5.7 Constructing Nonlinear Models Find an exponential model Find a logarithmic model Find a logistic model Select a model

Types of Nonlinear Data

Example: Modeling atmospheric pressure (1 of 4) As altitude increases, air pressure decreases. The atmospheric pressure P in millibars (mb) at a given altitude x in meters: Altitude and Air Pressure x(m) 5000 10,000 15,000 20,000 25,000 30,000 p(mb) 1013 541 265 121 55 26 12 a. Make a scatterplot of the data. What type of function might model the data?

Example: Modeling atmospheric pressure (2 of 4) Altitude and Air Pressure x(m) 5000 10,000 15,000 20,000 25,000 30,000 p(mb) 1013 541 265 121 55 26 12 a. A scatterplot of the data is shown. A decreasing exponential function might model the data.

Example: Modeling atmospheric pressure (3 of 4) b. Select ExpReg to find a ≈ 1104.9 and b ≈ 0.99985

Example: Modeling atmospheric pressure (4 of 4) c. f(23,000) = 1104.9(0.99985)23,000 f(23,000) ≈ 35.1 millibars

Example: Modeling interest rates (1 of 4) The table lists the interest rates for certificates of deposit. Time (months) 1 3 6 9 24 36 60 Yield (%) 0.25 0.39 0.74 0.80 1.25 1.40 1.50 a. Make a scatterplot of the data. What type of function might model these data? b. Use least-squares regression to find a formula f(x) = a + b ln x that models the data. c. Graph f and the data in the same viewing rectangle.

Example: Modeling interest rates (2 of 4) Solution Time (months) 1 3 6 9 24 36 60 Yield (%) 0.25 0.39 0.74 0.80 1.25 1.40 1.50 a. A scatterplot of the data is shown. The data increase but gradually level off. A logarithmic modeling might be appropriate.

Example: Modeling interest rates (3 of 4) b. Select LnReg to find: f(x) = 0.143 + 0.334 ln x

Example: Modeling interest rates (4 of 4) c. Here’s a graph of f and the data.

Logistic Model (1 of 2) In real life, populations of bacteria, insects, and animals do not continue to grow indefinitely. Initially, population growth may be unrestricted, and modeled by exponential growth. Then as resources become more scarce, their rate of growth begins to slow. After a region has become heavily populated or saturated, the population usually levels off because of limited resources.

Logistic Model (2 of 2) This type of growth may be modeled by a logistic function represented by where a, b, and care positive constants. The graph of f is referred to as a sigmoidal curve.

Example: Modeling logistic growth (1 of 6) One of the earliest studies about population growth was done using yeast plants in 1913. A small amount of yeast was placed in a container with a fixed amount of nourishment. The units of yeast were recorded every 2 hours. Time 2 4 6 8 10 12 14 16 18 Yeast 9.6 29.0 71.1 174.6 350.7 513.3 594.8 640.8 655.9 661.8

Example: Modeling logistic growth (2 of 6) a. Make a scatterplot of the data. Describe the growth. b. Use least-squares regression to find a logistic function f that models the data. c. Graph f and the data in the same viewing rectangle. d. Approximate graphically the time when the amount of yeast was 200 units.

Example: Modeling logistic growth (3 of 6) Solution Time 2 4 6 8 10 12 14 16 18 Yeast 9.6 29.0 71.1 174.6 350.7 513.3 594.8 640.8 655.9 661.8 a. When the units of yeast are small, the growth is rapid, or exponential. Then the rate of growth slows. The limited amount of nourishment causes this leveling off.

Example: Modeling logistic growth (4 of 6) b. Select Logistic to find:

Example: Modeling logistic growth (5 of 6) c. Here’s a graph of f and the data. The fit for the real data is remarkably good.

Example: Modeling logistic growth (6 of 6) d. The graphs of Y1 = f(x) and Y2 = 200 intersect near (6.29, 200). The amount of yeast reached 200 units after about 6.29 hours.

Selecting a Model In real-data applications, a modeling function is seldom given. Many times we must choose the type of modeling function and then find it using least-squares regression. Thus far in this section, we have used exponential, logarithmic, and logistic functions to model data.

Example: Modeling highway design (1 of 5) To allow enough distance for cars to pass on two-lane highways, engineers calculate minimum sight distances between curves and hills. The table shows the minimum sight distance y in feet for a car traveling at x miles per hour. x(mph) 20 30 40 50 60 65 70 y(ft) 810 1090 1480 1840 2140 2310 2490

Example: Modeling highway design (2 of 5) a. Find a modeling function for the data. b. Graph the data and your modeling function. c. Estimate the minimum sight distance for a car traveling at 43 miles per hour. x(mph) 20 30 40 50 60 65 70 y(ft) 810 1090 1480 1840 2140 2310 2490

Example: Modeling highway design (3 of 5) Solution x(mph) 20 30 40 50 60 65 70 y(ft) 810 1090 1480 1840 2140 2310 2490 a. A scatterplot of the data is shown. The data appear to be (nearly) linear.

Example: Modeling highway design (4 of 5) a. Use linear regression to obtain f(x) = 33.93x + 113.4.

Example: Modeling highway design (5 of 5) b. The data and f are graphed. Function f gives a good fit. c. f(43) = 33.93(43) + 113.4 ≈ 1572 feet

Example: Modeling asbestos and cancer (1 of 6) The table lists the number N of lung cancer cases occurring within a group of asbestos insulation workers with a cumulative total of 100,000 years of work experience, with their first date of employment x years ago. X (years) 10 15 20 25 30 N (cases) 6.9 25.4 63.6 130 233

Example: Modeling asbestos and cancer (2 of 6) a. Find a modeling function for the data. b. Graph the data and your modeling function. c. Estimate the number of lung cancer cases for x = 23 years. Interpret your answer. X (years) 10 15 20 25 30 N (cases) 6.9 25.4 63.6 130 233

Example: Modeling asbestos and cancer (3 of 6) Solution X (years) 10 15 20 25 30 N (cases) 6.9 25.4 63.6 130 233 a. A scatterplot of the data is shown. We will use a power function to model the data.

Example: Modeling asbestos and cancer (4 of 6) a. Use PwrReg to obtain

Example: Modeling asbestos and cancer (5 of 6) b. The data and f are graphed. Function f gives a good fit.

Example: Modeling asbestos and cancer (6 of 6)