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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: (2)

(a) (b) (c) (d)

(3) What is written in exponential form?(3)

(a) 3x = a (b) a3 = x (c) ax = 3 (d) 3a = x

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

(a) (b) (c) (d)

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

(a) log 3 + 2 log 2(b) log 6 + log 6(c) log 3 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

(7) The expression is equivalent to:

(7)

(a)

(b)

(c) (d)

(8) The expression is equivalent to:

(8)

(a) (b)

(c)

(d)

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

(a) (b) (c) (d)

(10) Which of the following equations is equivalent to ?(10)

(a) (b) (c) (d)

Precalculus Name:__________________________________Lesson- properties, equations with exponents and

power and exponential functions Date:___________________________________

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Objectives: use the properties of exponents solve equations containing rational exponents examine power and exponential functions

Do Now: Use the exponential properties to simplify and rewrite the following expressions:

(1)

(2)

(3)

(4)

(5)

(6)

(7)

__________________________________________________________________________________________

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. 2. 3.

More Examples (each relates to “Process 2”):4. 5. 6.

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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.

Precalculus Name:__________________________________Lesson- Graphing exponential functions, exponential

growth and decay Date:___________________________________

Objectives: graph exponential functions

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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 x 20.

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__________________________________________________________________________________________(2) Suppose the value of a computer depreciates at a rate of 25% a year. Determine the value of a laptop

computer two years after it has been purchased for $3,750.

(3) Mexico has a population of about 100 million people, and it is estimated that the population will double in 21 years. If population growth continues at the same rate, what will be the population in:(a) 15 years(b) 30 years(c) graph the population growth for 0 time 50

__________________________________________________________________________________________(4) A researcher estimates that the initial population of honeybees in a colony is 500. They are increasing at a

rate of 14% per week. What is the expected population in 22 weeks?

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(5) In 1990, Exponential City had a population of 700,000 people. The average yearly rate of growth is 5.9%. Find the projected population for 2010.

(6) Find the projected population of each location in 2015:

(a) In Honolulu, Hawaii, the population was 836,231 in 1990. The average yearly rate of growth is 0.7%.

(b) The population in Kings County, New York has demonstrated an average decrease of 0.45% over several years. The population in 1997 was 2,240,384.

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Precalculus Name:____________________________________Lesson- More exponential function graphs,

Population growth, half-life

Objectives: graph exponential functions use exponential functions to determine population growth and half-life decay

(1) The population of Los Angeles County was 9,145,219 in 1997. If the average growth rate is 0.45%, predict the population in 2010. Graph the equation for 0 time 20.

(2) Radioactive gold 198 (198Au), used in imaging the structure of the liver, has a half-life of 2.67 days. If the initial amount is 50 milligrams of the isotope, how many milligrams (rounded to the nearest tenth) will be left over after:

(a) ½ day (b) 1 week

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(3) If a farmer uses 25 pounds of insecticide, assuming its half-life is 12 years, how many pounds (rounded to the nearest tenth) will still be active after:

(a) 5 years(b) 20 years

(4) In 2000, the chicken population on a farm was 10,000. The number of chickens increased at a rate of 9% per year. Predict the population in 2005.Graph the equation for 0 time 15.

(5) If Kenya has a population of about 30,000,000 people and a doubling time of 19 years and if the growth continues at the same rate, find the population (rounded to the nearest million) in:

(a) 10 years(b) 30 years

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Precalculus Name:__________________________________Lesson- Compound Interest

Date:___________________________________

Objectives: 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:

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__________________________________________________________________________________________(1) If Charlie invested $1,000 in an account paying 10% compounded monthly, how much will be in the

account at the end of 10 years?

(2) Mike would like to have $20,000 cash for a new car 5 years from now. How much should be placed in an account now if the account pays 9.75% compounded weekly?

(3) Suppose $2,500 is invested at 7% compounded quarterly. How much money will be in the account in:(c) ¾ year(d) 15 years

__________________________________________________________________________________________(4) Suppose $4,000 is invested at 11% compounded weekly. How much money will be in the account in:

(e) ½ year(f) 10 years

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(5) How much money must Cindy invest for a new yacht if she wants to have $50,000 in her account that earns 5% compounded quarterly after 15 years?

