[email protected] MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 1 Bruce Mayer, PE Chabot College...

[email protected] • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§4.1 ax

Functions



Review §

Any QUESTIONS About• §3.5 → Applied Optimization

Any QUESTIONS About HomeWork• §3.5 → HW-17

3.5

http://www.google.com/url?sa=i&source=images&cd=&cad=rja&docid=4fd_u7ErafVHkM&tbnid=xQe-3kC6wfL7kM:&ved=0CAgQjRwwAA&url=http://www.getacopywriter.com/blog/friday-funnies-episode-8-s-e-oh-no&ei=x4rkUejxLOjsiQKR8oC4Dw&psig=AFQjCNFnipEZp-3Fpkdu4U9FFb1ui36yMQ&ust=1374018631783452



§4.1 Learning Goals

Define exponential functions Explore properties of the natural

exponential function Examine investments involving

continuous compounding of interest

http://www.google.com/url?sa=i&rct=j&q=continuous+compounding&source=images&cd=&cad=rja&docid=fXu41b_DUyMoTM&tbnid=IsM6ipgorKYtxM:&ved=0CAUQjRw&url=http://erehweb.wordpress.com/&ei=y4vkUZi9OOf0iQKQ5YCgBw&psig=AFQjCNEQZD4yO5Qdc61-K9vfNJjlv6hhjg&ust=1374018812949762



Exponential Function

A function, f(x), of the form

f x ax , a 0 and a1,

is called an EXPONENTIAL function with BASE a.

The domain of the exponential function is (−∞, ∞); i.e., ALL Real Numbers



Recall Rules of Exponents

Let a, b, x, and y be real numbers with a > 0 and b > 0. Then

ax ay axy ,ax

ayax y ,

ab x axbx ,

ax y axy ,

a0 1,

a x 1

ax

1

a

x

.

yxaa yx thenif

Product Rule

Quotient Rule

Product to a Power Rule

Power to a Power Rule

Zero Power Rule

Negative Power Rule

Equal Powers Rule



Evaluate Exponential Functions

Example a. Let f x 3x 2. Find f 4 . Solution a. f 4 34 2 32 9

Example b. Let g x 210x. Find g 2 .

Solution b. g 2 210 2 21

102 21

100 0.02

b. g 2 210 2 21

102 21

100 0.02



Evaluate Exponential Functions

Example

Solution

c. Let h x 1

9

x

. Find h 3

2

.

c. Let h 3

2

1

9

3

2 9 1

3

2 93

2 27

c. Let h 3

2

1

9

3

2 9 1

3

2 93

2 27

-1 -0.5 0 0.5 1 1.5 20

1

2

3

4

5

6

7

8

9

x

y =

f(x)

= (

1/9

)x

MTH15 • Bruce Mayer, PE

XYf cnGraph6x6BlueGreenBkGndTemplate1306.m



Solve Exponential Equation

Solve the following for x

Using the Transitive Property

Need to state 2187 in terms of a Base-3 to a power

Using the Equal Powers Rule

187 2when3 32

xfxf x

xfxf x 187 23 32

73187 2

7333 2732

xx

2442 xxx



Example Graph y = f(x) =3x

Graph the exponential fcn: ( ) 3 .xf x

Make T-Table,& Connect Dots

x y

01

–12

–23

13

1/39

1/927

x

y

-5 -4 -3 -2 -1 1 2 3 4 5

4

3

6

2

5

1

-1

-2

78



Example Graph Exponential

Graph the exponential fcn:

Make T-Table,& Connect Dots

1( ) .

3

xf x

x y

01

–12

–2–3

11/33

1/99

27 • This fcn is a REFLECTION of y = 3x

3xy

x

y

-5 -4 -3 -2 -1 1 2 3 4 5

4

3

6

2

5

1

-1

-2

78

1( )

3

xy f x




Graph the exponential fcn: y1

2

x

.

