Precalculus Name:____________________________________ Prequiz- Logs and Exponential Functions Date:_____________________________________ MULTIPLE CHOICE (INDICATE YOUR ANSWERS IN THE SPACE PROVIDED): (1) The expression (-3x2y3)3 is equivalent to: (1) (a) -9x6y9 (b) -27x5y6 (c) -27x6y9 (d) -3x5y6

(2) Simplify: )3()3( 4a2a � (2) (a) Error! Bookmark not defined. 4aa2

3 �� (b) 4a33 � (c) 6a23 � (d) a4a3

3 �

(3) What is axlog3 written in exponential form? (3) (a) 3x = a (b) a3 = x (c) Error! Bookmark not defined.ax = 3 (d) 3a = x

(4) The equation y = ax expressed in logarithmic form is: (4)

(a) ylogx a (b) yloga x (c) alogx y (d) xlogy a

(5) The expression log 12 is equivalent to: (5)

(a) log 3 + 2 log 2 (b) log 6 + log 6 (c) log 3 x log 4 (d) log 3 – 2 log 2

(6) The expression log 4x is equivalent to: (6) (a) 4 log x

(b) log 4 + log x (c) (log 4)(log x) (d) log x4

2

(7) The expression balog

3

is equivalent to:

(7)

(a) ¸̧¹

·¨̈©

§blogalog3

(b) )bloga(log31

�

(c) )bloga(log3 � (d) blogalog3 �

(8) The expression blog21alog � is equivalent to:

(8) (a) )balog( �

(b) balog

(c) � � ¸¹·

¨©§ blog

21alog

(d) ablog

(9) If A = pr2, which equation is true? (9) (a) rlog2pAlog �

(b) )r(logp2Alog (c) rlog2logplogAlog �� (d) rlog2plogAlog �

(10) Which of the following equations is equivalent to 5log33log73logx � ? (10) (a) 3x7 53

(b) 125)7x( 3 � (c) 37x 53 � (d) 1521x3 �

3

Precalculus Name:__________________________________ Lesson- properties, equations with exponents and power and exponential functions Date:___________________________________ Objectives: x use the properties of exponents

x solve equations containing rational exponents x examine power and exponential functions

Do Now: Use the exponential properties to simplify and rewrite the following expressions: (1) x yx aa

(2) � � yxa

(3) � � xab

(4) ¸¹·

¨©§

x

ba

(5) y

x

aa

(6) �xa

(7) 0a

__________________________________________________________________________________________

In Small Groups: Use each example in the “Do Now” to arrive at general rules as they apply to monomials with exponents.

Using Exponential Function Properties to Solve for x: Process 1 Process 2 Examples (each relates to “Process 1”): 1. 2214 44 �� xx 2. 1215 164 �� xx 3. 493

2 � xx

4

More Examples (each relates to “Process 2”): 4. 814 x 5. 41 �x 6. 32)12( 5 � �x Power function: exponential function:

Small Group Activity

On your graphing calculator, simultaneously graph: y = 0.5x, y = 0.75x, y = 2x, y = 5x

(1) What is the range of each exponential function?

(2) What is the behavior of each graph?

(3) Do the graphs have any asymptotes?

(4) (a) What point is on the graph of each function?

(b) Why?

Characteristics of graphs of y = nx

n > 1 0 < n < 1

domain

range

y-intercept

behavior

horizontal asymptote

vertical asymptote

Extension: Graph the exponential functions y = 2x, y = 2x + 3, and y = 2x – 2 on the same set of axes.

Compare and contrast the graphs using a table similar to the one above.

5

Precalculus Name:__________________________________ Lesson- Graphing exponential functions, exponential growth and decay Date:___________________________________ Objectives: x graph exponential functions

x use exponential functions to determine growth and decay

Using Exponential Functions for Real World Applications: Exponential growth: Exponential decay: Exponential Growth or Decay: N = N0 (1 + r)t

(1) Write a formula that represents the average growth of the population of a city with a rate of 7.5% per year. Let x represent the number of years, y represent the most recent total population of the city, and A is the city’s population now. What is the expected population in 10 years if the city’s population now is 22,750 people? Graph the function for 0 d x d 20.

