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Chapter 6 Exponential Functions Chapter 6 Exponential Functions 1 To this point we have explored the behavior of linear functions that have graphs that are lines, e.g. Figure 6.1. Figure 6.1 You noticed that this function increased at a constant rate throughout its domain. You also explored the behavior of polynomial functions whose graphs were not lines but curves that increased and decreased in sections of their domain. Examples of polynomials graphs are seen in Figures 6.2 and 6.3. Figure 6.2 Figure 6.3 Chapter 6 Exponential Functions 2 Chapter 0 Example 7 presents data on the value of gold in given years. These values are increasing, with no indication of a period of decrease in value, but not at a constant rate. This brings us to yet another type of mathematical model for consideration and study, the Exponential Model. The price of gold is only one example of a commodity that may exhibit the pattern seen in Figure 0.7. Sales of a new product typically start at a low level, and as word spreads about the product, pick up at a rapid rate. If the product is successful, this pattern continues for some time. Thus, the monthly sales might look like this. Figure 6.4 The function that models this type of growth is not a polynomial. Why? _______________________________________________________________________________________________ _______________________________________________________________________________________________ The behavior described above is modeled by an exponential function. It is this family of functions that will be explored in this chapter. Chapter 6 Exponential Functions 3 We will begin our discussion by considering a chessboard. We will use the board in the following way. On day one of our exploration we will place 2 pennies on the lower left hand square. 2 On day two we will double the number of pennies and place them on the square to the right of the already occupied square. On day three we will double the number of pennies used on the previous day and place them on the square to the right of the already occupied Chapter 6 Exponential Functions 4 squares. This continues until we fill the bottom row, at which time we rise to the next row and repeat. 512 … … 2 4 8 16 32 64 128 256 The question that we wish to answer is, “How many pennies are placed on the square marked x?” x x = ____________________________________________________________________ Chapter 6 Exponential Functions 5 You may have already noticed that relationship described above is a function. Why? _____________________________________________________________________________________ ____________________________________________________________________________________________ If you answered each day, represented by a square on the chessboard, is associated with only one cash value, you are correct. Let us now examine the process above using a table. Table 6.1 x (Day) f(x) (# of pennies) 1 2 2 4 = 2 × 2 3 8 = 4 × 2 = 2 × 2 × 2 4 16 = 8 × 2 = 2 × 2 × 2 × 2 5 32 = 16 × 2 = 2 × 2 × 2 × 2 × 2 : Remember that a multiplication like 2 × 2 × 2 × 2 can be expressed 24. This suggests that the function modeling the amount of pennies on any particular square is f(x) = 2x . A function of this type f(x) = bx is called an exponential function, where b is called the base of the function. The table above can be rewritten as: Chapter 6 Exponential Functions 6 Table 6.2 x (Day) f(x) (# of pennies) 1 2 2 4 = 2 × 2 = 22 3 8 = 22 × 2 = 23 4 16 = 23 × 2 = 24 5 32 = 24 × 2 = 25 : Examining this table we see that as one moves down the f(x) column the next value of f(x) is obtained by multiplying the previous value of f(x) by 2. Question: What happens to the exponent when f(x) is multiplied by 2? ______________________________________________________________________________________________ Question: What is the value of f(x) on day x? ___________________________________________ Question: What is the value of f(x) the day after x? ____________________________________ Now we will read the table from the bottom up. Question: Find a mathematical operation will transform 32 into 16? ______________________________________________________________________________________________ Question: Will the same operation transform 16 into 8? If no, find an operation that performs both tasks. ______________________________________________________________________________________________ Chapter 6 Exponential Functions 7 You should now be convinced that when reading up the table the value of f(x) in the row above is obtained by dividing the row value of the function in the row below by 2, e.g. f(4) = f(5) ÷ 2 = 32 ÷ 2 = 16. Question: What happens to the exponent when a value of f(x) = 2x is divided by 2? _______________________________________________________________________________________________ The answer to this question allows us to make some interesting discoveries after filling in the blanks in the table below. Table 6.3 x f(x) f(x) = f(x + 1) ÷ 2 -n : -3 -2 -1 1 ÷ 2 0 2 ÷ 2 1 2 4 ÷ 2 2 4 8 ÷ 2 3 8 16 ÷ 2 4 16 32 ÷ 2 5 32 : n Did you notice that 20 = 1? Explain why this is the