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

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

http://player.discoveryeducation.com/index.cfm?guidAssetId=9EB40A20-22E1-415C-9EDD-713506F72B7B&blnFromSearch=1&productcode=US

http://player.discoveryeducation.com/index.cfm?guidAssetId=B2960069-CECB-4FD7-909A-9D43BDB1E5B9&blnFromSearch=1&productcode=US

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

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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.


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….


• 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.


Examples

e3 · e4

e7

10e3

5e2

2e3-2

2e

(3e-4x)2

9e(-4x)2

9e-8x

9 e8x


More Examples!

24e8

8e5

3e3

(2e-5x)-2

2-2e10x

e10x

4


Using a calculator• Evaluate e2 using

a graphing calculator

• Locate the ex button

• you need to use the second button

7.389


Evaluate e-.06 with a calculator


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


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)


Graphing cont.• Graph y = e-x

(0,1)(1,.368)


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)


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


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 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

Documents

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