# If b > 1, then f is an increasing function. I.e. f(x) increases in value as x increases. f(x) = 2 x. Note from the property of exponents that

Save this PDF as:

Size: px
Start display at page:

Download "If b > 1, then f is an increasing function. I.e. f(x) increases in value as x increases. f(x) = 2 x. Note from the property of exponents that"

## Transcription

1 An Exponential Function with base b is a function of the form: fx) = b x, where b > 0, b is a real number. We know the meaning of b r if r is a rational number. What if r is irrational? What we do is we approximate the value of b r by using rational approximate for r. For example, to approximate 5 π, we may approximate it as 5 3., 5 3., 5 3.5, In advance mathematics one can define the value of 5 π to be the limit of these approximations. For now, just realize that such approximation can be used to approximate the value of b r for any positive number b and any real number r. Since b x can be defined for all real numbers r, the domain of an exponential function is all real numbers. The range of b x is all real numbers greater than 0. Note the difference between an exponential function and a power function. x is a power function. In this function, the exponent is the constant. x is an exponential function. In this function, the base is the constant but the exponent is the variable input). An exponential function is always positive. And if in addition 0 < b <, f is a decreasing function. That is fx) decreases as x increases. fx) = ) x

2 If b >, then f is an increasing function. I.e. fx) increases in value as x increases. fx) = x Note from the property of exponents that x = x = ) x In other words, the exponential functions fx) = b x and gx) = b) x are reflections of each other with respect to the y axis. Simple Interest When you borrow money from a bank, you have to pay interest based on a particular interest rate. The amount you borrow is called the principal. If the interest is being calculated based on the principal only, we call this simple interest. The following formula gives the relationship of these quantities: Let P = principal, the amount borrowed. Let r = the simple annual interest rate. r the rate at which interest will be paid if calculated after year. r is usually given as a percentage, but when used in calculation, r will be written as a decimal. Let t = the amount of time, in terms of years, passed when interest is being

3 calculated. Let I = interest. Then we have: I = P rt. Example: You borrowed \$3500 for 5 months, for an interest rate of 6%. How much interest will you have to pay? Ans: P = \$ 3500, r = 6% = t = 5 = 5 =.5 I = = 6.5. You will have to pay \$6.5 in interest. If you borrowed P in principal, on an annual simple interest rate of r, then after t years, how much total, A, will you have to pay? In interest problems like this, the total amount at the end of the period, A, is the future value, and P the present value. According to the simple interest formula, you will have to pay I = P rt in interest. Of course, you will still have to pay off the original amouunt, the principle P, so the total A you will have to pay is: A = P + P rt = P + rt) where A is the total you have to pay or the future value). Exmaple: You deposited \$8000 into an account that pays 3.5% annual simple interest. After 8 months, how much should you have in your account? Ans: P = \$8000, r = 3.5% = 0.035, t = 8 months =.5 years. A = ) = 80 You will have total of \$80 in your account. \$0 is the interest. Compound Interest: If you deposit money into a bank and earns an interest), after your interest is being paid, if you do not take out the money, the next time you earn interest, you will earn interest on the original amount, P, and the interest from the first period, too. When you earn interest on the interest that you earned from a previous period, you are being paid compound interest. In calculating compound interest, since you will receive interest on interest that is being paid from the pervious period, how often you are being paid interest makes a difference. Example: You invested \$000 into a compound interest account that pays % rate, compounded quarterly. How much will you have at the end of two years? Ans: By interest being compounded quarterly, this means that the interest you

4 earn is being calculated and paid) every 3 months 3 months is of a year). Using the same terminology, if we say interest is compounded monthly, that means interest will be calculated and paid each month; compounded daily means interest will be calculated and paid each day...etc. At the end of the first three months, you will be paid interest. The time elasped since you last received interest is 3 months, or t = year. A = P + rt) = )) = 0 At the end of 6 months, you will )) be paid interest again, this time, you will already have A = in your account. This is your new principal. We use this number in the formula to calculate the new balance: )) [ )] [ A = = )] 3.60 At the end of 9 months, you will be paid interest again, this time using yet a new principal: [ )] [ )] [ )] 3 A = = Looking at the pattern, we see that for each additional compounding period, the power of the expression + rt), increased by one. At the end of two years, interest would be have been compounded 8 times, the total amount we will have is: [ )] 8 A = We have the following formula: If a principal amount P is earning an annual interest rate of r, and interest is compounded m times a year, then the total at the end of t years will be given by: A = P + r ) mt m Example: \$0, 000 is being invested into an account that pays r = 8% nominal interest. How much will be in the account after 5 years if interest is compounded a. annualy? b. semi-annually? c. quarterly? d. monthly? e. daily? Ans: a. If interest is compounded annually, then r m = 0.08 = In 5 years interest will be compounded 5 times, so mt = 5, using the formula we have:

