6 IntroductionThere are many examples of exponential and logarithmic models in real life.The five most common types of mathematical models involving exponential functions or logarithmic functions are as follows.Exponential growth model: y = aebx, b > 0Exponential decay model: y = ae–bx, b > 0
7 Introduction 3. Gaussian model: y = ae 4. Logistic growth model: Logarithmic models: y = a + b ln x,y = a + b log10x
8 IntroductionThe basic shapes of these graphs are shown in Figure 3.32.Figure 3.32
10 Please read the next three slides, but do not copy them down. Exponential Growth and DecayPlease read the next three slides, but do not copy them down.
11 Example 1 – DemographyEstimates of the world population (in millions) from 2003 through 2009 are shown in the table. A scatter plot of the data is shown in Figure (Source: U.S. Census Bureau)Figure 3.33
12 Example 1 – Demographycont’dAn exponential growth model that approximates these data is given byP = 6097e0.0116t, 3 t 9where P is the population (in millions) and t = 3 represents Compare the values given by the model with the estimates shown in the table. According to this model, when will the world population reach 7.1 billion?
13 Example 1 – SolutionThe following table compares the two sets of population figures.From the table, it appears that the model is a good fit for the data. To find when the world population will reach billion, letP = 7100in the model and solve for t.
14 Example 1 – Solution 6097e0.0116t = P 6097e0.0116t = 7100 cont’d6097e0.0116t = P6097e0.0116t = 7100e0.0116t Ine0.0116t In0.0116t t 13.1According to the model, the world population will reach billion in 2013.Write original equation.Substitute 7100 for P.Divide each side by 6097.Take natural log of each side.Inverse PropertyDivide each side by
15 Please read the next three slides, but do not copy them down. Gaussian ModelsPlease read the next three slides, but do not copy them down.
16 Gaussian ModelsThe graph of a Gaussian model is called a bell-shaped curve.The average value for a population can be found from the bell-shaped curve by observing where the maximum y-value of the function occurs. The x-value corresponding to the maximum y-value of the function represents the average value of the independent variable—in this case, x.
17 Example 4 – SAT ScoresIn 2009, the Scholastic Aptitude Test (SAT) mathematics scores for college-bound seniors roughly followed the normal distributiony = e–(x – 515)226,912, 200 x 800where x is the SAT score for mathematics. Use a graphing utility to graph this function and estimate the average SAT score. (Source: College Board)
18 Example 4 – SolutionThe graph of the function is shown in Figure On this bell-shaped curve, the maximum value of the curve represents the average score. Using the maximum feature of the graphing utility, you can see that the average mathematics score for college bound seniors in 2009 was 515.Figure 3.37
22 Logarithmic ModelsOn the Richter scale, the magnitude R of an earthquake of intensity I is given bywhere I0 = 1 is the minimum intensity used for comparison. Intensity is a measure of the wave energy of an earthquake.
23 Example 6 – Magnitudes of Earthquakes In 2009, Crete, Greece experienced an earthquake that measured 6.4 on the Richter scale. Also in 2009, the north coast of Indonesia experienced an earthquake that measured 7.6 on the Richter scale. Find the intensity of each earthquake.Solution:Because I0 = 1 and R = 6.4, you have106.4 = 10 log10I = I
24 Example 6 – Solution For R = 7.6 you have 107.6 = 10 log10I 107.6 = I cont’dFor R = 7.6 you have107.6 = 10 log10I = I