(6) Carol won $5,000 in a raffle. She would like to invest her winnings in a money market account that provides an APR of 6% compounded quarterly. Does she have to invest all of it in order to have $9,000 in the account at the end of 10 years? Show your work and explain your answer.

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Precalculus Name:__________________________________Lesson: Exponential Functions with base e

Date:___________________________________

Objective: use exponential functions with base e

Euler Savings Bank provides a savings account that earns compounded interest at a rate of 100%. You may choose how often to compound the interest, but you can only invest $1 over the course of one year.

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Exponential Growth or Decay (in terms of e ) : N = N0 ekt

(1) According to Newton, a beaker of liquid cools exponentially when removed from a source of heat. Assume that the initial temperature Ti is 90F and that k = 0.275.

(a) Write a function to model the rate at which the liquid cools.

(b) Find the temperature T of the liquid after 4 minutes (t)

(c) Graph the function and use the graph to verify your answer in part (b)

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(2) Suppose a certain type of bacteria reproduces according to the model B = 100 e0.271 t , where t is the time in hours.

(a) At what percentage rate does this type of bacteria reproduce?

(b) What was the initial number of bacteria?

(c) Find the number of bacteria (rounded to the nearest whole number) after:(i) 5 hours(ii) 1 day(iii) 3 days

(3) A city’s population can be modeled by the equation y = 33,430e0.0397 t , where t is the number of years since 1950.

(a) Has the city experienced a growth or decline in population?

(b) What was the population in 1950?

(c) Find the projected population in 2010.

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Precalculus Name:__________________________________HW- Compound Interest

Date:___________________________________

(1) If you invest $5,250 in an account paying 11.38% compounded continuously, how much money will be in the account at the end of:

(a) 6 years 3 months(b) 204 months


(a) 5.5 years(b) 12 years

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__________________________________________________________________________________________(3) A promissory note will pay $30,000 at maturity 10 years from now. How much should you be willing to

pay for the note now if the note gains value at a rate of 9% compounded continuously?

(4) Suppose Niki deposits $1,500 in a savings account that earns 6.75% interest compounded continuously. She plans to withdraw the money in 6 years to make a $2,500 down payment on a car. Will there be enough funds in Niki’s account in 6 years to meet her goal? Explain your answer.

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Precalculus Name:__________________________________Lesson- Continuous Compound Interest

Date:___________________________________

Objective: use exponential functions to determine continuously compounded interest

Continuously Compounded Interest: A = Pert

(1) Tim and Kerry are saving for their daughter’s college education. If they deposit $12,000 in an account bearing 6.4% interest compounded continuously, how much will be in the account when she goes to college in 12 years?

(2) Paul invested a sum of money in a certificate of deposit that earns 8% interest compounded continuously. If Paul made the investment on January 1, 1995, and the account was worth $12,000 on January 1, 1999, what was the original amount in the account?

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(3) Compare the balance after 30 years of a $15,000 investment earning 12% interest compounded continuously to the same investment compounded quarterly.

(4) Given the original principal, the annual interest rate, the amount of time for each investment, and the type of compounded interest, find the amount at the end of the investment:

(a) P = $1,250; r = 8.5%; t = 3 years; compounded semi-annually

(b) P = $2,575; r = 6.25%; t = 5 years 3 months; compounded continuously

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Precalculus Name:__________________________________HW- Compound Interest

Date:___________________________________


(a) 6 years 3 months(b) 204 months


(a) 5.5 years(b) 12 years

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__________________________________________________________________________________________(3) A promissory note will pay $30,000 at maturity 10 years from now. How much should you be willing to

pay for the note now if the note gains value at a rate of 9% compounded continuously?

(4) Suppose Niki deposits $1,500 in a savings account that earns 6.75% interest compounded continuously. She plans to withdraw the money in 6 years to make a $2,500 down payment on a car. Will there be enough funds in Niki’s account in 6 years to meet her goal? Explain your answer.