5 A = 0, ) b. If interest is compounded semi-annually, then r m = r = 0.08 = 0.0. In 5 years interest will be compounded 0 times two times a year times 5 years), so mt = 0, using the formula again gives: A = 0, ) c. If interest is compounded quarterly, then r m = r = 0.08 = 0.0. In 5 years interest will be compounded 0 times four times a year times 5 years), so mt = 0, we have: A = 0, ) d. If interest is compounded monthly, then r m = r = In 5 years interest will be compounded 60 times twelve times a year times 5 years), so mt = 60, we have: A = 0, ) e. If interest is compounded daily, then r m = r 365 = In years interest will be compounded 85 times 365 times a year times 5 years), so mt = 85, we have: A = 0, ) We see that the more often interest is compounded, the more money we get at the end. What if we compound interest more often, say compounded every hour, or every minute, or every second? Will we get infinite amount of money, or will the amount we get approach a particular number? Looking at the formula A = P + r ) mt m If we let x = m r, then m = xr, r m = m/r =, then the formula becomes: x A = P + x) xrt [ = P + ) x ] rt x As we compound more often, m keeps on increasing while r remains the same, so x also keeps on increasing.

6 If we look at the expression, + x) x, and try some large values of x to see what is the value of the expression, we give a table of the value of the expression for some large values of x: x 0 00, 000 0, , 000, 000, ) x x + 0) ) ) ) ) ) It appears that the value of the expression + x) x approaches a particular number as x becomes incresingly large. This is indeed the case. It can be shown that the limit of this expression, when the value of x becomes arbitrarily large we say that the value of x increases without bound, or that x approaches infinity), the value of the expression: + x) x approaches a particular irrational) real number. This number is very important in mathematics and science, and is represented by the letter e and has an approximate value of e The exponential function using e as its base is called the natural expenential function. In other words, the natural exponential function is the function fx) = e x Using e to replace the above expression, we get the formula: A = P e rt This is the formula used when interest is compounded continuously. In reality, we can not really compound interest continuously. The formula gives an upperbound on how much interest can be earned regardless of how often interest is being compounded.

7 Example: \$0000 is being invested into an account that pays r = 8% nominal interest. How much will be in the account after 5 years if interest is compounded continuously? Ans: P = 0000, r = 0.08, t = 5 We use the formula for continuous compound interest: A = 0000e 0.085) In other words, given the above condition, you can never earn more than \$ no matter how often interest is compounded. Logarithmic Functions: An exponential function is a one-to-one function. It has in inverse. However, the expression for the inverse of an exponential function cannot be solved by any algebraic means, therefore we do not have an algebraic expression for it. We just define such a function and study its property knowing that it is the inverse of b x Definition: gx) = log b x read log base b of x ) is the inverse function of fx) = b x log b is the name of the function. x is the argument input) to log b, and the value output) is log b x. Since the range of an exponential function is all positive real numbers, the domain of a log function is all positive real numbers. Because of the fact that log b x is the inverse of b x, we have this by definition: log b b x ) = x for all x b log b x = x for all x > 0 The following two equations are equivalent: y = b x log b y = x The logarithm with base e, which is the inverse function of e x, is given a special name, the natural logarithm, and written ln. I.e. log e x = ln x

8

9 Example: Evaluate: log 6 Ans: Set log 6 = y, this is the same as asking: y = 6, what is y? From observation we see that y =, so log 6 = Example: log 5 5 =? Ans: Since 5 3 = 5, log 5 5 = 3 Example: log x = 5, what is x? Ans: 5 = x, so x = 3 log x 00 =, what is x? This is the same as saying, x = 00, so x = 0

10 Because of the fact that logs are inverse functions of the exponential functions, they have many properties that are similar to that of the exponential functions, and can be easily proved using the definition: Properties of Logarithm For any real number r, any base b > 0, a > 0, any x > 0, y > 0, we have: log b xy) = log b x + log b y ) x log b = log y b x log b y log b x r = r log b x Example: ln0) = ln 5) = ln) + ln5) Example: log 3 3xy) = log 3 3) + log 3 x) + log 3 y) = + log 3 x) + log 3 y) ) Example: ln = ln) ln5) 5 Example: log x 5 ) = 5 log x) Example: log b x 3 y ) = log b )+log b x 3 )+log b y ) = log b )+3 log b x)+ log b y) Change of Base Formula: log a x = log b x log b a Example: log 5) = log 0 5 log 0 Example: log 5 3) = ln3) ln5) The change of base formula says that we can easily change from log of one base to another, so the choice of log of which base to use usually for convenience only. We like to use natural log for most of our studies because that is the most conveninent one in math and science. Also make sure not to misuse the properties of logs, for example, the property of logarithm does not give us this: logx + y) = log x + log y Here s some example of how to incorrectly use the property of log: log b x + 5) = log b x) + log b 5) log b x 3) = log b x ) log b 3) log b x) 3 = 3 log b x)

11 To solve an equation involving exponential function, one would need to use logarithm, and to solve an equation involving logarithm, one uses exponential functions. E.g. Solve 3 x = 0 Ans: We take the log of both side log 3 3 x = log 3 0 Since log and exponential functions are inverse, the left hand side is just x, we have x = log 3 0 One may approach the same problem by taking the ln of both sides and use properties of logarithm: 3 x = 0 ln3 x ) = ln 0 x ln 3 = ln 0 ln 0 x = ln 3

