithmic Functions
A3.2.2
mathbook
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 ay = 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 Exp. form
•log216 = 4
•log1010 = 1
•log31 = 0
•log10 .1 = -1
•log2 6 ≈ 2.585
•24 = 16•101 = 10•30 = 1•10-1 = .1•22.585 = 6
Evaluate without a calculator
•log381 =
•Log5125 =
•Log4256 =
•Log2(1/32) =
•3x = 81•5x = 125•4x = 256•2x = (1/32)
4
34
-5
Evaluating logarithms now you try some!
•Log 4 16 = •Log 5 1 =•Log 4 2 =•Log 3 (-1) =• (Think of the graph of y=3x)
20
½ (because 41/2 = 2) undefined
You should learn the following general forms!!!
•Log a 1 = 0 because a0 = 1
•Log a a = 1 because a1 = 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 b x is the inverse of
•f(x) = bx
•f(g(x)) = x and g(f(x)) = x•Exponential and log functions
are inverses and “undo” each other
•So: g(f(x)) = logbbx = x• f(g(x)) = blog
bx = x
•10log2 = •Log39x =•10logx =•Log5125x =
2Log3(32)x =Log332x=2x
x3x
Finding Inverses
• Find the inverse of:
•y = log3x• By definition of logarithm, the inverse
is y=3x • 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) Switch the x &
y
•ex = y + 1 Write in exp form
•ex – 1 = y solve for y