case. ___________________________________________________________________________________________ Chapter 6 Exponential Functions 8 Did you notice that 2-1 = ? ___________________________________________________________ Then did you notice that 2-2 = ? _____________________________________________________ Thinking about this observation you will discover that this observation can be expressed as 2-2 = . Moving up a row in the table we see that 2-3 = . Question: The observation above can be generalized to 2-n = ______________________ We will now look at the function f(x) = 3x. Question: Fill in the missing values in the table below. Table 6.4 x f(x) = 3x f(x) -n = : -3 = -2 = -1 = 0 = 1 = You should notice that there is nothing special about the numbers 2 or 3. The observations above can be applied to any base of an exponential function. € 1 2 € 1 4 € 1 22 € 1 23 € 3___ € 1 3____ € 3___ € 1 3____ € 3___ € 1 3____ € 3___ € 1 3____ € 3___ € 3___ Chapter 6 Exponential Functions 9 Question: If f(x) = bx then: f(1) = _______________________________________________ a different expression for f(-1) = __________________________________________ f(0) = _______________________________________________ a different expression for f(0) = __________________________________________ f(-n) = ______________________________________________ a different expression for f(-n) = _________________________________________ Now that we have learned something about exponential functions it is time to go back and create a model for the gold price data presented in Chapter 0 Example 7. In this example we are exploring the manner in which the price of gold has changed since the year 2000. Thus we can think of the price of gold in 2001 as the price of gold 1 year after the initial observation, the price in 2002 as the price 2 years after the initial observation, etc., etc. With this in mind we will once again slightly modify the method of finding a regression model we have become familiar with in earlier chapters. Use “Add Lists & Spreadsheets” with variables named “year” and “price” enter this data. Enter the years as the years after 2000 and the price as the price of gold in that year. When all the data is entered, plot the data adding the variables “year” along the horizontal axis and “price” on the vertical axis. Chapter 6 Exponential Functions 10 Figure 6.5 How does the scatterplot in Figure 6.5 differ from the one seen in Figure 0.7? __________________________________________________________________________________________ __________________________________________________________________________________________ Although the price of gold increases slowly at first, eventually the prices start increasing faster, looking more like an exponential graph than a linear or polynomial one. Hit: Menu Choose 4: Analyze Choose 6: Regression Choose 8: Show Exponential. Chapter 6 Exponential Functions 11 Figure 6.6 The calculator gives the best-fitting exponential model for the data to be: f(x) = 229.523(1.17803)x What does this model predict as the price of gold in 2006? __________________________ What was the price of gold in 2006? ___________________________________________________ Do you consider this a good prediction? _______________________________________________ What does this model predict as the price of gold in 2000? __________________________ What was the price of gold in 2000? ___________________________________________________ Do you consider this a good prediction? _______________________________________________ Derivative of the Exponential Function At this time you know that you can find the derivative of the exponential function f(x) = 229.523(1.17803)x using the 4:Calculus menu of the TI- Nspire. • Hit menu • Choose 4:Calculus • Choose 1:Derivative • Fill in Boxes Appropriately (Figure 6.7) Chapter 6 Exponential Functions 12 Figure 6.7 This is an interesting answer, it is the base raised to the power x multiplied by a different factor. Will the same be true for other exponential functions? Question: Find the derivative of � � = 5 7.2 � f ‘ (x) = ____________________________________________________________ Question: Find the derivative of � � = 12 107.5 � f ‘ (x) = ____________________________________________________________ Question: Find the derivative of � � = 0.4 3.2 � f ‘ (x) = ____________________________________________________________ Has the same pattern held? _____________________________________________________ Question: Find the derivative of � � = 5 7 � f ‘ (x) = ____________________________________________________________ Here we are introduced to something we have not seen before ln(7). The ln, natural logarithm, function will be discussed in a latter chapter. For the moment we will be content with the knowledge that the natural logarithm of any positive number exists and is a Real number. Chapter 6 Exponential Functions 13 The value of ln(7) can be found by: • Hit ctrl • Hit the key ex (preceding this by ctrl evaluates the ln above the key) • Enter 7 in the parentheses • Hit ctrl enter We can now use the definition of the derivative to not only show that the generalization suggested by the examples above is in fact true but to provide a formula for finding these