12 E.g. Solve log x + ) = 3 Ans: We turn this into an equation involving exponential function: log x + ) = 3 3 = x + 6 = x + x = 6 E.g. Solve log x + ) + log x + ) = Ans: We use property of log to combine the left hand side: log x + ) + log x + ) = log x + )x + )) = log x + 5x + ) = = x + 5x + = x + 5x + x + 5x = 0 x = 0 or x = 5 Notice that x = 5 is an extraneous solution, so our only solution is x = 0

13 Application to compound interest: Example: You invested \$8000 into an account that pays 5% nominal rate. How long before you will have \$000 if interest is compounded: a. quarterly? b. monthly? Ans: We are solving for time t in the formula: A = P + r ) mt m a. We are told that the nominal rate, r, is r = 5% = If compounded quarterly, then m =. You started with P = You want to end up with A = 000, so we have the equation: 000 = ) t This is an exponential equation which can be solved using logarithm: = ) t 3 = ) t ) [ 3 ln = ln ) ] t ln ) 3 t = = t ln ln ) 3 ln ) ) ) ln 3 t = ln ) It takes about 8 years and two months for the \$8000 to become \$000 Example: You have some money to invest into an account that pays 6% nominal interest rate compounded monthly. How long would it take you to double your investment? Ans: It does not really matter the amount you started with. The time it takes you to double your investment is dependent only on the interest rate and the ways interest is compounded. If you started with initial investment P, you want to end up with future value of A = P. r = 0.06, m =, we want to solve for t:

14 P = P ) t = ) t [ ln = ln ) ] t ln = t ln ) ln t = ln ) It takes about and a half years to double your investment. Example: You want to deposit some money into an account that pays 3% nominal interest rate compounded monthly. If you want to be able to withdraw \$50000 to buy a car in 5 years, how much must you deposit now? Ans: We are trying to find P, given A = 50000, r = 0.05, t = 5, m = : = P ) 5) P = ) ) Exponential Growth and Decay: There are many situations where we may use an exponential function as a model. For example, the growth of a population may be modeled as an exponential growth, the amound of radioactive material remain after a certain period of time may be modeled as an exponential decay. An exponential growth or decay is defined by the equation: y = y 0 e kt In this equation, t is time, y 0 and k are constants. y 0 is the initial value of the population, or amount of radioactive material present). If k > 0, the equation represents an exponential growth, and k is the growth constant. If k < 0, then the equation represents an exponential decay, and k is the decay constant. Example: A biologist starts with 000 bacteria in a petri dish. After three hours, the population has increased to, 600. Assuming that the population of the bacteria increases exponentially, provide an equation that express the number of

15 bateria as a function of time. Ans: Since we assume that the rate of growth of the bacteria is exponential, we use the equation y = y 0 e kt, where y is the number of bacteria at time t. At time t = 0, the initial population size is y = y 0 e k0) =, 000 y 0 =, 000 This is the initial population, y 0. After t = hours, the population y has grown to, 600, we then set up the equation:, 600 = 000e k) e k = ek = 3 0 k = ln.3) ln.3) k = 0.3 The equation we can use to model this population growth is: y = 000e 0.3t How long does it take this bacteria culture to double in size from the original population size? Ans: To double in size, the bateria will have to increase to 000, we set y = 000 to solve the equation: 000 = 000e 0.3t e 0.3t = 0.3t = ln) t = ln) It takes approximately 5.8 hours for the population to double in size. This number, the amount of time it takes for a population to double in size in an exponential growth, is called the doubling time. This time is independent of the amount of the initial population. If you started with 50 bacteria, it will take 5.8 hours to double to 00 bacteria. If you started with 5000 bacteria, it will take the same 5.8 hours to double to 0,000 bacteria. Example: 50 grams of certain type of radioactive material is placed inside a detector. After 0 days, it is observed that only 7 grams of the radioactive material is left. Assuming that the radioactive material decays exponentially, find an equation that describes the amount of material left as a function of time t in days). Ans: We let y = the amount of radioactive material present at any time t. At t = 0 we have 50 grams, so we have: y = y 0 e k0) = 50 y 0 = 50 The initial amount of radioactive material is 50 grams, as we know. After t = 0 days, y = 7 grams left, we have:

### Investing Section 5.5

MATH 11009: Exponential Functions & Investing Section 5.5 Exponential Function: If b > 0, b 1, then the function f(x) = b x is an exponential function. The constant b is called the base of the function

### Chapter 12_Logarithms Word Problems

The applications found here will mostly involve exponential equations. These equations will be solved using logarithms and their properties. Interest Problems Compound Interest If we start with a principal

### Chapter 8. Exponential and Logarithmic Functions

Chapter 8 Exponential and Logarithmic Functions This unit defines and investigates exponential and logarithmic functions. We motivate exponential functions by their similarity to monomials as well as their