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Precalculus Name:____________________________________Lesson- Properties of a logs, rewriting

Exponential functions as logarithms, log graphs Date:_____________________________________

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

Do Now: Solve for x: 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: using the algebraic techniques we have learned so far. Therefore, we must try an alternative technique.

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

Properties of Logs

, where x > 0

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

1. 4.

2. 5.

3. 6.

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__________________________________________________________________________________________What is a Natural Logarithm?

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

Properties of Logs

, where x > 0

Example:Convert each into logarithmic form Convert each into logarithmic form1. 4. 2. 5.

3. 6.

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

Precalculus Name:__________________________________Lesson/HW- Simplify log expressions, common logs, evaluate

Date:___________________________________

Objectives: simplify expressions using the properties of logarithmic functions define common logarithms evaluate expressions involving logarithms

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x

y

Problem Set: write the following expressions in simpler logarithmic forms:

(1) (2)

(3) (4)

(5) (6)

(7) Use logarithmic properties to find the value of x (without using a calculator):

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Write each expression in terms of a single logarithm with a coefficient of one:

(8) (9)

(10) (11)

(12) (13)

Common Logarithm:

Change of Base Formula:

Given loga n, evaluate each logarithm to four decimal places:

(14) (15) (16)

Extension: Given y = logb n, what can you determine about the log value (y) based on b and n?

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Precalculus Name:____________________________________Lesson/HW- Properties of Logarithmic Functions,

Simplifying logarithmic expressions Date:_____________________________________

Objective: examine properties of logarithmic functions simplify expressions using the properties of logarithmic functions

Use the properties of logarithmic functions to solve for x:

(1) (2)

(3) (4)

Use the properties of logarithmic functions to simplify each expression:

(5) (6)

(7) (8)

(9) (10)

(11) (12)

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Write the following expressions in simpler logarithmic forms:

(13) (14)

(15) (16)

(17) (18)

Precalculus Name:____________________________________27

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 to the nearest ten-thousandths place.

__________________________________________________________________________________________

I. Graph each of the following on the same set of coordinate axes and answer the following questions.1.2.3.4.

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.1.2.3.4.

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

Precalculus Name:__________________________________

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Lesson- Natural Log Word ProblemsDate:___________________________________

Objectives: solve real-world applications with natural logarithmic functions

Do Now:

Laura won $2,500 on a game show. She would like to invest her winnings in an account that earns an interest rate of 12% compounded continuously. Does she have to invest all of it in order to have $4,000 in the account at the end of 4 years to put a down payment on a new sailboat? Show your work and explain your answer.

(1) Ana is trying to save for a new house. How many years, to the nearest year, will it take Ana to triple the money in her account if it is invested at 7% compounded annually?

(2) At what annual percentage rate (to the nearest hundredth of a percent) compounded continuously will $6,000 have to be invested to amount to $11,000 in 8 years.

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__________________________________________________________________________________________(3) In 1990, Exponential City had a population of 142,000 people. In what year will the city have a

population of about 200,000 people if it was growing at an exponential rate of k = 0.014?

(4) If $5,000 is invested at an annual interest rate of 5% compounded quarterly, how long will it take the investment to double?

(5) What was the annual interest rate (to the nearest hundredth of a percent) of an account that took 12 years to double if the interest was compounded continuously and no deposits or withdrawals were made during the 12-year period?

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Precalculus Name:__________________________________Lesson- More natural log word problems

Date:___________________________________

Objective: solve real-world applications with natural logarithmic functions

(1) If a car originally costs $18,000 and the average rate of depreciation is 30%, find the value of the car to the nearest dollar after 6 years.

(2) How many years, to the nearest year, will it take for the balance of an account to double if it is gaining 6% interest compounded semiannually?

(3) When Rachel was born, her mother invested $5,000 in an account that compounded 4% interest monthly. Determine the value of this investment when Rachel is 25 years old.