6

7

Precalculus

8

9

10

Precalculus Name:______________________________ Lesson: Linear and exponential models Date:_______________________________ Objective: x write linear and exponential functions to model depreciation Def.: Practical Domain: (1) At Happy Motors, a car salesperson offers a sale on two types of cars: Car A has a sticker price of

$17,000, but depreciates in value exponentially at an average rate of 20% each year. Car B also has a sticker price of $17,000, but depreciates to $2,000 in 10 years, which represents a linear depreciation.

(a) Write an exponential function A(t) that relates the value of Car A in dollars to time t in years.

(b) Write a linear function B(t) that relates the value of Car B in dollars to time t in years.

(c) What is the depreciated value of both cars after 3 years?

(d) In 2004, Rodd decides to buy Car A and Todd decides to buy Car B. Will their cars ever have the same value again, and if so, during what year(s)?

(e) Sketch the graphs and indicate the WINDOW used.

11

(2) Rachel purchased a new stereo system for $2,000 and is assumed to have a depreciated value of $800 after eight years, which represents a linear depreciation during this period. In the same year, her best friend Joshua bought his new stereo system for $2,500 and is assumed to have an average exponential depreciation rate of 17.5% per year.

(a) Write a linear function R(t) that relates the value of Rachel’s stereo in dollars to time t in years.

(b) Write an exponential function J(t) that relates the value of Joshua’s stereo in dollars to time t in years.

(c) What is the depreciated value of each stereo after 1 year?

(d) Rachel and Joshua both bought their stereos in 2003, but spent different amounts of money. Will their stereos ever be the same value again, and if so, during what year(s)?

(e) Sketch the graphs and indicate the WINDOW used. (3) Ms. Ingram wants to buy a new computer that will retain its value. She has a choice of buying the Alpha

Computer for $5,000 or the Beta Computer for $4,000. The Alpha Computer is assumed to have a depreciated value of $1,400 after three years, which represents a linear depreciation. The Beta Computer depreciates in value exponentially at an average rate of 30% each year.

(a) Write a linear function A(t) that relates the value of the Alpha Computer in dollars to time t in

years.

(b) Write an exponential function B(t) that relates the value of the Beta Computer in dollars to time t in years.

(c) If Ms. Ingram wants to be able to sell this computer after 4 years, which model will have a higher value?

(d) Sketch the graphs and indicate the WINDOW used.

12

Precalculus Name:__________________________________ Lesson- Compound Interest Date:___________________________________ Objectives: x use exponential functions to determine compound interest

Do Now:

(1) A laser printer was purchased for $300 in 2001. If its value depreciates at a rate of 30% a year, determine how much it will be worth in 2007.

(2) Rates can be compounded in different increments per year. Exponential growth occurs how often if the

rate is compounded:

annually:

bi-annually:

quarterly:

monthly:

weekly:

daily:

The general equation for exponential growth is modified for finding the balance in an account that earns compound interest.

Compound Interest: tn

nr1PA ¸¹·

¨©§ �

9 13

1014

11

Precalculus

15

12 16

1317

20

Precalculus

21

22

Precalculus

23

24

Precalculus Name:____________________________________ Lesson- Properties of a logs, rewriting Exponential functions as logarithms, log graphs Date:_____________________________________ Objective:

x To learn what a logarithm is x To learn the properties of logs x To learn to rewrite an exponential function as a logarithm x Graphing logs x

Do Now: Solve for x: 193 � xx and check. _________________________________________________________________________________________ What is a logarithm? Logarithms are inverses of exponential functions. Logarithms are functions because exponential functions are one-to-one functions. We cannot solve an equation like: xy 2 using the algebraic techniques we have learned so far. Therefore, we must try an alternative technique. Rule: xytoequivalentisbx b

y log The log to the base b is the exponent to which b must be raised to obtain x.

Properties of Logs 01logb

1blogb

xblog xb

xb xlog b , where x > 0 NlogMlogMNlog bbb �

NlogMlogNMlog bbb �

MlogpMlog bp

b Example: Convert each into logarithmic form Convert each into logarithmic form

1. xy 2 4. 215log25

2. 93 5. cba log

A different way of looking at an exponential function.

25

3. 1551 � 6. 2

91log3 � ¸¹·

¨©§

__________________________________________________________________________________________ What is a Natural Logarithm? Rule: xytoequivalentisbx b

y log The log to the base b is the exponent to which b must be raised to obtain x.