### Week 2: Exponential Functions

Week 2: Exponential Functions Goals: Introduce exponential functions Study the compounded interest and introduce the number e Suggested Textbook Readings: Chapter 4: 4.1, and Chapter 5: 5.1. Practice Problems:

### Logarithmic and Exponential Equations

11.5 Logarithmic and Exponential Equations 11.5 OBJECTIVES 1. Solve a logarithmic equation 2. Solve an exponential equation 3. Solve an application involving an exponential equation Much of the importance

### MAT12X Intermediate Algebra

MAT12X Intermediate Algebra Workshop I - Exponential Functions LEARNING CENTER Overview Workshop I Exponential Functions of the form y = ab x Properties of the increasing and decreasing exponential functions

### Unit 4 - Student Notes

Unit 4 - Student Notes Section 2.6, 5.1 - Compositions & Inverses I. Compositions (2.6) 1. Forming a composition of functions is another way to create a new function from two other functions. The composition

### Financial Mathematics

Financial Mathematics 1 Introduction In this section we will examine a number of techniques that relate to the world of financial mathematics. Calculations that revolve around interest calculations and

### 6 Rational Inequalities, (In)equalities with Absolute value; Exponents and Logarithms

AAU - Business Mathematics I Lecture #6, March 16, 2009 6 Rational Inequalities, (In)equalities with Absolute value; Exponents and Logarithms 6.1 Rational Inequalities: x + 1 x 3 > 1, x + 1 x 2 3x + 5

### n=number of interest periods per year (see the table below for more information) q=q(t)=q 0 e rt

Compound interest formula P = the principal (the initial amount) A= r=annual interest rate (expressed as a decimal) n=number of interest periods per year (see the table below for more information) t=number

### 8.3 Applications of Exponential Functions

Section 83 Applications of Exponential Functions 789 83 Applications of Exponential Functions In the preceding section, we examined a population growth problem in which the population grew at a fixed percentage

### Date Per r Remember the compound interest formula At () = P 1+ The investment of \$1 is going to earn 100% annual interest over a period of 1 year.

Advanced Algebra Name CW 38: The Natural Base Date Per r Remember the compound interest formula At () = P + where P is the principal, r is the annual interest rate, n is the number of compounding periods

### Exponential growth and decay

Exponential growth and decay Suppose you deposit 100 dollars in a bank account that receives 5 percent interest, compounded annually. This means: at the end of each year, the bank adds to your account

### Pre-Session Review. Part 2: Mathematics of Finance

Pre-Session Review Part 2: Mathematics of Finance For this section you will need a calculator with logarithmic and exponential function keys (such as log, ln, and x y ) D. Exponential and Logarithmic Functions

### 8.7 Exponential Growth and Decay

Section 8.7 Exponential Growth and Decay 847 8.7 Exponential Growth and Decay Exponential Growth Models Recalling the investigations in Section 8.3, we started by developing a formula for discrete compound

### Exponential and Logarithmic Functions

Exponential and Logarithmic Functions Exponential Functions Overview of Objectives, students should be able to: 1. Evaluate exponential functions. Main Overarching Questions: 1. How do you graph exponential

### Lesson A - Natural Exponential Function and Natural Logarithm Functions

A- Lesson A - Natural Exponential Function and Natural Logarithm Functions Natural Exponential Function In Lesson 2, we explored the world of logarithms in base 0. The natural logarithm has a base of e.

### For all a > 0, there is a unique real number n such that a = 10 n. The exponent n is called the logarithm of a to the base 10, written log 10 a = n.

Derivation Rules for Logarithms For all a > 0, there is a unique real number n such that a = 10 n. The exponent n is called the logarithm of a to the base 10, written log 10 a = n. In general, the log

### MATH 109 EXAM 2 REVIEW

MATH 109 EXAM REVIEW Remarks About the Exam: The exam will have some multiple choice questions ( 4-6 questions) AND some free response questions with multiple parts where you will have to show your work

### Review for Calculus Rational Functions, Logarithms & Exponentials

Definition and Domain of Rational Functions A rational function is defined as the quotient of two polynomial functions. F(x) = P(x) / Q(x) The domain of F is the set of all real numbers except those for

### CHAPTER 5: Exponential and Logarithmic Functions

MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College 5.2 CHAPTER 5: Exponential and Logarithmic Functions 5.1 Inverse Functions 5.2 Exponential Functions and Graphs 5.3 Logarithmic

### Mathematics of Finance. Learning objectives. Compound Interest

Mathematics of Finance Section 3.2 Learning objectives Compound interest Continuous compound interest Growth and time Annual percentage yield (APY) Compound Interest Compound interest arises when interest

### Math 1101 Exam 3 Practice Problems

Math 0 Exam 3 Practice Problems These problems are not intended to cover all possible test topics. These problems should serve as an activity in preparing for your test, but other study is required to

### Midterm I Review:

Midterm I Review: 1.1 -.1 Monday, October 17, 016 1 1.1-1. Functions Definition 1.0.1. Functions A function is like a machine. There is an input (x) and an output (f(x)), where the output is designated

