Exponential Functions

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End of slide show, click to exit.

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The Savings Bond�

Exponential Growth<= /a>

The Savings Bond�

Exponential Functio= n

Rules for Exponents=

Population Growth

Population Growth

Exponential Decay

Modeling Radioactive Decay

Modeling Radioactive Decay

Exponential Growth, Exponential Decay

The number e

The number e

Compound Interest

Example

Example Cont�d<= /b>

Example Cont�d<= /b>

Example Cont�d<= /b>

Assignment<= /font>
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Sec 1.3

Exponential Functio= ns

Goals

To determine the do= main, range, and graph of an exponential function.

To solve problems i= nvolving exponential growth and decay.

To use exponential = regression equations to solve problems.

The Savings Bond�

Mr. Maierhofer was = given a $75 savings bond in 1965.� It was originally purchased for $37.50 and was to be worth the $75 after about 10 years.� He didn�t cash it in, and= he found out in the summer of 2006 that it was fully matured (no longer earn= ing interest).� If the interest rate = of the bond was 6.89%, how much was the bond worth when he cashed it in last sum= mer?

Exponential Growth<= /a>

This problem is an = example of an exponential growth problem

Compound interest p= roblems are modeled by the function

y =3D P � a^x

P =3D the initial i= nvestment amount

a =3D (1 + the annu= al interest rate (as a decimal))

x is the number of = years

The Savings Bond�

So what was Mr. Mai= erhofer�s savings bond worth?

Originally purchase= d in the summer of 1965 for $37.50

Earned 6.89% intere= st

Was cashed in the s= ummer of 2005

y =3D P � a^x

Exponential Functio= n

Definition of Expon= ential Function

Let a be a positive= real number other than 1.� The function:

f(x) =3D a^x

is the exponential = function with base a

Rules for Exponents=

a^x � ay =3D a^x+y

(a^x)y =3D (a^y)^x =3D a^xy

a^x � bx =3D (ab)^x

Population Growth

By about what perce= nt is the population increasing annually?

Population Growth

Population growth c= an also be modeled by an exponential function.

The population of t= he world was 4936 million in 1986.� It is incr= easing by about 1.8% per year.� What wil= l the population be in 2010?

Exponential Decay

Exponential functio= ns can also be used to model a decrease over time, such as radioactive decay.<= /td>

Half-life: the amou= nt of time it takes for half of a radioactive substance to change from its radioacti= ve state to its nonradioactive state.

Modeling Radioactive Decay

Suppose the half-li= fe of a certain radioactive substance is 20 days and that there are 5 grams prese= nt initially.� When will there be on= ly 1 gram of the substance remaining?

How much after 20 d= ays?

After 40 days?

Modeling Radioactive Decay

After t days?

When will there be = only 1 gram of the substance remaining?� Use a graphing calculator to solve it graphically.

Exponential Growth, Exponential Decay

Compound interest i= nvestments, population growth, and radioactive decay are all examples of exponential growth and decay.

The function y =3D = k � a^x, k > 0 is a model for exponential growth if a > 1.

It is a model for e= xponential decay if

�� 0 < a < 1

Example 3

Predicting the Popu= lation: Use the population data table to estimate the population for the year 1990.� Compare it with the actual population= of approximately 250 million in 1990.

Example 3

Let x=3D0 be 1880, = x=3D1 is 1890, etc.

Enter the years int= o L1 and the population into L2 (STAT, 1 Edit)

Set up your window<= /font>

Turn on stat plot (= 2^nd Y=3D)

Do an exponential r= egression (STAT, CALC, 0 ExpReg)

Store that exp reg = in y1 (VARS, 5 Stats, EQ, 1 RegEQ)

Graph

How could we use th= e graph to find the population in 1990?

The number e

e is defined as the= value the function f(x)=3D(1+1/x)^x approaches as x approaches infinity

Plug this equation = in your calculator and look at the table as x gets �big�.