derivatives. It can then be shown that the derivative package in the TI-Nspire derives the same formula as the definition. See Figure 6.8. Figure 6.8 Chapter 6 Exponential Functions 14 Homework 1. Find the exponential model for the price of gold using the year as the predictor variable rather than the years since 2000. Model: ____________________________________________________________________ 2. Use the model you found in problem 1 above to predict the price of gold in the year 2006. Predicted price of gold in 2006: _______________________________________ 3. Is this prediction the same as the one for the model f(x) = 229.523(1.17803)x Yes or No? _________________________ 4. Conjecture a reason for your finding in problem 3. __________________ ____________________________________________________________________________ ____________________________________________________________________________ 5. The Consumer Price Index (CPI), using 1967 as the base year (CPI = 100), has had the values given in the table below: Year 1915 1920 1925 1930 1935 1940 1945 1950 1955 1960 CPI 30.4 60.0 52.5 50.0 41.1 42.0 53.9 72.1 80.2 88.7 Year 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 CPI 94.5 116.3 161.2 248.8 322.2 391.4 456.5 528.8 599.7 669.6 The scatterplot of this information using 0 for the year 1915, 5 for 1920, etc is seen is seen in Figure 6.9. Chapter 6 Exponential Functions 15 Figure 6.9 Figure 6.9 shows that the CPI increases slowly at first. Eventually these values start increasing faster, making the plot look more like an exponential graph than a linear or polynomial one. 6. Find the exponential model for this data. Exponential Model: ______________________________________________________________ Figure 6.10 7. Does your model look like the one in Figure 6.10? ___________________________ 8. Use your model to estimate the CPI in 1945. ________________________________ 9. Did your model prdict that f(30) ≈ 71.2? ___________ If No check your work. Historical Note: You can see that the model fits the data fairly well, considering the unusual conditions during the Great Depression causes the model to overestimate the CPI through the 1930's Chapter 6 Exponential Functions 16 and changing economic condition has the model underpredicting the CPI after 1980. 10. Find the rate of change in the CPI in the year 2010. ____________________________________________________________________________________________ If instead of continuing to grow exponentially, CPI increases at the same rate for a while, the CPI value can be predicted by the tangent line, which is the linear function that has the same rate of increase that it had at the last measured point. 11. Find the equation of the line tangent to the CPI function above at the year 2010. ____________________________________________________________________________________________ ____________________________________________________________________________________________ ____________________________________________________________________________________________ 12. Use your tangent line to predict the CPI in 2013. _____________________________ 13. Use the exponential function to predict the CPI in 2013. _____________________ 14. What did you observe? __________________________________________________________ The reason for this is that although the exponential model does model the history of the CPI well, it was underestimating the actual CPI in 2010 (and 2013). By using the actual CPI for 2010 and estimating the value using the rate of change at that time probably gives a more realistic result. The tangent line may give a better estimate for a short time after 2010, but eventually the CPI will probably curve upward like the exponential curve and the tangent line will fall below the correct value. One step back, several steps forward….. Chapter 6 Exponential Functions 17 Simple Interest You may remember the formula for simple interest I = Prt where P is the amount of the initial, and only, deposit , r is the annual interest rate, and t is the time, in years, of the investment. Example 6.1: Find the interest earned on a 2-year investment of $100 deposit account paying 2% simple interest . Find the amount of money, A, In the account at that time. I = 100× .02 × 2 = $4 A = 100 + 4 = $104 Example 6.2: Find the interest earned on a 9-month investment of $100 deposit account paying 2% simple interest . Find the amount of money, A, In the account at that time. I = 100× .02 × = $1.50 A = 100 + 1.50 = $101.50 or These examples demonstrate that the final amount in the account is the initial deposit plus the interest earned. A = P + I = P + Prt = P(1 + rt) Compound Interest Simple interest is the foundational idea of compound interest, the more commonly used type of interest earned in savings accounts. The difference between simple and compound interest will be demonstrated through examples. Example 6.3: Find the maturity value of $100 deposited at 2% simple € 9 12 Chapter 6 Exponential Functions 18 interest for 1 year, 2 years, 3 years. A = 100(1 + .02 × 1) = $102.00 A = 100(1 + .02 × 2) = $104.00 A = 100(1 + .02 × 3) = $106.00 Thus, we see the account earned an additional $2.00 