Properties of Logs 01ln 1ln b

xex ln

xe x ln , where x > 0 NMMN lnlnln �

NMNM lnlnln �

MpM p lnln Example: Convert each into logarithmic form Convert each into logarithmic form 1. xey 4. x 5ln 2. xe 5. cb ln

3. 11 � ee

6. 2ln � y

Example: Graph each of the following on the same set of axes using the graphing calculator.

yxxy

xy

y

x

2

2

log.4log.3

2.22.1

yxxy

exey

y

x

ln.8ln.7

.6

.5

x

y

26

Precalculus

ORJE[�\�

������ ����� �������

27

�

�

28

Precalculus

29

30

Precalculus Name:____________________________________ Activity- Graphing Log Equations **Will be collected and graded (separate paper) Date:_____________________________________ Objective: To learn how to graph log equations that are not of base 10 or e. DO NOW: Find 7log3 to the nearest ten-thousandths place. __________________________________________________________________________________________ I. Graph each of the following on the same set of coordinate axes and answer the following questions. 1. )1(log2 � xy 2. )1(log3 � xy 3. )1(log4 � xy 4. )1(log5 � xy a. What are some notable similarities and differences among the graphs? b. What appears to happen as the base gets larger and larger? II. Graph each of the following on the same set of coordinate axes and answer the following questions. 1. )1(log2 � xy 2. )2(log2 � xy 3. )3(log2 � xy 4. )4(log2 � xy a. What are some notable similarities and differences among the graphs? b. What appears to happen as the constant in the binomial changes? HW p313 #77-85 odd, 97-100 all

31

Precalculus

32

33

Precalculus

34

35

36

Unit 6: Exponential & Logarithmic Functions Definitions, Properties & Formulas

Properties of Exponents

Property Definition Product baba xxx �

Quotient bab

a

xxx � , where x z 0

Power Raised to a Power (xa)b = xab Product Raised to a Power (xy)a = xa ya

Quotient Raised to a Power a

aa

yx

yx

¸̧¹

·¨̈©

§ , where y z 0

Zero Power x0 = 1, where x z 0

Negative Power nn

x1x � , where x z 0

Rational Exponent nn

1

xx for any real number x t 0 and any integer n > 1

and when x < 0 and n is odd

Exponential Growth/Decay

N = N0 (1 + r)t

where: N is the final amount, N0 is the initial amount, t is the number of time periods, and r is the average rate of growth(positive) or decay(negative) per time period

Compound Interest (Periodic)

tn

nr1PA ¸¹·

¨©§ �

where: A is the final amount, P is the principal investment, r is the annual interest rate, n is the number of times interest is compounded each year, and t is the number of years

Exponential Growth/Decay (in terms of e)

N = N0 ekt

where: N is the final amount, N0 is the initial amount, t is the number of time periods, and k (a constant) is the exponential rate of growth(positive) or decay(negative) per time period

Continuously Compounded

Interest

A = Pert

where: A is the final amount, P is the principal investment, r is the annual interest rate, and t is the number of years

37

Logarithmic Functions

are inverses of exponential functions

? a logarithm is an exponent!

Common Logarithms

when no base is indicated, the base is assumed to be 10

xlogxlog 10

x10yxlog y l ?

Change of Base Formula

alognlognlog

b

ba

where a, b, and n are positive numbers, and a z1, b z1

Natural Logarithms

instead of log, ln is used; these logarithms have a base of e

ln x xloge

? ln x = y xey l

all properties of logarithms also hold for natural logarithms

Properties of Logarithmic Functions

If b, M, and N are positive real numbers, b z 1, and p and x are real numbers, then:

Definition Examples

01logb written exponentially: b0 = 1

1blogb written exponentially: b1 = b

xblog xb written exponentially: bx = bx

xb xlog b , where x > 0 710 7log 10

NlogMlogMNlog bbb � xlog9logx9log 333 �

zlogylogyzlog51

51

51

�

NlogMlogNMlog bbb �

5log2log52log 444 �

xlog7logx7log 888 �

MlogpMlog bp

b 6logx6log 2

x2

ylog4ylog 54

5

NlogMlog bb if and only if M = N )2x5(log)4x3(log 66 � �

)2x5()4x3( � �?