Construct SideWays T-Table

x −3 −2 −1 0 1 2 3

y = (1/2)x 8 4 2 1 1/2 1/4 1/8

Plot Points and Connect Dots with Smooth Curve




As x increases in the positive direction, y decreases towards 0



Exponential Fcn Properties

Let f(x) = ax, a > 0, a ≠ 1. Then

A. The domain of f(x) = ax is (−∞, ∞).

B. The range of f(x) = ax is (0, ∞); thus, the entire graph lies above the x-axis.

C. For a > 1 (e.g., a = 7)i. f is an INcreasing function; thus, the graph

is RISING as we move from left to right

ii. As x→∞, y = ax increases indefinitely and VERY rapidly




Let f(x) = ax, a > 1, a ≠ 1. Then iii. As x→−∞, the values of y = ax get

closer and closer to 0.

D. For 0 < a < 1 (e.g., a = 1/5 = 0.2)i. f is a DEcreasing function; thus, the graph

is falling as we scan from left to right.

ii. As x→−∞, y = ax increases indefinitely and VERY rapidly

iii. As x→ ∞, the values of y = ax get closer and closer to 0




Let f(x) = ax, a > 0, a ≠ 1. Then

E. Each exponential function f is one-to-one; i.e., each value of x has exactly ONE target. Thus:

i. – The Basis of the Equal Powers Rule

ii. f has an inverse

2121 xxaa xx




Let f(x) = ax, a > 0, a ≠ 1. Then

F. The graph f(x) = ax has no x-intercepts • In other words, the graph of f(x) = ax

never crosses the x-axis. Put another way, there is no value of x that will cause f(x) = ax to equal 0

G. The x-axis is a horizontal asymptote for every exponential function of the form f(x) = ax.



ExponentialFcn ≠ PowerFcn

The POWER Function is the Variable (x) Raised to a Constant Power; e.g.:

• Note that PolyNomials are simply SUMS of Power Functions:

The EXPONENTIAL Function is a Constant Raised to a Variable Power (x); e.g.:

9/4087.0137 xxxx

1238 xxxx

x

xxx

99

57310



ExponentialFcn ≠ PowerFcn

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3-10

-8

-6

-4

-2

0

2

4

6

8

10

x

y =

f(x)

MTH15 • Power & Exponential Fcns

XYf cnGraph6x6BlueGreenBkGndTemplate1306.m

xy 3

3xy

The Exponential is NEVER Negative



Example Bacterial Growth

A technician to the Great French MicroBiologist Louis Pasteur noticed that a certain culture of bacteria in milk doubled every hour.

Assume that the bacteria count B(t) is modeled by the equation

B t 20002t ,• Where t is time in hours




Given Bacterial Growth Equation B t 20002t ,

Find:a) the initial number of bacteria,

b) the number of bacteria after 10 hours; and

c) the time when the number of bacteria will be 32,000.




a) INITIALLY time, t, is ZERO → Sub t = 0 into Growth Eqn:

B0 B 0 200020 20001 2000

b) At Ten Hours Sub t = 10 into Eqn:

b. B 10 2000210 2,048,000




c) Find t when B(t) = 32,000

Thus 4 hours after the starting time, the number of bacteria will be 32k

32000 20002t

16 2t24 2t

4 t



The Value of the Natural Base e The number e, an irrational number, is

sometimes called the Euler constant. Mathematically speaking, e is the fixed

number that the expression

approaches e as n gets larger & larger

The value of e to 15 places:

e = 2.718 281 828 459 045

n

n

11



The “Natural” base e The most “common” base for people is

10; e.g., 7.3x105

However, analysis of physical; i.e., Natural, phenomena leads to base e

Check the Definition Graphically

0.495% less than the actual e-Value

718.21

1lim

n

n ne

0 10 20 30 40 50 60 70 80 90 1001

1.25

1.5

1.75

2

2.25

2.5

2.75

n

y =

f(n

) =

(1

+ 1

/n)n

MTH15 • e Value

0 1 2 3 4 5 6 7 8 9 101

1.25

1.5

1.75

2

2.25

2.5

2.75

n

MTH15 • e Value

BMay er • 16Jul13BMay er • 16Jul13

215297048138294 2.100

11

100

100

e



MA

TL

AB

Co

de

% Bruce Mayer, PE% MTH-15 • 16Jul13% XYfcnGraph6x6BlueGreenBkGndTemplate1306.m%% The Limitsxmin = 0; xmax = 100; ymin = 1; ymax = 2.75;% The FUNCTIONx = linspace(xmin,xmax,1000); y = (1 +1./x).^x;% % The ZERO Lineszxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ymin ymax];%% the 6x6 Plotaxes; set(gca,'FontSize',12);whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Greensubplot(1, 2, 1)plot(x,y, 'LineWidth', 4),axis([xmin xmax ymin ymax]),... grid, xlabel('\fontsize{14}n'), ylabel('\fontsize{14}y = f(n) = (1 + 1/n)^n'),... title(['\fontsize{16}MTH15 • e Value',]),... annotation('textbox',[.75 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'BMayer • 16Jul13','FontSize',7)hold onplot([xmin, xmax], [2.7182818, 2.7182818], '--m', 'LineWidth', 2)set(gca,'XTick',[xmin:10:xmax]); set(gca,'YTick',[ymin:.25:ymax])hold off%%xmin1 = 0; xmax1 = 10; ymin1 = 1; ymax1 = 2.75;% The FUNCTIONn = linspace(xmin,xmax,1000); z = (1 +1./n).^n;% % The ZERO Lines%% the 6x6 Plotaxes; set(gca,'FontSize',12);whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Greensubplot(1, 2, 2)plot(n,z, 'LineWidth', 4),axis([xmin1 xmax1 ymin1 ymax1]),... grid on, xlabel('\fontsize{14}n'),... title(['\fontsize{16}MTH15 • e Value',]),... annotation('textbox',[.75 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'BMayer • 16Jul13','FontSize',7)hold onplot([xmin1, xmax1], [2.7182818, 2.7182818], '--m', 'LineWidth', 2)set(gca,'XTick',[xmin1:1:xmax1]); set(gca,'YTick',[ymin1:.25:ymax1])



The NATURAL Exponential Fcn

The exponential function

with base e is so prevalent in the sciences that it is often referred to as THE exponential function or the NATURAL exponential function.

f x ex



Compare 2x, ex, 3x Several

ExponentialFunctionsGraphically• Note that

EVERY Exponetial intercepts the y-Axisat x = 1




Graph f(x) = 2 − e−3x

SOLUTIONMake T-Table,Connect-Dots

22

1.951

1

−18.09

−401.43

y = f(x)

0

−1

−2

x



Exponential Growth or Decay

Math Model for “Natural” Growth/Decay:

A t A0ekt

A(t) = amount at time t A0 = A(0), the initial, or time-zero, amount

k = relative rate of • Growth (k > 0); i.e., k is POSITIVE • Decay (k < 0); i.e., k is NEGATIVE

t = time



Exponential Growth

An exponential GROWTH model is a function of the form

00 keAtA kt

where A0 is the population at time 0, A(t) is the population at time t, and k is the exponential growth rate • The doubling time is the amount of time

needed for the population to double in size

A0

A(t)

t

2A0

Doubling time

kteAA 0



Exponential Decay

An exponential DECAY model is a function of the form

00 keAtA kt

where A0 is the population at time 0, A(t) is the population at time t, and k is the exponential decay rate • The half-life is the amount of time needed

for half of the quantity to decay

A0

A(t)

t½A0

Half-life

kteAA 0



Example Exponential Growth

In the year 2000, the human population of the world was approximately 6 billion and the annual rate of growth was about 2.1 percent.

Using the model on the previous slide, estimate the population of the world in the years

a) 2030

b) 1990




SOLUTION a) Use year 2000 as t = 0 Thusfor 2030 t = 30

A0 6

k 0.021

t 30

A t 6e 0.021 30

A t 11.265663

The model predicts there will be 11.26 billion people in the world in the year 2030




SOLUTION b) Use year 2000 as t = 0 Thusfor 1990 t = −10

The model postdicted that the world had 4.86 billion people in 1990 (actual was 5.28 billion).