### CCSS: F.IF.7.e., F.IF.8.b MATHEMATICAL PRACTICES: 3 Construct viable arguments and critique the reasoning of others

7.1 Graphing Exponential Functions 1) State characteristics of exponential functions 2) Identify transformations of exponential functions 3) Distinguish exponential growth from exponential decay CCSS:

### Compound Interest. If you begin with P dollars (the principal), at the end of the first quarter you would have P dollars plus 2% of P dollars:

Compound Interest Compound Interest Money invested at compound interest grows exponentially. Banks always state annual interest rates, but the compounding may be done more frequently. For example, if a

### Exponential Functions and Their Graphs

Exponential Functions and Their Graphs Definition of Exponential Function f ( x) a x where a > 0, a 1, and x is any real number Parent function for transformations: f(x) = b a x h + k The graph of f moves

### Grade 7 & 8 Math Circles. Finance

Faculty of Mathematics Waterloo, Ontario N2L 3G1 Introduction Grade 7 & 8 Math Circles October 22/23, 2013 Finance A key point in finance is the time value of money, a concept which states that a dollar

### CHAPTER 3. Exponential and Logarithmic Functions

CHAPTER 3 Exponential and Logarithmic Functions Section 3.1 (e-book 5.1-part 1) Exponential Functions and Their Graphs Definition 1: Let. An exponential function with base a is a function defined as Example

### 1. At what interest rate, compounded monthly, would you need to invest your money so that you have at least \$5,000 accumulated in 4 years?

Suppose your grandparents offer you \$3,500 as a graduation gift. However, you will receive the gift only if you agree to invest the money for at least 4 years. At that time, you hope to purchase a new

### Exponential Functions Video Lecture. Section 5.3

Exponential Functions Video Lecture Section 5.3 Course Learning Objectives: 1)Graph exponential functions and use such graphs to solve applied problems and to understand the significance of attributes

### Math 120 Final Exam Practice Problems, Form: A

Math 120 Final Exam Practice Problems, Form: A Name: While every attempt was made to be complete in the types of problems given below, we make no guarantees about the completeness of the problems. Specifically,

### With compound interest you earn an additional \$128.89 (\$1628.89 - \$1500).

Compound Interest Interest is the amount you receive for lending money (making an investment) or the fee you pay for borrowing money. Compound interest is interest that is calculated using both the principle

### Exponential growth and decay

September 9, 2013 Many quantities in nature change according to an exponential growth or decay function of the form P = P 0 e kt, where P 0 is the initial quantity and k is the continuous growth or decay.

### Section 8.3 Notes- Compound Interest

Section 8.3 Notes- Compound The Difference between Simple and Compound : Simple is paid on your investment or principal and NOT on any interest added Compound paid on BOTH on the principal and on all interest

### MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Math 110 Review for Final Examination 2012 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Match the equation to the correct graph. 1) y = -

### Exponential Functions. Exponential Functions and Their Graphs. Example 2. Example 1. Example 3. Graphs of Exponential Functions 9/17/2014

Eponential Functions Eponential Functions and Their Graphs Precalculus.1 Eample 1 Use a calculator to evaluate each function at the indicated value of. a) f ( ) 8 = Eample In the same coordinate place,

### Solving Compound Interest Problems

Solving Compound Interest Problems What is Compound Interest? If you walk into a bank and open up a savings account you will earn interest on the money you deposit in the bank. If the interest is calculated

### Annually Semiannually Monthly. Quarterly Weekly Daily

Pre-Calculus Chapter 4 Modeling/Applications Name: Sections 4.4 & 4.5 Nov. 2014 Exponential Growth Model: Exponential Decay Model: Compounding Interest: Annually Semiannually Monthly Quarterly Weekly Daily

### 7.5 Exponential Growth and Decay

7.5 Exponential Growth and Decay Mark Woodard Furman U Fall 2010 Mark Woodard (Furman U) 7.5 Exponential Growth and Decay Fall 2010 1 / 11 Outline 1 The general model 2 Examples 3 Doubling-time and half-life

### LOGARITHMIC FUNCTIONS

CHAPTER LOGARITHMIC FUNCTIONS The heavenly bodies have always fascinated and challenged humankind. Our earliest records contain conclusions, some false and some true, that were believed about the relationships

### Section 4.5 Exponential and Logarithmic Equations

Section 4.5 Exponential and Logarithmic Equations Exponential Equations An exponential equation is one in which the variable occurs in the exponent. EXAMPLE: Solve the equation x = 7. Solution 1: We have

### Exponential Growth and Decay

Exponential Growth and Decay 28 April 2014 Exponential Growth and Decay 28 April 2014 1/24 This week we ll talk about a few situations which behave mathematically like compound interest. They include population

### Compounded Interest. Covers

Compounded Interest Covers Section 3.3 Exponential Growth & Continuously Compounded Interest Section 4.3 Example: Continuously Compounded Interest (requires Integration) Section 3.3: Exponential Growth

### Class Notes, Math 110 Winter 2008, Sec 2 / Whitehead 1/18 Sections

20080225 Class Notes, Math 110 Winter 2008, Sec 2 / Whitehead 1/18 Review of Simple Interest You borrow \$100 at an annual interest rate of 5%. How much would you owe at the end of 1 year Symbol/equation

### 5.1 Simple and Compound Interest

5.1 Simple and Compound Interest Question 1: What is simple interest? Question 2: What is compound interest? Question 3: What is an effective interest rate? Question 4: What is continuous compound interest?