Look at the graph a= s x gets �big�.

e =3D2.71828182845904523536028747135

The number e

The exponential fun= ctions y =3D e^x and y =3D e^-x are often used as models of exponential growth or decay.

Compound Interest

Interest can be com= pounded annually, monthly, daily, or even continuously.

If interest is comp= ounded continuously, we use the model y =3D P � e^rt

P =3D initial inves= tment

r =3D interest rate=

t =3D time in years= .

Example

If Harmony invests = $2500 in a savings account with a 7% interest rate compounded annually, how long wil= l it take until Harmony�s account has a balance of $5000?

Example Cont�d<= /b>

What if it is compo= unded monthly?

Example Cont�d<= /b>

What if it is compo= unded daily?

Example Cont�d<= /b>

What if it is compo= unded continuously?

Assignment<= /font>

Sec 1.3, p. 24 -26:= 1-6, 7, 11-16, 23-29 odd, 30-38 even

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------=_NextPart_01C7F06B.3E621250 Content-Location: file:///C:/EC825262/CalcCh1Sec31B_files/pres.xml Content-Transfer-Encoding: quoted-printable Content-Type: text/xml; charset="utf-8" ------=_NextPart_01C7F06B.3E621250 Content-Location: file:///C:/EC825262/CalcCh1Sec31B_files/v3_slide0001.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="windows-1252" Exponential Functions

Exponential Functions

------=_NextPart_01C7F06B.3E621250 Content-Location: file:///C:/EC825262/CalcCh1Sec31B_files/v3_master04_background.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlh3AFlAXcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAADb AWQBgQAAAAAzZv///wECAwL/lI+py+0Po5y02ouz3rz7D4biSJbmiabqyrbuC8fyTNf2jef6zvf+ DwwKh8Si8YhMKpfMpvMJjUqn1Kr1is1qt9yu9wsOi8fksvmMTqvX7Lb7DY/L5/S6/Y7P6/f8vv8P GCg4SFhoeIiYqLjI2Oj4CBkpOUlZaXmJmam5ydnp+QkaKjpKWmp6ipqqusra6voKGys7S1tre4ub q7vL2+v7CxwsPExcbHyMnKy8zNzs/AwdLT0XUG19jZ2tvc3d7f0NHi4+Tl5ufo6erl6eu+7+Dh8v P09fb8/dfq+/z9/v/38vH8CBBAsaPEhvmsKFDBs6fAgxosSJFCtavIgxo8aN/xw7evwIMqTIkSRL mjyJMqXKlSxbunwJM6bMmTRr2ryJM6fOnTx7+vwJNKjQoUSLGj2KNKnSpUybOn0KNarUqVSrWr2K NavWrVy7ev0KNqzYsWTLmj2LNq3atWzbun0LN67cuXTr2r0rC6HevXz77hXoN7DgwYTRAS6MOLFi wocXO34M2R/eyZQrW76MObPmzZw7e/4MOrTo0aRLmz6NOrXq1axbu34NO7bs2bRr276NO7fu3bx7 +/4NPLjw4cSLGz+OPLny5cybO38OPbr06dSrW7+OPbv27dy7e/8OPrz48eTLmz+PPr369ezbu38P P778+fTr27+PP7/+/fz7+7z/D2CAAg5IYIEGHohgggouyGCDDj4IYYQSTkhhhRZeiGGGGm7IYYce fghiiCKOSGKJJp6IYooqrshiiy6+CGOMMs5IY4023ohjjjruyGOPPv4IZJBCDklkkUYeiWSSSi7J ZJNOPglllFJOSWWVVl6JZZZabslll15+CWaYYo5JZplmnolmmmquyWabbr4JZ5xyzklnnXbeiWee eu7JZ59+/glooIIOSmihhh6KaKKKLspoo44+CmmkktJWAAA7 ------=_NextPart_01C7F06B.3E621250 Content-Location: file:///C:/EC825262/CalcCh1Sec31B_files/v3_slide0002.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="windows-1252" Exponential Functions

<= /td> = <= /td> <= /td> = <= /td>

w To determine the domain, range, and graph of

an exponential function.

w To solve problems involving exponential

growth and decay.

w To use exponential regression equations to

solve problems.