A0 6

k 0.021

t 10

A t 6e 0.021 10

A t 4.8635055



Compound Interest Terms

INTEREST ≡ A fee charged for borrowing a lender’s money is called the interest, denoted by I

PRINCIPAL ≡ The original amount of money borrowed is called the principal, or initial amount, denoted by P• Then Total AMOUNT, A, that accululates in

an interest bearing account if the sum of the Interest & Principal → A = P + I




TIME: Suppose P dollars is borrowed. The borrower agrees to pay back the initial P dollars, plus the interest amount, within a specified period. This period is called the time (or time-period) of the loan and is denoted by t.

SIMPLE INTEREST ≡ The amount of interest computed only on the principal is called simple interest.




INTEREST RATE: The interest rate is the percent charged for the use of the principal for the given period. The interest rate is expressed as a decimal and denoted by r.

Unless stated otherwise, it is assumed the time-base for the rate is one year; that is, r is thus an annual interest rate.



Simple Interest Formula

The simple interest amount, I, on a principal P at a rate r (expressed as a decimal) per year for t years is

I Prt.



Example Calc Simple Interest

Rosarita deposited $8000 in a bank for 5 years at a simple interest rate of 6%

a) How much interest-$’s will she receive?

b) How much money will she receive at the end of five years?

SOLUTION a) Use the simple interest formula with:

P = 8000, r = 0.06, and t = 5



Example Calc Simple Interest

SOLUTION a) Use Formula

I Prt

I $8000 0.06 5 I $2400

SOLUTION b) The total amount, A, due her in five years is the sum of the original principal and the interest earned

AP IA$8000 $2400

A$10, 400



Compound Interest Formula

AP 1 rn

nt

A = $-Amount after t years P = Principal (original $-amount) r = annual interest rate (expressed as a

decimal) n = number of times interest is compounded

each year t = number of years



Compare Compounding Periods

One hundred dollars is deposited in a bank that pays 5% annual interest. Find the future-value amount, A, after one year if the interest is compounded:

a) Annually.

b) SemiAnnually.

c) Quarterly.

d) Monthly.

e) Daily.




SOLUTIONIn each of the computations that follow, P = 100 and r = 0.05 and t = 1. Only n, the number of times interest is compounded each year, is changing. Since t = 1, nt = n∙1 = n.

a) AnnualAmount: AP 1

r

n

n

A100 1 0.05 $105.00




b) Semi Annual Amount:

AP 1r

n

n

A100 10.05

2

2

$105.06

AP 1r

4

4

A100 10.05

4

4

$105.09

c) Quarterly Amount:




d) Monthly Amount: AP 1

r

12

12

A100 10.05

12

12

$105.12

AP 1r

365

365

A100 10.05

365

365

$105.13

e) Daily Amount:



Continuous Compound Interest

The formula for Interest Compounded Continuously; e.g., a trillion times a sec.

A = $-Amount after t years P = Principal (original $-amount) r = annual interest rate (expressed as a

decimal) t = number of years

rtPeA



Example Continuous Interest

Find the amount when a principal of $8300 is invested at a 7.5% annual rate of interest compounded continuously for eight years and three months.

SOLUTION: Convert 8-yrs & 3-months to 8.25 years. P = $8300 and r = 0.075 thenuse Formula

APert

A$8300e 0.075 8.25

A$15, 409.83



Compare Continuous Compounding

Italy's Banca Monte dei Paschi di Siena (MPS), the world's oldest bank founded in 1472 and is today one the top five banks in Italy

If in 1797 Thomas Jefferson Placed a Deposit of $450k in the MPS bank at an interest rate of 6%, then find the value $-Amount for the this Account in 2010; 213 years Later




SIMPLE Interest

AP Prt P 1 rt A$450,000 1 0.06 213 A$6.201 million.

YEARLY Compounding

AP 1 r t $450,000 1 0.06 213

A$1.105 1011

A$110.5 million.