### 3. Exponential and Logarithmic functions

3. ial and s ial and ic... 3.1. Here are a few examples to remind the reader of the definitions and laws for expressions involving exponents: 2 3 = 2 2 2 = 8, 2 0 = 1, 2 1 = 1 2, 2 3 = 1 2 3 = 1 8, 9 1/2

### Teaching & Learning Plans. Introducing e. Leaving Certificate Syllabus Higher level

Teaching & Learning Plans Introducing e Leaving Certificate Syllabus Higher level The Teaching & Learning Plans are structured as follows: Aims outline what the lesson, or series of lessons, hopes to achieve.

### Section 1. Logarithms

Worksheet 2.7 Logarithms and Exponentials Section 1 Logarithms The mathematics of logarithms and exponentials occurs naturally in many branches of science. It is very important in solving problems related

### Warm Up. Warm Up. Accelerated Pre-Calculus Exponential Functions and Their Graphs Mr. Niedert 1 / 17

Warm Up Warm Up Plug in points (of your choosing) and sketch the graph of each function on the same coordinate plane. Use different colors if possible. a f (x) = 2 x b g(x) = 4 x c h(x) = 2 x d j(x) =

### \$496. 80. Example If you can earn 6% interest, what lump sum must be deposited now so that its value will be \$3500 after 9 months?

Simple Interest, Compound Interest, and Effective Yield Simple Interest The formula that gives the amount of simple interest (also known as add-on interest) owed on a Principal P (also known as present

### Compound Interest and Continuous Growth

Warm Up 1 SECTION 6.3 Compound Interest and Continuous Growth 1 Use the compound interest formula to calculate the future value of an investment 2 Construct and use continuous growth models 3 Use exponential

### REVIEW SHEETS COLLEGE ALGEBRA MATH 111

REVIEW SHEETS COLLEGE ALGEBRA MATH 111 A Summary of Concepts Needed to be Successful in Mathematics The following sheets list the key concepts that are taught in the specified math course. The sheets present

### 4.6 Exponential and Logarithmic Equations (Part I)

4.6 Eponential and Logarithmic Equations (Part I) In this section you will learn to: solve eponential equations using like ases solve eponential equations using logarithms solve logarithmic equations using

### Time Value of Money. MATH 100 Survey of Mathematical Ideas. J. Robert Buchanan. Department of Mathematics. Fall 2014

Time Value of Money MATH 100 Survey of Mathematical Ideas J. Robert Buchanan Department of Mathematics Fall 2014 Terminology Money borrowed or saved increases in value over time. principal: the amount

### Chapter 7 Outline Math 236 Spring 2001

Chapter 7 Outline Math 236 Spring 2001 Note 1: Be sure to read the Disclaimer on Chapter Outlines! I cannot be responsible for misfortunes that may happen to you if you do not. Note 2: Section 7.9 will

### Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression.

MAC 1105 Final Review Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. 1) 8x 2-49x + 6 x - 6 A) 1, x 6 B) 8x - 1, x 6 x -

### Unit 8: Exponential & Logarithmic Functions

Date Period Unit 8: Eponential & Logarithmic Functions DAY TOPIC 1 Eponential Growth Eponential Decay 8. Properties of Eponential Functions; Continuous Compound Interest ( e ) 4 8. Logarithmic Functions;

### Solutions to Exercises, Section 4.5

Instructor s Solutions Manual, Section 4.5 Exercise 1 Solutions to Exercises, Section 4.5 1. How much would an initial amount of \$2000, compounded continuously at 6% annual interest, become after 25 years?

### 12.6 Logarithmic and Exponential Equations PREPARING FOR THIS SECTION Before getting started, review the following:

Section 1.6 Logarithmic and Exponential Equations 811 1.6 Logarithmic and Exponential Equations PREPARING FOR THIS SECTION Before getting started, review the following: Solve Quadratic Equations (Section

### EXPONENTIAL FUNCTIONS

CHAPTER 7 CHAPTER TABLE OF CONTENTS 7- Laws of Exponents 7- Zero and Negative Exponents 7- Fractional Exponents 7-4 Exponential Functions and Their Graphs 7-5 Solving Equations Involving Exponents 7-6

### WEEK #2: Cobwebbing, Equilibria, Exponentials and Logarithms

WEEK #2: Cobwebbing, Equilibria, Exponentials and Logarithms Goals: Analyze Discrete-Time Dynamical Systems Logs and Exponentials Textbook reading for Week #2: Read Sections 1.6 1.7 2 Graphical Analysis