38

Properties of Logarithmic Functions If b, M, and N are positive real numbers, b z 1, and p and x are real numbers, then:

Definition Examples

01logb written exponentially: b0 = 1

1blogb written exponentially: b1 = b

xblog xb written exponentially: bx = bx

xb xlog b , where x > 0 710 7log 10

NlogMlogMNlog bbb � xlog9logx9log 333 �

zlogylogyzlog51

51

51

�

NlogMlogNMlog bbb �

5log2log52log 444 �

xlog7logx7log 888 �

MlogpMlog bp

b 6logx6log 2

x2

ylog4ylog 54

5

NlogMlog bb if and only if M = N )2x5(log)4x3(log 66 � �

)2x5()4x3( � �?

Common Errors:

NlogMlogNlogMlog

bbb

b �z NMlogNlogMlog bbb �

NlogMlog

b

b cannot be simplified

NlogMlog)NM(log bbb �z� MNlogNlogMlog bbb �

)NM(logb � cannot be simplified

Mlogp)M(log bp

b z p

bb MlogMlogp p

b )M(log cannot be simplified

39

Precalculus Name:____________________________________ Review- Exponential and Logarithmic Functions part 1 Date:_____________________________________

ANSWER THE FOLLOWING QUESTIONS ON A SEPARATE SHEET OF PAPER AND SHOW ALL WORK!

Write each expression in terms of simpler logarithmic forms:

(1) yxlog 5b

(2) 7

54

b uslog

(3) 8b c1log

(4) pnmlog

35

b

Given loga n, evaluate each logarithm to four decimal places: (5) 42log3 (6) 5log

21 (7) 00098.0log6

Solve each equation and round answers to four decimal places where necessary: (8) 3xlog2 �

(9) 36logxlog4log 555 �

(10) x5.0e751000

(11) 2xlog 6 �

(12) x491log7

(13) 214logx

(14) 5.2710x

(15) )3xlog(2log5logxlog �� �

(16) 12logxlog �

(17) 3xlog4

(18) 2log3)x5(log 99 �

(19) 1xlog20log �

(20) x4002.12

(21) 25.1e x25

(22) 2)5xlog()10xlog( ���

(23) xlog36log21216log 666 �

40

Precalculus Name:____________________________________ Review- Exponential and Logarithmic Functions part 2 Date:_____________________________________ SHOW ALL WORK: (1) Anthony is an actuary working for a corporate pension fund. He needs to have $14.6 million grow to $22

million in 6 years. What interest rate (to the nearest hundredth of a percent) compounded annually does he need for this investment?

(2) The number of guppies living in Logarithm Lake doubles every day. If there are four guppies initially: c. Express the number of guppies as a function of the time t. d. Use your answer from part (a) to find how many guppies are present after 1 week? e. Use your answer from part (a) to find, to the nearest day, when will there be 2,000 guppies?

abc

41

SHOW ALL WORK: (3) The relationship between intensity, i, of light (in lumens) at a depth of x feet in Lake Erie is given by

x00235.012ilog � . What is the intensity, to the nearest tenth, at a depth of 40 feet?

(4) Tiki went to a rock concert where the decibel level was 88. The decibel is defined by the formula

0iilog10D , where D is the decibel level of sound, i is the intensity of the sound, and i0 = 10 -12 watt per

square meter is a standardized sound level. Use this information and formula to find the intensity of the sound at the concert.

42

SHOW ALL WORK: (5) How many years, to the nearest year, will it take the world population to double if it grows continuously at

an annual rate of 2%.

(6) Bank A pays 8.5% interest compounded annually and Bank B pays 8% interest compounded quarterly. If you invest $500 over a period of 5 years, what is the difference in the amounts of interest paid by the two banks?

(7) Determine how much time, to the nearest year, is required for an investment to double in value if interest is earned at the rate of 5.75% compounded quarterly.

(8) Jenny wants to buy a new flat screen television that will retain its value. She has a choice of buying the Luxury Brand television for $3,000 or the Expensive Brand television for $3,500. The Luxury Brand is assumed to have a depreciated value of $1,200 after three years, which represents a linear depreciation. The Expensive Brand depreciates in value exponentially at an average rate of 37.5% each year.

(a) Write a linear function L(t) that relates the value of the Luxury Brand television in dollars to time t

in years.

(b) Write an exponential function E(t) that relates the value of the Expensive Brand television in dollars to time t in years.

(c) If Jenny wants to be able to sell this flat screen television after 5 years, which model will have a higher value? Explain your answer.