Quarterly Compounding

Continuous Compounding

GigaBucks) (145.3 B 3.145$

103.145$4

0.061$450,0001

9

4213

A

A

rPA rt

GigaBucks) (159.8 B 8.159$

108.159$

e$450,0009

21306.0

A

A

PeA rt



Account $Value for $450k invested at 6% Interest for 213 Years

159.80

145.30

0.11

0.01

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170

Co

nti

nu

ou

sQ

ua

rte

rly

Ye

arl

yS

imp

le

Inte

res

t C

om

po

un

din

g

Account Value ($B)M55_Sec9_1_Compare_Compounding_0810.xls



Effective Interest rate → APR

To help people compare simple, MultiPeriod-compounded, and continuous-compounded Interest rates, ALL advertised interest rates are stated in the effective Annual Percentage Rate, or APR or re

APR is the simple annual interest, re , that produces the same Change in $-Value in ONE year



Effective Interest rate → APR

APR Defined• For MultiPeriod Compounding

at k times per year• For Continuous

Compounding– Where r is the stated, or nominal, interest rate

When Assessing a Loan or a Savings Instrument the Consumer should consider ONLY the APR for comparisons

k

rre 1

1 re er



WhiteBoard Work

Problems From §4.1• P71 → Beer-Lambert (absorption) Law

See AlsoENGR45

http://www.google.com/url?sa=i&rct=j&q=beer-lambert%20law&source=images&cd=&cad=rja&docid=U6CQ--xLKasHuM&tbnid=jOBsgM3C19b7dM:&ved=0CAUQjRw&url=http://pharmaxchange.info/press/2012/04/ultraviolet-visible-uv-vis-spectroscopy-%E2%80%93-derivation-of-beer-lambert-law/&ei=Q7DlUYrNMYrwiwLv74HoDQ&bvm=bv.49405654,d.cGE&psig=AFQjCNEfOe1YozXEbRR5YCpmSJ68s0xp-Q&ust=1374093529381697



All Done for Today

Very GoodCar Loan

RateBut what about the Purchase

$-Price, and Loan Fees?

http://www.google.com/url?sa=i&rct=j&q=annual+percentage+rate&source=images&cd=&cad=rja&docid=M6ThUgUA9QAj9M&tbnid=OaJfydc_QUnxHM:&ved=0CAUQjRw&url=http://www.thetruthaboutcreditcards.com/what-is-credit-card-apr/&ei=B7PlUca2Lq7siwKpnIEY&bvm=bv.49405654,d.cGE&psig=AFQjCNGfGc-AZwt8GPShzxzWHpN4wpXjew&ust=1374094092556509



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

Appendix

–

srsrsr 22



ConCavity Sign Chart

a b c

−−−−−−++++++ −−−−−− ++++++

x

ConCavityForm

d2f/dx2 Sign

Critical (Break)Points Inflection NO

InflectionInflection







http://www.google.com/url?sa=i&rct=j&q=derivative+power+rule&source=images&cd=&cad=rja&docid=NjbfKj_JZBSzwM&tbnid=jJkxz7X095T-NM:&ved=0CAUQjRw&url=http://mathworld.wolfram.com/Derivative.html&ei=TfPWUe_nA6akiQKZ64DIDw&bvm=bv.48705608,d.cGE&psig=AFQjCNGw-AM0Kc2tARyUGs-zPM7iK-qH3g&ust=1373127702659906

http://www.google.com/url?sa=i&rct=j&q=related+rates+calculus&source=images&cd=&cad=rja&docid=wSGtOAIrRxIoIM&tbnid=l4Q1f5xFVNRraM:&ved=0CAUQjRw&url=http://www.cantonschools.org/~lforastiere/ap%20calculus?Templates=Printable&ei=2TLcUY-gOa2UjALp4IHgDw&bvm=bv.48705608,d.cGE&psig=AFQjCNH42r6LfHpjrPyJXK2DJrvdwHWZIQ&ust=1373471717102765

http://www.google.com/url?sa=i&rct=j&q=related+rates+calculus&source=images&cd=&cad=rja&docid=nuuUeFKHM5pqjM&tbnid=x7raEhpCGJ8edM:&ved=0CAUQjRw&url=http://designatedderiver.wikispaces.com/Games+and+fun+stuff&ei=IzPcUYPxBaOLiwLjs4GYAQ&bvm=bv.48705608,d.cGE&psig=AFQjCNH42r6LfHpjrPyJXK2DJrvdwHWZIQ&ust=1373471717102765

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