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PreCalculus

PreCalculus. 1.9 – Mathematical Modeling. Introduction. Techniques for fitting models to data: direct and inverse variation and least squares regression. The resulting models are either polynomial functions or rational functions. Example 1 – A Mathematical Model.

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PreCalculus

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  1. PreCalculus 1.9 – Mathematical Modeling

  2. Introduction • Techniques for fitting models to data: direct and inverse variation and least squares regression. The resulting models are either polynomial functions or rational functions.

  3. Example 1 – A Mathematical Model • The numbers of insured commercial banks y (in thousands) in the United States for the years 1995 to 1999 are shown in the table below: YearInsured commercial banks, y 1995 9.94 1996 9.53 1997 9.14 1998 8.77 1999 8.58 A linear model that approximates this data is y = -0.348t + 11.63 for 5 < t < 9, where t is the year, with t = 5 corresponding to 1995. Plot the actual data and the model on the same graph. How closely does the model represent the data?

  4. Direct Variation • There are two basic types of linear models. The more general model has a y-intercept that is nonzero: y = mx + b, where b ≠ 0 • The simpler model is y = kx has a y-intercept that is zero. In the simpler model, y is said to vary directly as x, or to be directly proportional to x. • Direct Variation – The following statements are equivalent: • y varies directly as x. • y is directly proportional to x. • y = kx for some nonzero constant k. • k is the constant of variation or the constant of proportionality.

  5. Example 2 – Direct Variation • Determine whether the tables express a direct variation. If so find the constant of variation and write an equation for the table.

  6. B) Give that y varies directly as x, and y = 0.18 when x = -6, find y when x = -9.

  7. C) In Pennsylvania, the state income tax is directly proportional to gross income. You are working in PA and your state income tax deduction is $42 for a gross monthly income of $1500. Find a mathematical model that gives the PA state income tax in terms of gross income.

  8. Direct Variation as an nth Power • Another type of direct variation relates one variable to a power of another variable. • For example, in the formula for the area of a circle A = π r2 , the area A is directly proportional to the square of the radius r. Note that for this formula, π is the constant of proportionality.

  9. Direct Variation as an nth Power – The following statements are equivalent: 1) y varies directly as the nth power of x. 2) y is directly proportional to the nth power of x. 3) y = kxn for some constant k.

  10. Example 3 – Direct Variation as nth Power • Determine whether the tables express a direct variation. If so find the constant of variation and write an equation for the table.

  11. B) A number y varies directly as the square of the number x. If y = 32 when x = 4, find y when x = 6.

  12. C) This distance a ball rolls down an inclined plane is directly proportional to the square of the time it rolls. During the first second, the ball rolls 8 feet. • Write an equation relating the distance traveled to the time. • How far will the ball roll during the first 3 seconds?

  13. Inverse Variation • Inverse Variation – The following statements are equivalent: 1) y varies inversely as x. 2) y is inversely proportional to x. 3) y = k / x for some constant k. • If x and y are related by an equation of the form y = k / xn , then y varies inversely as the nth power of x (or y is inversely proportional to the nth power of x).

  14. Example 4 – Inverse Variation A) Determine whether the tables express an inverse variation. If so, find the constant of variation.

  15. B) A number y varies inversely as x. If y = 24, when x = 3, what is the y when x = 8? C) A number y varies inversely as the square of x. If y = 2, when x = 6, what is y when x = 3?

  16. D) A gas law states that the volume of an enclosed gas varies directly as the temperature and inversely as the pressure, as shown in the figure. The pressure of a gas is 0.75 kilogram per square centimeter when the temperature is 294 K and the volume is 8000 cubic centimeters. • Write an equation relating pressure, temperature, and volume. • Find the pressure when the temperature is 300 K and the volume is 7000 cubic centimeters.

  17. Joint Variation • In Example 4, note that when a direct variation and an inverse variation occur in the same statement, they are coupled with the word “and.” To describe two different direct variations in the same statement, the word jointly is used.

  18. Joint Variation – The following statements are equivalent: 1) z varies jointly as x and y. 2) z is jointly proportional to x and y. 3) z = kxy for some constant k. • If x, y, and z are related by an equation of the form z = kxnym , then z varies jointly as the nth power of x and the mth power of y.

  19. Example 5 – Joint Variation • A number y varies directly as x and z. If y = 6 when x = 8 and z = 3, what is y when x = 12 and z = 4?

  20. B) A number y varies jointly as x and z. If y = 5 when x = 3 and z = 10, what is y when x = 15 and z = 4?

  21. C) A number z varies directly as m and n, and inversely as the square of p. If z = 30 when m = 5, n = 4 and p = 2, what is p when z = 15, m = 5, and n = 32?

  22. D) The simple interest for a certain savings account is jointly proportional to the time and the principal. After one quarter (3 months), the interest on a principal of $5000 is $43.75. • Write an equation relating the interest, principal, and time. • Find the interest after three quarters.

  23. Least Squares Regression and Graphing Utilities • The Sum of Square Differences – The sum of the squares of the differences between actual data values and model values. • Least Squares Regression Line – The “best-fitting” linear model is the one with the least sum of square differences. This best fitting linear model is called the least squares regression line.

  24. Example 6 – Finding a Least Squares Regression Line • The amounts p (in millions of dollars) of total annual prize money awarded at the Indianapolis 500 race from 1993 to 2001 are shown in the table. Construct a scatter plot that represents the data and find a linear model that approximates the data. Year Prize Money, p 1993 7.68 1994 7.86 1995 8.06 1996 8.11 1997 8.61 1998 8.72 1999 9.05 2000 9.48 2001 9.62

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