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The Savings Bond�

<= /td> = = <= /td>

w Mr. Maierhofer was given a $75 savings bond in

1965.� It was originally purchased for $37= .50 and

was to be worth the $75 after about 10 years.� He

didn�t cash it in, and he found out in the summer of

2006 that it was fully matured (no longer earning

interest).� If the interest rate of the bond was 6.89%,

how much was the bond worth when he cashed it in

last summer?

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Exponential Growth

<= /td> <= /td> <= /td> <= /td> <= /td> <= /td> <= /td> = <= /td>

w This problem is an example of an exponential

growth problem

w Compound interest problems are modeled by

the function

n y =3D P � a^x

n P =3D the initial investment amount

n a =3D (1 + the annual interest rate (as a decimal))

n x is the number of years

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The Savings Bond�

<= /td> <= /td> <= /td> = = <= /td>

So what was Mr. Maierhofer�s savings bond worth?

w Originally purchased in the summer of 1965 for

$37.50

w Earned 6.89% interest

w Was cashed in the summer of 2005

w y =3D P � a^x

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Exponential Function

<= /td> <= /td> <= /td> <= /td> = <= /td>

Definition of Exponential Function

w Let a be a positive real number other than 1.

The function:

n f(x) =3D a^x

w is the exponential function with base a

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Rules for Exponents

=

a^x � a^y = =3D a^x+y

(a^x)^y =3D= (a^y)^x =3D a^xy

a^x � bx =3D (ab)^x

<= /td> =

If a>0 and b>0, the following

hold for all real numbers x

and = y.

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Population Growth

<= /td> =

By about what percent is the population

inc= reasing annually?

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Population Growth

<= /td> = = <= /td>

w Population growth can also be modeled by an

exponential function.

w The population of the world was 4936 million in

1986.� It is increasing by about 1.8% per = year.

What will the population be in 2010?

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Exponential Decay

<= /td> = = <= /td> <= /td> <= /td>

w Exponential functions can also be used to

model a decrease over time, such as

radioactive decay.

w Half-life: the amount of time it takes for half

of a radioactive substance to change from its

radioactive state to its nonradioactive state.

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Modeling Radioactive Decay

<= /td> = <= /td> = <= /td>

w Suppose the half-life of a certain radioactive

substance is 20 days and that there are 5

grams present initially.� When will t= here be

only 1 gram of the substance remaining?

w How much after 20 days?

w After 40 days?

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Modeling Radioactive Decay

<= /td> <= /td> =

w After t days?

w When will there be only 1 gram of the substance remaining?

Use a graphing calculator to solve it graphically.

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<= /td> =

Exponential Growth,

Exponential Decay

<= /td> <= /td> = <= /td> <= /td>

w Compound interest investments, population

growth, and radioactive decay are all

examples of exponential growth and decay.

w The function y =3D k � a^x, k > 0 = is a model for

exponential growth if a > 1.

w It is a model for exponential decay if

�� 0 < a < 1

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Example 3

<= /td> <= /td> <= /td> <= /td> <= /td>

w Predicting the Population:

Use the population data

table to estimate the

population for the year

1990.� Compare it with the

actual population of

approximately 250 million

in 1990.

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file:///C:/EC825262/CalcCh1Sec31B_files/v3_slide0022.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="windows-1252" Exponential Functions

Example 3

<= /td> <= /td> <= /td> <= /td> <= /td> <= /td> <= /td>

w Let x=3D0 be 1880, x=3D1 is 1890, etc.

w Enter the years into L1 and the

population into L2 (STAT, 1 Edit)

w Set up your window

w Turn on stat plot (2^nd Y=3D)

w Do an exponential regression

(STAT, CALC, 0 ExpReg)

w Store that exp reg in y1 (VARS, 5

Stats, EQ, 1 RegEQ)

w Graph

w How could we use the graph to

find the population in 1990?