in interest during the second and third years. From this we see that the interest earned during each of these years was 100 × .02 × 1. That is, the interest earned during the second and third year was based on the original investment amount. In a compound interest account interest earned is added to the account at specific points in its history. If an account is compounded annually interest is added to the account at the end of every year and that new amount is used to calculate the next year’s earned interest. If an account is compounded quarterly interest is added to the account at the end of every quarter and that new amount is used to calculate the next quarter’s earned interest. Example 6.4: Find the maturity value of $100 deposited for 3-years at 2% compounded annually. Year 1: A = 100(1 + .02 × 1) = $102.00 Year 2: A = 102(1 + .02 × 1) = $104.04 Year 3: A = 104.04(1 + 104.04 × 1) = $106.12 Notice that compounding annually earns an additional $0.12 in interest. Example6.5: Now consider the investment of $100 deposited for 2-years at 2% compounded quarterly. Chapter 6 Exponential Functions 19 Quarter 1: Quarter 2: Quarter 3: Quarter 4: Quarter 5: Quarter 6: Quarter 7: Quarter 8: You should notice that compounding quarterly earned $0.03 more than compounding annually. The maturity value after Quarter 2 was found using the following calculation: Noticing that tells us that � = 100 1 + .!" ! ! . A similar argument tells us that at the end of Quarter 3 the value of the account will be � = 100 1 + .!" ! ! . € A =100 1+ .02 × 1 4 # $ % & ' ( =100 1+ .02 4 # $ % & ' ( = $100.50 € A =100.50 1+ .02 4 " # $ % & ' = $101.0025 € A =100.0025 1+ .02 4 " # $ % & ' = $101.5075125 € A =101.5075125 1+ .02 4 " # $ % & ' = $102.0150501 € A =102.0150501 1+ .02 4 " # $ % & ' = $102.5251253 € A =102.5251253 1+ .02 4 " # $ % & ' = $103.0377509 € A =103.0377509 1+ .02 4 " # $ % & ' = $103.5529397 € A =103.5529397 1+ .02 4 " # $ % & ' = $104.07 € A =100.50 1+ .02 4 " # $ % & ' = $101.0025 € 100.50 =100 1+ .02 4 " # $ % & ' Chapter 6 Exponential Functions 20 Since two quarters is one-half of a year and three quarters is three-quarters of a year the pattern seen above suggests a general formula for the maturity value of a compound interest account: where P = the initial investment, r = the annual interest rate, n = number of compounding periods per year, and t = number of years since initial investment. Example 6.6: Find the maturity value of $100 deposited for 5-years at 4% compounded quarterly. Example 6.7: Find the maturity value of $100 deposited for 5-years at 4% compounded monthly. Example 6.8: Find the maturity value of $100 deposited for 5-years at 4% compounded daily. This entire discussion informs us that compound interest is practical example of a exponential function in everyday life. € A = P 1+ r n " # $ % & ' nt € A =100 1+ .04 4 " # $ % & ' 4×5 = $122.02 € A =100 1+ .04 12 " # $ % & ' 12×5 = $122.10 € A =100 1+ .04 365 " # $ % & ' 365×5 = $122.14 Chapter 6 Exponential Functions 21 Homework 15. Explain why the compound interest formula is an example of an exponential model. ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ The exponential model for the price of gold since 2000 was found to be f(x) = 229.523(1.17803)x. 16. Interpret the meaning of the number 229.523 in the context of this model. _____________________________________________________________________________________ _____________________________________________________________________________________ 17. Interpret the meaning of the decimal .17803 in the context of this model. _____________________________________________________________________________________ _____________________________________________________________________________________ The exponential model for the CPI since 1915 was found to be f(x) = 26.3667(1.03369)x. 18. Interpret the meaning of the number 26.3667 in the context of this model. _____________________________________________________________________________________ _____________________________________________________________________________________ 19. Interpret the meaning of the decimal .03369 in the context of this model. _____________________________________________________________________________________ _____________________________________________________________________________________ Chapter 6 Exponential Functions 22 An Unexpected but Frequently used Base for Exponential Models In many applications, the most common base used is a number known as e. The importance of this number lies in its relation to the rate of natural growth (or decay) of many populations. The symbol e is the name given to the number that the function approaches asymptotically as x becomes large. This function can be thought of as a compound interest problem in which $1.00 is invested for 1-year at a 100% interest rate compounded x times a year. Table 6.5 illustrates the maturity value of the $1.00 investment at various compounding periods. Table 6.5 x 1 2 10 2.59374 100 2.70481 1000 2.71692 10,000 2.71815 100,000 271827 Figure 6.11 The graph and table above show that this number is near 2.718. They also suggest that e € 1+ 1 x " # $ % & ' x € 