Exponential and3.5   Logarithmic Models
IntroductionThe five most common types of mathematical modelsinvolving exponential functions or logarithmic functions are:...
Exponential Growth and DecayGrowth: y = aebx, b > 0   Decay: y = aebx, b < 0                                               3
Example 1 – DemographyEstimates of the world population (in millions) from 2003through 2009 are shown in the table. A scat...
Example 1 – Demography                                    cont’dAn exponential growth model that approximates these datais...
Example 1 – Solution                                             cont’d              P = 6097e0.0116t        Write origina...
Logistic Growth Models                         7
Logistic Growth ModelsSome populations initially have rapid growth, followed by adeclining rate of growth, as indicated by...
Logistic Growth ModelsOne model for describing this type of growth pattern is thelogistic curve given by the functionwhere...
Example 2 – Spread of a VirusOn a college campus of 5000 students, one student returnsfrom vacation with a contagious flu ...
Example 2 – Solutiona. After 5 days, the number of students infected is                54.b. Classes are canceled when th...
Example 2 – Solution                                      cont’d             1 + 4999e –0.8t = 2.5                      e ...
Gaussian Model          ( x b)                2        y   ae       c                    13
Gaussian ModelsThis type of model is commonly used in probability andstatistics to represent populations that are normally...
Gaussian ModelsThe average value for a population can be found from thebell-shaped curve by observing where the maximumy-v...
Example 3 – SAT ScoresIn 2009, the Scholastic Aptitude Test (SAT) mathematicsscores for college-bound seniors roughly foll...
Example 3 – SolutionOn this bell-shaped curve, the maximum value of the curverepresents the average score. Using the maxim...
Logarithmic Modelsy = a + b ln x      y = a + b log10x                                       18
Example 4 - MeteorologyIn meteorology, the relationship between the height H of aweather balloon (measured in km) and the ...
Newton’s Law of CoolingTt  Tm  (T0  Tm )e , k  0             kt   Tt = temperature of object at time t   Tm = temperat...
Example 5 – Cooling Heated ObjectAn object is heated to 100°C and is then allowed to cool ina room whose air temperature i...
TI-84 Exponential Regression Modelon TI-84         y  ab                    x       a = initial value       b = ratio of ...
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3.5 EXP-LOG MODELS

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3.5 EXP-LOG MODELS

  1. 1. Exponential and3.5 Logarithmic Models
  2. 2. IntroductionThe five most common types of mathematical modelsinvolving exponential functions or logarithmic functions are:1. Exponential growth model: y = aebx, b > 02. Exponential decay model: y = aebx, b < 03. Logistic growth model:4. Gaussian model: y = ae5. Logarithmic models: y = a + b ln x y = a + b log10x 2
  3. 3. Exponential Growth and DecayGrowth: y = aebx, b > 0 Decay: y = aebx, b < 0 3
  4. 4. Example 1 – DemographyEstimates of the world population (in millions) from 2003through 2009 are shown in the table. A scatter plot of thedata is shown below. (Source: U.S. Census Bureau) 4
  5. 5. Example 1 – Demography cont’dAn exponential growth model that approximates these datais given by P = 6097e0.0116t, 3  t  9where P is the population (in millions) and t = 3 represents2003.a) According to this model, when will the world population reach 7.1 billion? 5
  6. 6. Example 1 – Solution cont’d P = 6097e0.0116t Write original equation. 7100 = 6097e0.0116t Substitute 7100 for P. 1.16451  e0.0116t Divide each side by 6097. In1.16451  Ine0.0116t Take natural log of each side. 0.15230  0.0116t Inverse Property t  13.1 Divide each side by 0.0116.According to the model, the world population will reach7.1 billion in 2013. 6
  7. 7. Logistic Growth Models 7
  8. 8. Logistic Growth ModelsSome populations initially have rapid growth, followed by adeclining rate of growth, as indicated by the graph below. Logistic Curve 8
  9. 9. Logistic Growth ModelsOne model for describing this type of growth pattern is thelogistic curve given by the functionwhere y is the population size and x is the time. Anexample is a bacteria culture that is initially allowed to growunder ideal conditions, and then under less favorableconditions that inhibit growth. A logistic growth curve is alsocalled a sigmoidal curve (or S curve). 9
  10. 10. Example 2 – Spread of a VirusOn a college campus of 5000 students, one student returnsfrom vacation with a contagious flu virus. The spread of thevirus is modeled bywhere y is the total number of students infected after days.The college will cancel classes when 40% or more of thestudents are infected.a. How many students are infected after 5 days?b. After how many days will the college cancel classes? 10
  11. 11. Example 2 – Solutiona. After 5 days, the number of students infected is  54.b. Classes are canceled when the number of infected students is (0.40)(5000) = 2000. 11
  12. 12. Example 2 – Solution cont’d 1 + 4999e –0.8t = 2.5 e –0.8t = In e –0.8t = In – 0.8t = In t = 10.14So, after about 10 days, at least 40% of the students will beinfected, and classes will be canceled. 12
  13. 13. Gaussian Model ( x b) 2 y ae c 13
  14. 14. Gaussian ModelsThis type of model is commonly used in probability andstatistics to represent populations that are normallydistributed. For standard normal distributions, the modeltakes the formThe graph of a Gaussian model is called a bell-shapedcurve. Try graphing the normal distribution curve with agraphing utility. Can you see why it is called a bell-shaped curve? 14
  15. 15. Gaussian ModelsThe average value for a population can be found from thebell-shaped curve by observing where the maximumy-value of the function occurs. The x-value correspondingto the maximum y-value of the function represents theaverage value of the independent variable—in this case, x. 15
  16. 16. Example 3 – SAT ScoresIn 2009, the Scholastic Aptitude Test (SAT) mathematicsscores for college-bound seniors roughly followed thenormal distribution ( x 515)2  y  0.0034e 26,912 , 200  x  800where x is the SAT score for mathematics.a) Use a graphing utility to graph this functionb) Estimate the average SAT score. 16
  17. 17. Example 3 – SolutionOn this bell-shaped curve, the maximum value of the curverepresents the average score. Using the maximum feature ofthe graphing utility, you can see that the average mathematicsscore for college bound seniors in 2009 was 515. 17
  18. 18. Logarithmic Modelsy = a + b ln x y = a + b log10x 18
  19. 19. Example 4 - MeteorologyIn meteorology, the relationship between the height H of aweather balloon (measured in km) and the atmosphericpressure p (measured in millimeters of mercury) is modeledby the function H  48  8 ln pa) Predict the height of a weather balloon when the atmospheric pressure is 560 millimeters of mercury.b) If the height of the balloon is 3 km, what is the atmospheric pressure?c) Graph this model. Does it look like a log graph? Explain. 19
  20. 20. Newton’s Law of CoolingTt  Tm  (T0  Tm )e , k  0 kt Tt = temperature of object at time t Tm = temperature of surrounding medium (room temp) T0 = initial temperature of heated object k = negative constant t = time 20
  21. 21. Example 5 – Cooling Heated ObjectAn object is heated to 100°C and is then allowed to cool ina room whose air temperature is 30°C.a) If the temp of the object is 80 C after 5 minutes, find the value of k.b) Determine the time that needs to elapse before the object is 75 C.c) Graph the relation found between temperature and time.d) Using the graph, determine the time that needs to elapse before the object is 75 C. 21
  22. 22. TI-84 Exponential Regression Modelon TI-84 y  ab x a = initial value b = ratio of successive y-values 22

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