Exponential Functions

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Exponential Functions

• According to legend, the inventor of the game of chess requested payment as follows:

• My request is a modest one. I humbly request one grain of rice on the first square on the chess board, two grains on the second square, four grains on the third square, and so on. Each square represents twice as may grains as the previous one.

• The king agreed to the request immediately, without doing any calculations. Do you think this was wise?

• A chess board has 64 squares. How many grains of rice are to be placed on the 64th square?

• If there were 1,000,000 people in the kingdom and each grain of rice weighs 0.00008 pound. The rice on the 64th square represents how many pounds of rice per person?

Investigate graphs of exponential functions

• Use your calculator to graph

x

x

y

y

2312

Exponential growth functions

10

ba

aby x

Exponential Growth

• 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:

Don’t confuse percent increase with growth factor! A percent increase of 200% actually means the quantity tripled! 1 + 2 = 3(1+r) is the growth factor, r is the percent increase

tray )1(

Modeling Exponential Growth

• A diamond ring was purchased twenty years ago for $500. The value of the ring increased by 8% each year. What is the value of the ring today?

48.330,2$08.1500

)08.1(500

)1(

20

20

tray

Compound Interest

• Consider an initial principle (P) deposited in an account that pays interest at an annual rate, r, compounded n times per year. The amount, A, in the account after t years can be modeled by this equation:

nt

nrPA )1(

Try these!

• You deposit $1000 in an account that pays 8% annual interest . Find the balance in the account at the end of the year if the interest is compounded annually? quarterly? daily?

The number “e”

• Natural base e - The Euler numberAs n gets larger the expression gets closer to 2.718281……

This is a famous number discovered by Leonhard Euler.

n

n

11

Exponential Decay

• If the function is of the form• and a>0 but 0<b<1 it is an exponential decay

function.

• As x increases, y approaches zero• As x decreases the function approaches

positive infinity • In your calculator try graphing

xabxf )(

xxf )2/1(5)(

Exponential Decay

tray )1(

When a real life quantity decreases by a fixed percent each yearThe amount (y) of the quantity after t years can be modeled by:

You by a car for $24,000 and the value of the car decreases16% each year. What is the value of the car after 5 years?

Evaluating natural base expressions

• A function of the form • is called a natural base exponential function.

• If a>0 and r>0 it is an exponential growth function

• If a>0 and r<0 it is an exponential decay function.

• Try graphing these in your calculator!

rxaexf )(

Using e in real life!

nt

nrPA )1(

If interest is compounded continuously the formula

becomes

If you deposit $1000 in an account that pays 8% interest Compounded continuously what will the balance be aftera year?

$1083.29

rtPeA

108.1000 eA

Warm up

• A customer purchases a tv for $800 using a credit card. The interest is charged on any unpaid balance at the rate of 18% per year compounded monthly. If the customer makes no payment for one year, how much is owed at the end of the year? $956.49

• A house was purchased for $90,000 in 1995. If the value of the home increases by 5% per year, what is it worth now?