### SCHOOL OF BUSINESS, ECONOMICS AND MANAGEMENT BUSINESS MATHEMATICS / MATHEMATICAL ANALYSIS

SCHOOL OF BUSINESS, ECONOMICS AND MANAGEMENT BUSINESS MATHEMATICS / MATHEMATICAL ANALYSIS Unit Five Moses Mwale e-mail: moses.mwale@ictar.ac.zm BBA 120 Business Mathematics Contents Unit 5: Mathematics

### Differential Equations

Differential Equations 1. Mr. Moneybags decides to open a bank account with an opening deposit of \$1000. Suppose that the account earns a nominal annual interest rate of 6%, compounded annually. 1) Assuming

### 4.1 INTRODUCTION TO THE FAMILY OF EXPONENTIAL FUNCTIONS

Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally 4.1 INTRODUCTION TO THE FAMILY OF EXPONENTIAL FUNCTIONS Functions Modeling Change: A Preparation for Calculus, 4th Edition,

### Saving Money, Making Money

Saving Money, Making Money Suppose you receive for graduation a gift of \$1,200 from your favorite relative. You are required to invest at least \$800 of the gift in a no-withdrawal savings program for at

### FINAL EXAM SECTIONS AND OBJECTIVES FOR COLLEGE ALGEBRA

FINAL EXAM SECTIONS AND OBJECTIVES FOR COLLEGE ALGEBRA 1.1 Solve linear equations and equations that lead to linear equations. a) Solve the equation: 1 (x + 5) 4 = 1 (2x 1) 2 3 b) Solve the equation: 3x

### Math 30-1: Exponential and Logarithmic Functions PRACTICE EXAM

Math 30-1: Exponential and Logarithmic Functions PRACTICE EXAM 1. All of the following are exponential functions except: y = 1 x y = 2 x y = 3 x 2. The point (-3, n) exists on the exponential graph shown.

### CHAPTER 4 Exponential and Logarithmic Functions

CHAPTER Eponential and Logarithmic Functions Section.1 Eponential Functions An eponential function with base a has the form f () =a where a is a constant, a > 0 and a = 1. The graph of an eponential function

### Also, compositions of an exponential function with another function are also referred to as exponential. An example would be f(x) = 4 + 100 3-2x.

Exponential Functions Exponential functions are perhaps the most important class of functions in mathematics. We use this type of function to calculate interest on investments, growth and decline rates

### Exponential and Logarithmic Functions

Chapter 4 Exponential and Logarithmic Functions 4. Exponential Functions. Use your HP48G. For e, punch, then e x, to get 3. e 7.389. For the TI-85, press nd e x then ENTER. Similarly e 0.35, e 0.05.05,

### 7. Finance. Maths for Economics. 1 Finance. Vera L. te Velde. 27. August Effective annual interest rates

Maths for Economics 7. Finance Vera L. te Velde 27. August 2015 1 Finance We now know enough to understand how to mathematically analyze many financial situations. Let s start with a simply bank account

### CHAPTER FIVE. Solutions for Section 5.1. Skill Refresher. Exercises

CHAPTER FIVE 5.1 SOLUTIONS 265 Solutions for Section 5.1 Skill Refresher S1. Since 1,000,000 = 10 6, we have x = 6. S2. Since 0.01 = 10 2, we have t = 2. S3. Since e 3 = ( e 3) 1/2 = e 3/2, we have z =

### The Mathematics of Savings

Lesson 14 The Mathematics of Savings Mathematics Focus: Algebra I, Algebra II, and Precalculus Mathematics Prerequisites: Prior to using this lesson, students should be able to: Use simple and compound

### College Algebra. Some of the vocabulary and skills to review are included in the following table:

MATH 11 College Algebra FINAL EXAM Review The Final Exam is cumulative. Problems will be similar to those you have seen on previous exams. The exam is worth 1 points. Formulas provided on Exam will also

### Lesson Exponential Growth and Decay

SACWAY STUDENT HANDOUT SACWAY BRAINSTORMING ALGEBRA & STATISTICS STUDENT NAME DATE INTRODUCTION. In our previous lesson, we experimented with the idea that a fixed- rate interest earning account could

### Name: Class: Date: 1. Find the exponential function f (x) = a x whose graph is given.

. Find the exponential function f (x) = a x whose graph is given. 2. Find the exponential function f (x) = a x whose graph is given. 3. State the domain of the function f (x) = 5 x. PAGE 4. State the asymptote

### 3-2 Logarithmic Functions

Evaluate each expression. 1. log 2 8 2. log 10 10 3. log 6 4. Inverse Property of Logarithms 5. log 11 121 esolutions Manual - Powered by Cognero Page 1 6. log 2 2 3 7. 8. log 0.01 9. log 42 Enter log

### 8.3. GEOMETRIC SEQUENCES AND SERIES

8.3. GEOMETRIC SEQUENCES AND SERIES What You Should Learn Recognize, write, and find the nth terms of geometric sequences. Find the sum of a finite geometric sequence. Find the sum of an infinite geometric