1+ 1 x " # $ % & ' x Chapter 6 Exponential Functions 23 can be found using the method of limits (Figure 6.12). Figure 6.12 Using ctrl enter rather than enter find Figure 6.13. Figure 6.13 Note that this value of e is an approximation since e is an irrational number with an infinite, non-repeating decimal representation. Chapter 6 Exponential Functions 24 Limits for Exponential Functions (Optional) Figure 6.14 contains some important limits for the Exponential Function ex. Figure 6.14 Figure 6.14 (cont.) From this we see that the larger a positive exponent becomes, the larger the exponential value will be and the more negative the exponent becomes, the closer the exponential value will be to zero. Chapter 6 Exponential Functions 25 The Continuous Growth Model In the compound interest model interest is added to an account at the end of a specified period of time. We saw that the advantage to shrinking the time of the compounding period. However, there is a limit to this advantage. This limiting argument is the basis for the continuous compounding, or Continuous Growth, model � = ����. In this model we assume that an account or population does not remain constant for a period of time before a change is implemented, but changes constantly. Think of the population of a city as an example of continuous growth. The population does not remain at x for a month and suddenly jump on the 1st of the next month and then once again remain constant until the next 1st of the month. The population is constantly changing. The formula the Continuous Growth Model emerges from the compound interest formula via the following argument. � = � 1 + � � �� Define � = � � , thus n = kr As � → ∞ we see � → ∞. � = � 1 + � �� ��� � = � 1 + 1 � � �� Figure 6.13 shows that as � → ∞, 1 + ! � � → � Therefore � = ���� the Continuous Growth Model. Be sure you understand that e is not a variable, but the name for a particular number, like 7 or π. Many exponential functions that describe natural growth of some kind use e for the base, and most of the following examples will also. The only disadvantage to using e for the Chapter 6 Exponential Functions 26 base is the fact that calculating powers of e is not convenient to do by hand. However, if you need to calculate a numerical value for a power of e, your calculator can do it. Example of Exponential Growth The new sales model could be described by a function like , where S gives the sales per week at week x. To start, the sales for this model are S(0) = 100 per week and after 5 weeks sales have risen to S(5) = 211.7 per week. Table 6.6 Notice how rapidly this function increases. Example: The number of bacteria in a culture was estimated to be 5,000 at 8 a.m. Three hours latter the culture contained 20,000 bacteria. Find the Growth model that can be used to predict bacteria population and predict the population at 2 p.m. 20,000 = 5,000��×! Figure 6.15 € S(x) =100e 0.15x Chapter 6 Exponential Functions 27 Model: � = 5000�.!"#� Estimated Population at 2 p.m. 5000�.!"#×! = 79,953 Example: You may have read that Dutch explorers purchased the island of Manhattan in 1624 for $24 worth of glass beads. This claim is now in dispute and you may find it interesting to read about the details. We, however, will not concern ourselves with facts but enter the world of fantasy. Suppose the Indians had demanded cash for the island and deposited the $24 in a bank paying continuous compound interest. If in 1660 the value of the account was $70.67, what was the value of the account in 2012? 70.67 = 24e36k Figure 6.16 Model: A = 24e.03t Value of Account in 2012 = 24e.03×388 = 2.7252 × 106 = $2,725,200 Chapter 6 Exponential Functions 28 Homework 20. Carbon-14 is a radioactive form of carbon found in all living plants and animals that slowly breaks down after the organism dies. Carbon-14 is used to determine the age of remains. The amount of Carbon-14 found in remains is modeled by the continuous exponential growth model. However, in cases such as this, when amounts decrease, the model is usually called the continuous exponential decay model. If it takes 100g of Carbon-14 2000 years to decay to 78g how long will it take for the 100g to decay to 50 grams? Chapter 6 Exponential Functions 29 More on the Derivatives of Exponential Functions It is interesting to note that if we consider the exponential function with base e we see that its derivative is k × ex. (See Figure 6.17) Figure 6.17 If k =1 then the derivative of € f (x) = ex is defined to be lim h→0 ex+h − ex h = ex . Thus, the exponential function € ex is equal to its own derivative. The rate of change of € ex at any x-value is equal to its own value at that point. We will now explore the more general exponential expression f (x) = keg( x) . Examples: Figure 6.18 The TI-Nspire easily finds derivatives of exponential functions. Chapter 6 Exponential Functions 30 Optional Please notice that: • 16�!� = 8�!� ∗ 2. Thus, if € f (x) = 8e 2x then f !(x) = 8e 2 x ∗ 2 • 60 ∗ �! ∗ �!�! = 5�!�! ∗ 12�!. Thus, if � � = 5�!�! then �′ � = 5�!�! ∗ 12�!. • −12��!!�! = 3�!!�! ∗ −4� . Thus, if € f (x) = 3e−2x 2 then f !