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The number e

<= /td> <= /td> = <= /td> <= /td> <= /td>

w e is defined as the value the function

f(x)=3D(1+1/x)^x approaches as x approaches

infinity

w Plug this equation in your calculator and look

at the table as x gets �big�.

w Look at the graph as x gets �big�.

w e =3D2.71828182845904523536028747135

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The number e

<= /td> = =

w The exponential functions y =3D e^x an= d y =3D e^-x

are often used as models of exponential

growth or decay.

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Compound Interest

<= /td> = <= /td> <= /td> <= /td> =

w Interest can be compounded annually,

monthly, daily, or even continuously.

w If interest is compounded continuously, we

use the model y =3D P � e^rt

w P =3D initial investment

w r =3D interest rate

w t =3D time in years.

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Example

<= /td> = <= /td> <= /td>

w If Harmony invests $2500 in a savings

account with a 7% interest rate compounded

annually, how long will it take until

Harmony�s account has a balance of $5000?

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Example Cont�d

<= /td>

w What if it is compounded monthly?

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Example Cont�d

<= /td>

w What if it is compounded daily?

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Example Cont�d

<= /td>

w What if it is compounded continuously?

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Assignment

<= /td> <= /td>

w Sec 1.3, p. 24 -26: 1-6, 7, 11-16, 23-29 odd,

30-38 even

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= ;

= ;

Sec 1.3

Exponential Functions

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Goals

wTo determine the domain, range, and graph of= an exponential function.

wTo solve problems involving exponential growth and decay.

wTo use exponential regress= ion equations to solve problems.