(d) Sketch the graphs and indicate the WINDOW used.

43

Precalculus Name:__________________________ Activity- Review of Expoenentials and Logs Date:___________________________ The accompanying diagrams contain exponential and logarithmic expressions and equations. When cut out, the 18 equilateral triangles fit together to form a large rhombus. For the triangles to create this shape, two expressions that are equivalent must be touching each other, sharing the same edge. All triangles must be used to complete the rhombus. There are expressions that have either the same or similar answers, so check your work and each pairing carefully; otherwise you may find triangles that do not fit properly.

SHOW ALL WORK ON A SEPARATE SHEET OF PAPER!

44

Precalculus Name:____________________________________ Test x 2 - Winter Project Date:_____________________________________ Objective: To use exponential & log functions to design a plan to save $1 million as quickly as possible. Research You will need: ~Job title, description, and salary ~Savings (assume interest rate is constant)/Investment Information ~Living expenses (including utilities, phone, groceries, entertainment, etc) ~Place to live (and amount of rent and renter’s insurance or mortgage and taxes ~Car/transportation expenses ~Miscellaneous expenses ~Prior debt (student loans, etc) Math You will need to include: ~Written explanation of your scenario (typed, double spaced, 12 point TNR font)

~exponential and logarithmic equations and their solutions or TVM Solver Data ~Graphs that model the rate of profit/income growth ~Written conclusion discussing the viability of your scenario Due Friday January 9, 2009 You will have (2) class sessions before the due date during which you may conduct research, ask questions of me, conduct mathematical computations, and/or work on the verbal portion of the project. We will also have (2) class sessions in a computer lab where we will:

1. Learn how to create MS Word documents consisting of mathematical equations 2. Be able to conduct research for our projects.

IDEAS? Student Loans- Savings Accounts- Transportation- Own/Rent House- Insurance- Jobs- Miscellaneous-

45

Precalculus Name:____________________________________ Model- Winter Project Calculations (Bland) Date:_____________________________________ Job title, description, and salary: Math Teacher, Teach Math, $4750 per month (used Median career value) Savings: 4% Savings account, deposit income – expense each month Expenses that don’t go away Electric, Gas, Oil: $400 per month Phone: $75 per month Groceries: $500 per month Entertainment: $350 per month Rent (including insurance): $1500 per month Car expenses (maintenance): $50 per month Car insurance: $100 per month Expense that expires after 5 years Car Payment: $333 per month Expense that expires after 20 years Student loans: $373 per month Prior Savings $50,000 I had two expenses that did not carry on forever. Therefore I decided to break my project up into phases. Phase I Phase II Phase III

50 dd t Years 205 d� t Years 20!t Years Income Expenditures $4750 $400

$75 $500 $350 $1500 $50 $100 $333 $373

Surplus of $1069/month Surplus of $1402/month Surplus of $1775/month Phase I: After 5 years, I now have a total of $131,923.4374 saved Phase II: After 20 years, I now have a total of $585,159.3123 saved How long will it take me to arrive at a savings of $1,000,000? I solved for N in TVM SOLVER and arrived at approximately 94.8568814 months beyond the 20th year. This gives me a total of approximately 27 years, 10 months, 25 days, 16 hours, 57 minutes and 16 seconds to arrive at $1,000,000 based on this information. **Note- Two very important things to be aware of: a. I never got a raise! Do you think you might? How much? When? and b. The costs in my scenario never increased! What about inflation? Higher taxes, etc?

Income Expenditures $4750 $400

$75 $500 $350 $1500 $50 $100

Income Expenditures $4750 $400

$75 $500 $350 $1500 $50 $100 $373

46

Precalculus Name:____________________________________ Lesson- Math on MSWORD Date:_____________________________________ Objective: To learn to use Microsoft Word to create math related documents. Example: 1. 2)2(2 2 �� xy 2.