(x) = 3e−2 x2 ∗(−4x) Do you see a pattern that is suggested in the examples above? ________________________ If you answered yes, what is the pattern you observed? ________________________________ _______________________________________________________________________________________________ If you answered no, examine the examples more closely and look for a pattern. If you are still not able to find a pattern discuss this problem with a member of the class that has found a pattern. Make up two exponential functions and test your conjectured pattern. Test 1: Function __________________________________________ Is the conjecture result equal to the TI-Nspire result? _________________________________ Test 2: Function __________________________________________ Is the conjecture result equal to the TI-Nspire result? _________________________________ The rule for the more general exponential expression f (x) = keg( x) is f !(x) = keg( x) ∗ g!(x). Was this your conjecture? ___________________ The proof that this version of the chain rule can be found in most standard Calculus textbooks intended for use by mathematics, engineering, and science majors. Chapter 6 Exponential Functions 31 Homework Exponential functions appear frequently in models that describe natural growth or learning and loss or decay. The function € f (x) = 80 − 60e−0.5x may serve as a model for the number of words a typist can type per minute, where x represents the amount of experience, in months, the typist has had. 21. What is the practical domain of this function? __________________________________________________________________________________________ 22. What is the y-intercept of the function? _____________________________________________ 23. Interpret the meaning of the y-intercept in the context of this problem. __________________________________________________________________________________________ 24. At what rate is the typist’s speed changing if he/she has had 4 months experience? ___________________________________________________________________________________________ ___________________________________________________________________________________________ 25. At what rate is the typist’s speed changing if he/she has had 24 months experience? ___________________________________________________________________________________________ ___________________________________________________________________________________________ 26. What does this last answer suggest? ___________________________________________________________________________________________ ___________________________________________________________________________________________ Chapter 6 Exponential Functions 32 27. Find € limx→∞ 80 − 60e−0.5x ( ) __________________________________________________________________ 28. Interpret the meaning of this answer in the context of the problem. __________________________________________________________________________________________ __________________________________________________________________________________________ Functions like this previous one are often useful in fields such as psychology or industrial management and are called learning curves. They are usually applied in situations that involve the learning of physical skills. Typically one starts doing a task with a certain level of skill, which usually improves rapidly with practice. However, once the person reaches a certain level of aptitude, the rate of improvement slows. Further improvement may be hard to discern when the skill has reached a level close to the individual's maximum performance level, which may differ from person to person. In Example 0.8 you saw the selling price of a car model decrease with age. Find the exponential model that predicts the selling price of a car at a given age. 29. Model: ____________________________________________________________________________ 30. Interpret the meaning of the coefficient of this model in the context of the problem. _____________________________________________________________________________________ _____________________________________________________________________________________ 31. Interpret the meaning of the decimal of the base of this model in the context of this problem. _________________________________________________________________________ _____________________________________________________________________________________ Chapter 6 Exponential Functions 33 Integration with Exponential Functions Suppose the Widgit Company weekly sales rate is modeled by € s(x) =100e 0.15x , where x is measured in weeks since starting sales operations. To find the total sales for the company in its 3rd month we can integrate the function from week 8 to week 12. (Figure 6.19) Figure 6.19 Therefore, sales in month 3 were approximately 1820 units. Approximate the total sales for year 2. __________________________________________________ Comment on the reasonableness of this result. _________________________________________ _______________________________________________________________________________________________ The model predicted that sales would be approximately 3.97 × 109 = 3,970,000,000, more than half the population of the entire world. This would be a most successful product indeed and such success is not very likely. Thus, it is probably unreasonable to expect that sales will continue to grow at an unrestricted exponential rate. There are usually naturally limiting factors that force an exponential growth model to slow. In our sales example these factors would include the desirability of the product - how many people would really want it?