------=_NextPart_01C7F06B.3E621250 Content-Location: file:///C:/EC825262/CalcCh1Sec31B_files/master03_image069.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwKLAXcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAK AokBgQAAAAAzZszMmePixwL/xI6py30Ao5y02ouz3rz7D4biSJbmiabqyobOCzftTNf2jef6zvf+ DwwKh8Si8YjUxZbMpPMJjUqn1Kr1is1qt1wS8+voisfksvmMTqvX7Lb7DY/L5/S6/Y7P6/f8vh8H FqjwR1hoeIiYqLjI2Oj4CBkpOUlZaXmJmekiyKnp+fnI2bkhUGp6ipqqKgDa6vq4Gitb+lprazib m3rL25unC0zrO0w8IyrIEQxczNx8pqzrLD2tBZ1LjZ0dZT2r7f1dEyA+Tl5ufo6err7O3u7+Dh8v P09fb3+Pn28OzqPv/w8woMCBBAsa1McPwrFABxs6fAgxosSJ7xLqoIgxo8aN/xw7irNoYOGXZNxi gTyJUkVJkylbugSxctXLmTQxxFRVM2ckkSNJ3USlMyjNn0CFGk1J9NTRpSCTmmIKtQ/PJj6dRr2a Z+oSklaxenXmVNjXscPCsiKLlpfZtGzTaI3BNWnbuaDW0r2LyS7evZP08v0r5S2MuEQBG27k97Di QokXOwYi+AXhn48rB4kcpqpcy5zxNO4M2hhmBpNvhj4d5zPq1WZUs37dxTXs2Vhk075tYTRpzYVx +64d9rfwKraHGxdS/LjyHsmXO8fR/Ln0GdGnc9a9W0N169xNbO8OPsT38ORLxywPG/sg3pTRs1af wPxK9/S9B6+PX/z9/Pzll//sfx18CPjHDYAG2rTfgQpGMN6C4DXoIFYCPsCeaREuCOGF0mWo4VIT DkCgNR0C9mGI0IxoIIcoGqfiijmVWOF5LubX4oy41WgjbTjmmBKM2iXII1k+ZrBjkCANiWBXRpJX 5JL8IHlBk052JuWU2kBpQZVWPqbllot16eVhYIYJ2JhkEoNlBWaeideabNLl5ps2mEWnknLOVGee Ft6Jp55+ishnn38OGs0FaQaKCKGKdoOoS4s+ilOjLUFK6VOSIlUppZdimumjnHkEaqiijkqqOoCV imqqqq56UEgCJsRqrLLOSus6rsIHa6267sorqaf2CmywwkJ0q3qbLtIppMf/npSsp4ZOuGwizS4a rUXTKlptQtcSmi0/2w7arYOHhrvXuOSei2666m5h7rruvgtvvPLOSy9f7daLb75z3KuvUPz2C3DA Aof7IYgDH4xwwgovzHDDYvzrcEIQR0xxxaJBa3HGGudQ8MYefwxyyCKPvGzHJJ+Mcsoqr8xyyy6/ bOTEMM9Mc80234xzzjrvTI3MPP+crc9AD72p0EQfjXTSSi/NdNOrGe101DxCLXXVVl+NtWEmZ811 115/XS7GYCtC9dhmn4122miKrXbbbr8N9y1lx013emzXfcfceHeh995+W9b334IPTnjhSARueGB3 J874aYg3DrlXj0dOeeWW/1+OuSSTZ855S5t3Dnroonf9+eimn4460aWnfjGurL8Oe+xWry577bbf 3vDWuO/OO4C09w7C78APn4jwxB+PfPLKK2388s4/D3300k9PPWSLV4999tpvz3333nPQ/PThf0/+ E+OXj74R56fPfvvuvw8/ues/P3/89rdu7P36j1H//v7/D8AASu56AiygAQ+IwAS6p38KbKADHwjB CEpwguRhIAUBaMEL+i+DGuygBz8IwhDqgYMibB8JS4jCFKowNwRcoQtfCMMbnDCGNKzhA2dowxzq MIA43KEPu9fD3QXxh0Ss3hCLiMQkKhGILTTgEZcoxCdCEXdSnKIVr4hF211VMYtc7KIXvwhGwm0x jGQsoxnPiMakjTGNl1sjGyvnxjfKcY50rGPI4mjHwuExj3zsI8726MdACpJlgBxdIQeJyKMdMpGM bKQju7XIR5otkp2jJOcsKUmvYfJyBQAAOw== ------=_NextPart_01C7F06B.3E621250 Content-Location: file:///C:/EC825262/CalcCh1Sec31B_files/slide0005.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Exponential Functions

The Savings Bond…

wMr. Maierhofer was given a $75 savings bond in 1965. It was originally purchased for $37.50 and was to be worth the $75 after about 10 years.= He didn’t cash = it in, and he found out in the summer of 2006 that it was fully matured (no longer earning interest). If the interest rate of the bond was 6.89%, how much was the bond worth when he cashed it in <= /span>last summer?

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Exponential Growth

wThis problem is an example of an exponential growth problem =

wCompound interest problems are modeled by the function <= /div>
ny =3D P • ax

nP =3D the initial investment amount

na =3D (1 + the annual interest rate (as a decimal)) = ;

nx is the number of years

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The Savings Bond…

w= So what was Mr. Maierhofer’s savings bond worth?

wOriginally purchas= ed in the summer of 1965 for $37.50

wEarned 6.89% interest

wWas cashed in the summer of 2005

wy =3D P • ax

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Exponential Function

wDefinition of Exponential Function

wLet a = be a positive real number other than 1. The function:

nf(x) =3D ax

wis the exponential function with base a

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Rules for Exponents

wax • a<= /i>y =3D ax+y

w

w

w

w

w(ax)y =3D= (ay)x =3D axy = ;

w=

wax • b<= /i>x =3D (ab)x

=

If a>0 and b>0, the following hold for all real numbers x and y.

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Population Growth

w= By about what percent is the population increasing annually?