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Section 3.5 Modeling with Exponential Logarithmic Functions Exponential Growth and Decay Example The equation A=A 0 ekt models growth of a deer population in a small local preserve. If the initial population is 14 deer and the population grows to 20 in 4 years, find the value of k, the rate of growth. Logistic Growth Models Example If the following logistic equation models the number of people who become infected with noro virus on a small cruise ship off the coast of Alaska, how many people will become ill by the second week if t is given in weeks. f(t)= 500 1 e 1.5t The Art of Modeling Graphing Calculator - Exponential Regression Press STAT then EDIT 1. Type in the values in List 1 (x values) and List 2 (y values). Then STAT, move over to CALC and press 0. You will get values for a, b, r and r2. The values of a and b go into the equation y=a*b^x or y=abx. When you use a graphing utility the value of r MAY appear. This r is called the correlation coefficient and is a measure of how well the model fits the data. The value of r is such that -1 r 1. A positive r means that as the x-values increase, so do the y-values. A negative r means that as the x-values increase, the y values decrease. The closer that r is to -1 or 1, the better the model fits the data. Example L1 L2 1 8 2 18 3 49 4 123 5 312 6 780 Find the exponential regression equation on your calculator using the data below. The data is for the population of black flies which increases exponentially in a favorable lab environment. L1 is the number of weeks and L2 is for the number of black flies in the laboratory experiment. See the next screen for continuation of example. Example-Graphing Calculator Continuing with this same data, we are now going to graph the equation that we got earlier in this exercise and we are going to graph the data. First make certain that your window is large enough so that you will be able to see the graphed points. Second, go to STAT PLOT and press ENTER on #1. Press ENTER on the word "On", for type of graph use the first one, make Xlist: L1, and Ylist:L2. Use either mark for the last option. Now return to the graph and see the data. Go to Y= and type in the equation that you got on the previous screen. Return to the graph and observe how close the curve is to the actual data. How can we obtain a logarithmic function that models the data? A graphing utility can be used to obtain a logarithmic model of the form y=a+b ln x. Because the domain of the logarithmic function is the set of positive numbers, zero must not be a value for x. The technique for finding that logarithmic equation is the same as for finding the exponential equation on a graphing calculator. On the STAT, CALC menu choose 9 for Logarithmic Regression. Expressing y=abx in Base e Example Rewrite y=3.82 (6.7) x in terms of base e. A=A 0 e kt mod els growth of gray squirrels in a small city park. If the initial population is 30 in 2000, and it grows to 38 in 1 year, what is the growth rate, k? (a) .236 (b) .023 (c) .135 (d) .126 1000 The equation f(x)= models a rare type of flu t 1 80e which is infecting a town school. How many students, f(x), will be sick in 2 weeks if t is time in weeks? (a) 102 (b) 36 (c) 85 (d) 94