- the size of the population to whom it is offered, and the ability to produce the needed number of units - including the availability of the resources needed to produce the item. Chapter 6 Exponential Functions 34 You can probably think of others. This thinking suggests the need for a modification of the exponential model. A function which serves as a good model for the growth of populations where there are limited natural resources, preventing the population to grow without bounds. A type of model that simulates the growth of a population that stays small until it becomes large enough to grow rapidly. It continues this growth until the stress caused by deficiencies in natural resources forces the population to limit its further growth. It serves well for the growth of populations in areas unaffected by immigration and emigration and for modeling sales or the spread of diseases or rumors. The graph below pictures the behavior described above. Figure 6.20 The function that produces this type of curve is called a logistic curve. Visual examination of the data in Table 0.9 and the scatterplot in Figure 0.9 suggest that the Platy population in the aquarium follows a logistic model. To find this model we will once again use the TI-Nspire regression procedure with a modification for logistic regression • B: ShowLogistic (d=0) as the final step. (See Figure 6.21) Chapter 6 Exponential Functions 35 Figure 6.21 Coming back to the Widgit Company we see that a logistic model that might better serve to predict weekly sales rate for this company is € l(x) = 10000 1+ 99e−0.17x . s(52) = 244,060 and l(52) = 9,858.65 a much more reasonable weekly sales rate. In Figure 6.22 we see that the total sales predicted by the exponential model and the logistic model are similar for the third month of operation. Figure 6.22 In Figure 6.23 we now see that the predicted total sales for year two of operations using the Chapter 6 Exponential Functions 36 logistic model is at least a number that is conceivably correct. Figure 6.23 The general form of the logistic growth model is given by the equation € P(t) = M 1+Ce−kt where M is the carrying capacity (maximum possible size), r is the growth rate, and t is the time. Chapter 6 Exponential Functions 37 Homework 32. Find € limt→∞ P(t). ________________________________________________________________ 33. Explain why L is called the carrying capacity. _____________________________ __________________________________________________________________________________ _________________________________________________________________________________ Influenza is spreading through a city according to the model € P(t) = 225000 1+ 971e−.12t , where t is measures in days. 34. How many people are projected to be infected during the first week of the outbreak? ________________________________________________________________________________________________ 35. How many people are projected to be infected during the month first of the outbreak June? ________________________________________________________________________________________________ 36. After how many days will half the city be infected? _____________________________________ _________________________________________________________________________________________________ 37. Describe a “real-world” situation you believe should be modeled by a logistic function. ________________________________________________________________________________________________ ________________________________________________________________________________________________ ________________________________________________________________________________________________ Chapter 6 Exponential Functions 38 38. Explain why think a logistic model would be appropriate. _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ 39. Find data on Internet usage between the years 1995 and 2011. Year 1995 1996 1996 1997 1998 1999 2000 2001 2002 Users Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 Users Fit a logistic model to this data. Discuss the appropriateness of the model. Model: ________________________________________________________________________________ Discussion: ___________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ Chapter 6 Exponential Functions 39 40. Pick a country that has a name starting with the first letter of either your first or last name. Country: ____________________________________ Find the population of that country in the years indicated in the table below. Year 1900 1910 1920 1930 1940 1950 Pop Year 1960 1970 1980 1990 2000 2010 Fit a logistic model to this data. Discuss the appropriateness of the model. Model: ________________________________________________________________________________ Discussion: ___________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________