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__________________________________________________________________________________________(4) The decay of carbon-14 can be described by the formula . Using this formula, how many

years, to the nearest year, will it take for carbon-14 to diminish to 1% of the original amount?

(5) In 2002, a farmer had 400 pigs on his farm. He estimated that this population of pigs will double in 15 years. If population growth continues at the same rate, predict the number of pigs in:

a. 2010b. 2030

(6) If the world population is about 6 billion people now and if the population grows continuously at an annual rate of 1.7%, what will the population be (to the nearest billion) in 10 years from now?

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__________________________________________________________________________________________(7) If $100 is invested in an account that has an interest of 7% compounded quarterly, how long will it take for

the balance to reach a value of $1,000?

(8) What interest rate (to the nearest hundredth of a percent) compounded monthly is required for an $8,500 investment to triple in 5 years?

(9) An optical instrument is required to observe stars beyond the sixth magnitude, the limit of ordinary vision. However, even optical instruments have their limitations. The limiting magnitude L of any optical telescope with lens diameter D, in inches, is given by the equation . Use this equation to find the following to the nearest tenth:

a. the limiting magnitude for a homemade 6-inch reflecting telescope.b. the diameter of a lens that would have a limiting magnitude of 20.6.

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Unit 6: Exponential & Logarithmic FunctionsDefinitions, Properties & Formulas

Properties of Exponents

Property DefinitionProduct

Quotient , where x 0

Power Raised to a Power (xa)b = xab

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

Quotient Raised to a Power , where y 0

Zero Power x0 = 1, where x 0

Negative Power , where x 0

Rational Exponent for any real number x 0 and any integer n > 1and 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) 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

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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

Change of Base Formula

where a, b, and n are positive numbers, and a 1, b 1

Natural Logarithms

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

ln x

ln x = y

all properties of logarithms also hold for natural logarithms

Properties of Logarithmic Functions

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

Definition Examples

written exponentially: b0 = 1

written exponentially: b1 = b

written exponentially: bx = bx

, where x > 0

if and only if M = N

Properties of Logarithmic Functions35

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

Definition Examples

written exponentially: b0 = 1

written exponentially: b1 = b

written exponentially: bx = bx

, where x > 0

if and only if M = N

Common Errors:

cannot be simplified



Precalculus Name:____________________________________Review- Exponential and Logarithmic Functions part 1

Date:_____________________________________

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

Write each expression in terms of simpler logarithmic forms:

(1) (2) (3) (4)

Given loga n, evaluate each logarithm to four decimal places:

(5) (6) (7)

Solve each equation and round answers to four decimal places where necessary:

(8) (9)

(10) (11)

(12) (13)

(14) (15)

(16) (17)

(18) (19)

(20) (21)

(22) (23)

Precalculus Name:____________________________________Review- Exponential and Logarithmic Functions part 2

Date:_____________________________________SHOW ALL WORK:

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(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?

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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

. 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

, 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.

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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.

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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!

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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.

ResearchYou 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)

MathYou 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

DueFriday 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 equations2. Be able to conduct research for our projects.

IDEAS?

Student Loans-

Savings Accounts-

Transportation-

Own/Rent House-

Insurance-

Jobs-

Miscellaneous-

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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 awayElectric, Gas, Oil: $400 per monthPhone: $75 per monthGroceries: $500 per monthEntertainment: $350 per monthRent (including insurance): $1500 per monthCar expenses (maintenance): $50 per monthCar insurance: $100 per month

Expense that expires after 5 yearsCar Payment: $333 per month

Expense that expires after 20 yearsStudent 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 Years Years 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 savedPhase 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?


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


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

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Precalculus Name:____________________________________Lesson- Math on MSWORD

Date:_____________________________________

Objective: To learn to use Microsoft Word to create math related documents.

Example:

1.

2.

On Your Own:

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

1. 2.

Given: Isosceles triangle CAT,,

bisects < CTA, and are drawn

Prove: <SCA SAC

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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:

Savings accounts Mortgages 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.

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

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.

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.

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.

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.

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

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

annuities 'due' (BEGIN).

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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 on 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?

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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




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