### Dimensional Analysis and Exponential Models

MAT 42 College Mathematics Module XP Dimensional Analysis and Exponential Models Terri Miller revised December 3, 200. Dimensional Analysis The purpose of this section is to convert between various types

### Section 4-4 The Exponential Function with Base e

4-4 The Exponential Function with Base e 83 73. Finance. A couple just had a new child. How much should they invest now at 8.% compounded daily in order to have \$4, for the child s education 7 years from

### Construction of the Real Line 2 Is Every Real Number Rational? 3 Problems Algebra of the Real Numbers 7

About the Author v Preface to the Instructor xiii WileyPLUS xviii Acknowledgments xix Preface to the Student xxi 1 The Real Numbers 1 1.1 The Real Line 2 Construction of the Real Line 2 Is Every Real Number

### Equations. #1-10 Solve for the variable. Inequalities. 1. Solve the inequality: 2 5 7. 2. Solve the inequality: 4 0

College Algebra Review Problems for Final Exam Equations #1-10 Solve for the variable 1. 2 1 4 = 0 6. 2 8 7 2. 2 5 3 7. = 3. 3 9 4 21 8. 3 6 9 18 4. 6 27 0 9. 1 + log 3 4 5. 10. 19 0 Inequalities 1. Solve

### Dimensional Analysis; Exponential and Logarithmic Growth/Decay

MAT 42 College Mathematics Module #5 Dimensional Analysis; Exponential and Logarithmic Growth/Decay Terri Miller Spring 2009 revised November 7, 2009. Dimensional Analysis The purpose of this section is

### FINANCIAL MATHEMATICS (2)

FINANCIAL MATHEMATICS (2) Learning Outcomes and Assessment Standards Learning Outcome 1: Number and number relationships When solving problems, the learner is able to recognise, describe, represent and

### 1. Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded:

Final Study Guide MATH 111 Sample Problems on Algebra, Functions, Exponents, & Logarithms Math 111 Part 1: No calculator or study sheet. Remember to get full credit, you must show your work. 1. Determine

### Official Math 112 Catalog Description

Official Math 112 Catalog Description Topics include properties of functions and graphs, linear and quadratic equations, polynomial functions, exponential and logarithmic functions with applications. A

### Annuities, Sinking Funds, and Amortization Math Analysis and Discrete Math Sections 5.3 and 5.4

Annuities, Sinking Funds, and Amortization Math Analysis and Discrete Math Sections 5.3 and 5.4 I. Warm-Up Problem Previously, we have computed the future value of an investment when a fixed amount of

### Section 4-7 Exponential and Logarithmic Equations. Solving an Exponential Equation. log 2. 3 2 log 5. log 2 1.4406

314 4 INVERSE FUNCTIONS; EXPONENTIAL AND LOGARITHMIC FUNCTIONS Section 4-7 Exponential and Logarithmic Equations Exponential Equations Logarithmic Equations Change of Base Equations involving exponential

### MATH 1103 Common Final Exam Multiple Choice Section Fall 2011

MATH 110 Common Final Exam Multiple Choice Section Fall 011 Please print the following information: Name: Instructor: Student ID: Section/Time: The MATH 110 Final Exam consists of two parts. These pages

### Linear functions are used to describe quantities that grow (or decline) at a constant rate. But not all functions are linear, or even nearly so.

Section 4.1 1 Exponential growth and decay Linear functions are used to describe quantities that grow (or decline) at a constant rate. But not all functions are linear, or even nearly so. There are commonly

### Hats 1 are growth rates, or percentage changes, in any variable. Take for example Y, the GDP in year t compared the year before, t 1.

1 Growth rates Hats 1 are growth rates, or percentage changes, in any variable. Take for example Y, the GDP in year t compared the year before, t 1. We have: Ŷ = Y Y = Y t Y t 1 Y t 1 = Y t Y t 1 1 Example

### Exponential and Logarithmic Models

Exponential and Logarithmic Models MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 2011 Objectives In this lesson we will learn to: recognize the five most common types of models

### The Time Value of Money

The Time Value of Money 1 Learning Objectives The time value of money and its importance to business. The future value and present value of a single amount. The future value and present value of an annuity.

### College Algebra. George Voutsadakis 1. LSSU Math 111. Lake Superior State University. 1 Mathematics and Computer Science

College Algebra George Voutsadakis 1 1 Mathematics and Computer Science Lake Superior State University LSSU Math 111 George Voutsadakis (LSSU) College Algebra December 2014 1 / 91 Outline 1 Exponential

### MATH 34A REVIEW FOR MIDTERM 2, WINTER 2012. 1. Lines. (1) Find the equation of the line passing through (2,-1) and (-2,9). y = 5

MATH 34A REVIEW FOR MIDTERM 2, WINTER 2012 ANSWERS 1. Lines (1) Find the equation of the line passing through (2,-1) and (-2,9). y = 5 2 x + 4. (2) Find the equation of the line which meets the x-axis