Math 111: Section 3.1 Exponential Growth and Decay Section 004

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Math 111: Section 3.1 Exponential Growth and Decay Section 004 An example of Exponential Growth If each bactrium splits into two bacteria every hour, then the population doubles every hour.

1 Math 111: Section 3.1 Exponential Growth and Decay Section 004 An example of Exponential Growth If each bactrium splits into two bacteria every hour, then the population doubles every hour. The question is: what s the number of bacteria after x hours? An example of Exponential Growth In last example, we modeled a bacteria population by the function f(x) = 10 2 x, where 10 is the initial population. The base 2 is called the growth f actor because the population is multiplied by the factor 2 in every time period. In this example, the time period is one hour. 1

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How to calculate growth factor: a a = f(x+1) f(x) The Growth Rate The growth rate of a population is the proportion of the population by which it increases during

3 How to calculate growth factor: a a = f(x+1) f(x) The Growth Rate The growth rate of a population is the proportion of the population by which it increases during one time period. In general, for an exponential growth model f(x) = Ca x, we have r = f(x+1) f(x) f(x) = f(x+1) f(x) 1 = Cax+1 Ca x 1 = a 1. So r = a 1 and a = 1 + r. 3

4 Modeling Exponential Decay: An example Soppose a patient is injected with 10mg of a therapeutic drug. It is known that 20% of the drug is expell by the body ecah hour, so after one hour 80% of the drug remains in the body. Let s calculate the amount of the grug remaining at the end of 1, 2, 3 hour: 4

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6 Math 111 Section 3.2 Compound Interest Section 004 In order to solve compound interest problems, you should be able to: use rules of exponents solve exponential equations There are several types of interest problems, compound interest, simple interest and continuously compound interest. Simple Interest Formula The simple interest I on a principal P at a simple interest rate r (expressed as a decimal) per year for t years is Example 1: (Calculating simple interest) I=Prt Lili has deposited $ 8000 in a bank for five years at a simple interest rate of 6%. a. How much interest will she receive? b. How much money will be in her account at the end of five years? Compound Interest Compound interest is interest added to the principal of a deposit or loan so that the added interest also earns interest from then on. This addition of interest to the principal is called compounding. Example 2: 1

7 If an amount P is invested at an annual interest rate r compounded n times each year, then the amount A(t) of the investment after t years is given by the formula: ( ) nt A(t) = P 1 + r n A represents the amount of money after a certain amount of time P represents the principle or the amount of money you start with r represents the annual interest rate and is always represented as a decimal t represents the amount of time in years n is the number of times interest is compounded in one year, for example: if interest is compounded annually then n = 1 if interest is compounded quarterly then n = 4 if interest is compounded monthly then n = 12 2

8 Example 3 Suppose Karen has $ 1000 that she invests in an account that pays 3.5% interest compounded quarterly. How much money does Karen have at the end of 5 years? Lets look at our formula and see how many values for the variables we are given in the problem. P = ( ) nt A(t) = P 1 + r n The interest rate is which must be changed into a decimal and becomes r =. The interest is compounded quarterly, or four times per years, which tells us that n = The money will stay in the account for 5 years so t = We have values for four of the variables. We can use this information to solve for A. ( ) (4)(5) A(t) = = So after 5 years, the account is worth $ Because we are dealing with money in these problems, it makes sense to round to two decimal places. Notice that the formula gives us the total value of the account at the end of the five years. This is not just the interest amount, it is the total amount. Since there are many variables in the equations, there are several ways that problems can be presented. Lets look at some other examples. William wants to have a total of $ 4000 in two years so that he can put a hot tub on his deck. He finds an account that pays 5% interest compounded monthly. How much should William put into this account so that hell have $ 4000 at the end of two years? Step 1: Determine which four values you know and can plug into the compound interest formula. Step 2: Plug in the numbers you know from Step One into the interest formula. Step 3: Solve the equation. Step 4: What is your answer? ($ ) 3

9 Suppose William, from our last example, only has $ 3500 to invest but still wants $ 4000 for a hot tub. He finds a bank offering 5.25% interest compounded quarterly. How long will he have to leave his money in the account to have $ (t = 2.56 years) Kelly plans to put her graduation money into an account and leave it there for 4 years while she goes to college. She receives $ 750 in graduation money that she puts it into an account that earns 4.25% interest compounded semi-annually. How much will be in Kellys account at the end of four years? A. $ 4.09 B. $ C. $ D. $ ABC Bank is offering to double your money! They say that if you invest with them at 6% interest compounded quarterly they will double your money. If you invest $1500 in the account, how long will it take to double your money. A years B years C years Answer: 1: $ : t = 2.56 years. 4