On Your Own:

116

)2(25

)2( 22

�

�� xy

3

4 33 25 5

C § · § ·¨ ¸ ¨ ¸© ¹ © ¹

1 5xxd d

Exit Ticket- MSWORD **Print out and hand in at the end of class

1. 2 212 36 36

6 6x x x

x� � �

y�

2. 0tan 59 = 15c

Given:

Isosceles triangle CAT, CT AT# , ST bisects < CTA, SC and SA are drawn

Prove: <SCA # SAC

47

Statistics Name:____________________________________ Lesson/HW- TVM Solver Date:_____________________________________ Objective: To learn how to use the TVM Solver on the TI-83/84 to determine exponential growth and decay

as they apply to: x Savings accounts x Mortgages x Student loan repayment

__________________________________________________________________________________________ Compound Interest Formula: Mini Example: _________________________________________________________________________________________ Fields in the TVM

The variables listed in the TVM solver are called 'fields'. Each variable represents a quantity associated with a common finanial concept or formula.

x N: This represents the number of compounding periods in the term of the investment, annuity or loan. This will always be a positive value.

x I%: This represents the 'nominal rate' for an investment, annuity or loan. This will always be a positive value. Note: We write the percent form here, not the fraction or decimal form of a percent.

x PV: This represents the 'present value' of an investment, loan or annuity. This number can be positive or negative. If the number is positive, then it indicated money was collected as in a loan. If the number is negative, then it represents money we paid out, as in an investment or loan where we are the lender.

x PMT: This represents the payment made to build an annuity or pay off a loan. The value will always be negative in these situations. If we have a 'payout' annuity, then the value will be positive. In either case, the value represents the payment per compounding period.

x FV: This represents the 'future value' of an investment, annuity or loan after N compounding periods have passed. This value will be positive or negative depending on the signs of PV and PMT.

x P/Y: This value represents the number of payments per year for annuities and loans. x C/Y: This represents the number of compounding periods per year. These must both be positive integers

greater than 1. x PMT: END BEGIN. This field allows one to set the TVM Solver for 'ordinary' annuities, (END), or

annuities 'due' (BEGIN).

48

Ex 1: Sue Simmons wants to re-finance her house. She currently owes $120,000 and closing costs will be $4,500. She gets a 30-year mortgage at 6% nominal interest. How large will her monthly payment be?

Ex 2: Mike makes an initial deposit in a new savings account of $10,000. If this account accrues interest at a

rate of 3.9% compounded monthly and Mike deposits $500 per month, how many years (to the nearest month) will it take him to have $1,000,000 in his account?

Ex 3: Professor X had $50,000 in outstanding student loans at a 6.5% interest rate upon finishing grad school.

If he plans on paying the loan off in 10 years, what will his monthly payment be? How much in total interest will he have paid at the end of the 10 years?

On Your Own 1. Duke plans on purchasing a 3BR house in Scarsdale for $700,000. He takes out a mortgage for

$750,000 to pay for realtor expenses, the first few months of utilities and taxes, and for some minor cosmetic work on the house. The mortgage he qualifies for is a 30 year loan at 9% nominal interest. What will his monthly payment be if he somehow got away with putting $0 as a down payment? What will his monthly payment be if he put $100,000 as a down payment? How much in total interest will be paid over the life of the 30 year mortgage in each case?

N= I%= PV= PMT= FV= P/Y= C/Y= PMT: END BEGIN

N= I%= PV= PMT= FV= P/Y= C/Y= PMT: END BEGIN

N= I%= PV= PMT= FV= P/Y= C/Y= PMT: END BEGIN

N= I%= PV= PMT= FV= P/Y= C/Y= PMT: END BEGIN

N= I%= PV= PMT= FV= P/Y= C/Y= PMT: END BEGIN

49

2. Revisit example 3 from the lesson. Professor X decides to consolidate his loans over a 20 year period at 6% interest. How much more in interest will he have paid than on the 10 year plan?

Extra Credit- 5 Test Points (all or nothing) Suppose you currently live with your parents but would like to purchase a house of your own. You add

up all your current monthly expenses and subtract them from your monthly net salary and discover a $1,500 surplus. You also have $40,000 in a savings account accruing interest at a 3.15% rate. You deposit your surplus of $1,500 each month for a year before purchasing a house. You apply for a 30 year mortgage and get approved for $550,000 at a 8.3% interest rate. You are unsure if you can afford a house that costs $550,000. Use the TVM Solver to determine how much of a mortgage you can afford to take out.

Savings Account TVM Mortgage TVM

N= I%= PV= PMT= FV= P/Y= C/Y= PMT: END BEGIN

N= I%= PV= PMT= FV= P/Y= C/Y= PMT: END BEGIN

N= I%= PV= PMT= FV= P/Y= C/Y= PMT: END BEGIN