THE FUN USE OF RATES

Exponential Function

Bizarro

Rules about exponents

)()( grapesbananasgrapesbananas aaa

grapes

bananasgrapesbananas

a

aa a≠0, can anyone tell me

why?

grapesbananasgrapesbananas aa )(

bananasbananasbananas tata )(

bananabanana

aa

1)(

Review

255552

16)2()2()2()2()2( 4

64)4()4()4()4( 3

Two 5’s

Four -2 ’s

Three -4’s

There’s even more….

Do you remember….

What is 6-2 ?When you have an exponent that is

negative….this is essentially what it means…

36

1

6

16

22

0625.516

81

11681

1681

8116

1681

11

811611

32

1

32

1

3

2

4

44

4

Review

But what about something like ?

Trick: if you have any number to the power of ½ , that’s the same thing as square rooting a number.

2

3

4

8)64()444()4(444 2

1

2

1

2

132

13

2

1

1

3

2

3

Three 4’s

Another way of looking at it

What about ?If you punched this into your calculator you’d get 11.

But why?

3

1

1331

3

1

3

1

)111111(1331 11

1211111)111111(1331 3

2

3

2

1331111111)111111(1331 3

3

3

3

So, theoretically, we use one out of the three 11’s

So, theoretically, we use two out of the three 11’s

So, theoretically, we use three out of the three 11’s

If you didn’t understand the last example, don’t fret. A lot of the time the numbers won’t work out as nicely as that did in the example and you will have to use your calculator anyways.

Please make sure you have at least a scientific calculator in class for this section. If you do not, I can recommend a wonderful calculator that doesn’t cost too much and it’s available at Bureau en Gros or even Zeller’s I think.

The exponential function is a function where the “power” is always changing. It is our x-value..

The function looks like:

where A = initial valuebase = amount of increase or

decreasex = time (always changing)

xbaseAy )(

So to the exponential function we go…

The exponential function that always “doubles” looks like this:

E.g. Table of values and graph look like this:

xy 2

x y

-2

2-2 = 0.25

-1

2-1 = 0.5

0 20 = 1

1 21 = 2

2 22 = 4

3 23 = 8

4 24 =16-3 -2 -1 0 1 2 3 4 5

0

2

4

6

8

10

12

14

16

18

20

22

24

26

So to the exponential function we go…

What happens to the function if we multiply it by a constant ?

E.g. Where A = 3 is the constantTable of values and graph look like this:

)2(3 xy

x y

-2 3(2-2) = 3(0.25) = 0.75

-1 3(2-1) = 3(0.5) = 1.5

0 3(20) = 3(1) = 3

1 3(21) = 3(2) = 6

2 3(22) = 3(4) = 12

3 3(23) = 3(8) = 24

4 3(24) = 3(16) = 48-3 -2 -1 0 1 2 3 4 5

0

4

8

12

16

20

24

There’s more?

Oh yes there is more…

If your base = a fraction, then instead of an increasing function we get a decreasing function:

Let’s look at an example…

Example

Suppose we had the function

Our table of values and graph would look like this:

x

y

2

1

x y

-2 (0.5)-2 = 4

-1 (0.5)-1 = 2

0 (0.5)0 = 1

1 (0.5)1 =0.5 = ½

2 (0.5)2 = 0.25 = ¼

3 (0.5)3 = 0.125 = 1/8

4 (0.5)4 = 0.0625 = 1/16

-3 -2 -1 0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

What do you notice?

From our last example we can deduce that the following graphs would be above and below the graph we just created. What do you notice?

-3 -2 -1 0 1 2 3 4 50

2

4

6

8

10

12

14

16

18x

y

2

14

x

y

2

1

x

y

2

125.0

When will I ever use exponentials?

Believe it or not, most things work on the concept of exponentials.

Exponential functions help find:

1. how much something will be worth in certain timeframe. Like an heirloom (my lego!!), a car, a house, minimum wage, inflation….