10 Math 111 Section 3.2 Simple vs Compound interest Section 004 Simple vs compound interest is not hard to understand. Basically, simple interest is interest paid on the original principal only. For example, 4000 dollars is deposited into a bank account and the annual interest rate is 8%. How much is the interest after 4 years? Use the following simple interest formula: I = prt where p is the principal or money deposited r is the rate of interest t is time We get: I = prt I = % 4 I = I = 1280 dollars However, coumpound interest is the interest earned not only on the original principal, but also on all interests earned previously In other words, at the end of each year, the interest earned is added to the original amount and the money is reinvested If we use compound interest for the situation above, the interest will be computed as follow: Interest at the end of the first year: I = I = 320 dollars Your new principal per say is now = 4320 Interest at the end of the second year: I = I = dollars Your new principal is now = Interest at the end of the third year: I = I = dollars Your new principal is now = Interest at the end of the fourth year: I = I = dollars Your new principal is now =

11 Total interest earned = = The difference in money between coumpond interest and simple interest is = As you can see, compound interest yield better result, so you make more money. Therefore, before investing your money, you should double check with your local bank if coumpound interest will be used. Having said that if you have a credit card and you owe money on it, you will pay less interest if the credit card company uses simple interest. However, they will never do something so foolish! I hope simple vs compound interest is well understood now! 2

12 1. Mr. Wiggins gives his daughter Celia two choices of payment for raking leaves: Choice 1: Two dollars for each bag of leaves Choice 2: Two cents for one bag, four cents for two bags, eight cents for three bags, and so on with the amount doubling for each additional bag. Bags Option 1 Option a. If Celia rakes five bags of leaves, should she opt for payment method 1 or 2? What if she rakes ten bags of leaves? b. How many bags of leaves does Celia have to rake before method 2 pays more than method 1? c. What kind of function is represented by option 1? Create a function, f(x), to represent this scenario.

13 d. What kind of function is represented by option 2? Create a function, g(x), to represent this scenario. e. Describe the differences in payment methods. f. Describe the difference in the way the payment grows in the table and on the graph.

14 Exponential growth versus linear growth I Mr. Wiggins gives his daughter Celia two choices of payment for raking leaves: 1. Two dollars for each bag of leaves filled, 2. She will be paid for the number of bags of leaves she rakes as follows: two cents for filling one bag, four cents for filling two bags, eight cents for filling three bags, and so on, with the amount doubling for each additional bag filled. a. If Celia rakes enough to five bags of leaves, should she opt for payment method 1 or 2? What if she fills ten bags of leaves? b. How many bags of leaves would Celia have to fill before method 2 pays more than method 1?

Math 111 Section 3.4 Graphs of Exponential Functions Section 004 The simplest exponential function is: f(x) = a x, a > 0, a 1. The reasons for the restrictions are simple.

17 Math 111 Section 3.4 Graphs of Exponential Functions Section 004 The simplest exponential function is: f(x) = a x, a > 0, a 1. The reasons for the restrictions are simple. If a 0, then when you raise it to a rational power, you may not get a real number. Example:. If a = 1, then no matter what x is, the value of f(x) is certainly not one-to-one. Rule of Exponents: a x = 1 a x = ( 1 a) x. Notice that if a > 1, then 1/a < 1. Example 1 Graphing Exponential Functions (a) f(x) = 3 x (b) g(x) = ( 1 3 )x. That is a pretty boring function, and it is Solution: Make a table of values for the exponential function. Use the x values -2,-1,0,1,2 and 3. Then graph on a coordinate plane. x f(x) x g(x) 1

18 Form the above exampel we see that the graph of an exponential function always lies entirely above the x- axis. The graph gets close to the x-axist, but never crosses it.this means the x- axis is a Horizontal asymptote of the graph of the function. Informally, an asymptote of a function is a line to which the graph of the function gets closer and closer as one travel along the line. 2

19 The Effect of Varying a or C Example 2 3

20 Example 3 4

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22 Finding an Exponential Function from a graph Example 7 Find the function f(x) = Ca x whose graph is given: 6

Exponential Growth and Decay

Exponential Growth and Decay Identifying Exponential Growth vs Decay A. Exponential Equation: f(x) = Ca x 1. C: COEFFICIENT 2. a: BASE 3. X: EXPONENT B. Exponential Growth 1. When the base is greater than