2. How much bacteria is left in your body after taking medication or how fast a disease can spread….(like H1N1)

3. Help predict Population growth of a certain area or city

The rule of this function has the same form as before:

But to make it more helpful for the types of problems listed in the last slide, we need a way to determine what “A” and the “base” are.

So the function looks like this:

xbaseAy )(

nx

Ay

compounded asinterest w the timesof #

ecreaseincrease/d of rate1

????Who’s confused?????

Let’s simplify the formula….

Let’s make this formula a little easier to understand:

A = initial amount you start withy = final amount you end up withi = rate of increase/decrease (as a decimal)n = # of times you compound your rate per unit of time (a lot of the time your n is going to equal 1)x = amount of “n”’s you had to use in order reach the final amount of timeLet’s look at some examples and I think you’ll understand

nx

n

iAy

1

Example

Suppose you bought a house for $210 000. Each year, the house increases in value by 3%. How much is the house worth in 16 years? nx

n

iAy )1(

161

1

03.01210000

y

16)03.01(210000 y16)03.1(210000y

) 604706439.1 ( 210000y35.336988y

So, in 16 years, you made a profit of $336,988.35 – $210,000 = $126,988.35Pretty sweet eh?

A = initial value = $210,000i = rate of increase = 3% = 0.03n = number of times we compound our rate per year = once a year = 1x = amount of times we calculate the increase is once each year for 16 years therefore x = 16

Another Example

I bought a 2006 Honda Accord last July (2008) for $15,000. Honda automobiles depreciate (goes down) in value a lot slower than most brands of car. Honda’s lose around 14% of their original value per year. How much is my car worth in 5 years after I bought it?

nx

n

iAy )1(

51

1

14.0115000$

y

5)14.01( 15000$ y5)86.0(15000$y

)470427017.0(15000$y41.7056$y

A = $15000 = initial valuei = rate of decrease = 14% = 0.14n = compounded (calculated) yearly n = 1x = amount of times we calculate our rate of decrease in 5 years = 5

Pretty crappy eh?

Another Example

Suppose 66 molecules of bacteria multiply every 3 minutes when exposed to sunlight. The amount of bacteria increases 2% every 3 minutes. How many molecules of bacteria are there after an hour (60 minutes)? nx

n

iAy )1(

201

1

02.0166

y

20)02.01(66 y20)02.1(66y

)485947296.1(66y98y

A = initial value = 66 bacteriai = rate of increase = 1.5% = 0.015n = 1

x = amount of times we calculated the rate of increase in the 1 hour time frame 20 times in one hour. (20 x 3 = 60 minutes) Impressive how fast something can

spread!!

Another Example

Between 2001 & 2006 the population of Montreal (Census Montreal Area) increased by 1.04%. If the population was 3,426,350 in 2001, how many people lived in Montreal in 2006?

nx

n

iAy )1(

51

1

0104.013426350

y

5)0104.01(3426350 y5)0104.1(3426350y

)053092907.1(3426350y265,608,3y

A = initial population = 3,426,350 peoplei = rate of increase = 1.04% = 0.0104

n = compounded (calculated) every year n = 1x = amount of times we calculated the rate of increase in the 5 year time frame = 5 times in 5 years.

Population could go up very fast or decrease fast, depending on the area.

Another Example

In the 1950’s Detroit Michigan’s population was peaked at 1.8 Million strong in the city center (because of the amount of jobs created by the car companies). But from that time, the population has been decreasing due to factories going elsewhere in the country and the world, and due to less demand of cars nowadays.

Suppose that the population went from 951 000 to 912 000 in the city center in one year. What is the rate of decrease in this one year?

nx

n

iAy )1(

11

11951000912000

i

1)1(951000912000 i

)1(951000

912000i

i196.004.096.01 i

A = initial population = 951 000 peoplei = rate of increase = ?????????????????n = compounded (calculated) every year n = 1x = amount of times we calculated the rate of increase in the 1 year time frame = 1 time in 1 year x = 1 year

y = final population = 912 000 people

So Detroit lost 4% of it’s population in one year

That’s a wrap!

That’s it! Now go do your homework!!!