L3-1 Lecture 3: Section 1.2 Linear Functions and Applications In Lecture 2, we explored linear equations of the form Ax + By = C. Solving for y, we have the equivalent form y = mx + b. This is a functional relationship: an equation which associates to a given value of the independent variable x exactly one value of the dependent variable y. Specific functions are designated by letters such as f, g, etc. For a given function f, we define f(x) to be the unique value of y associated with a given x value. Def. A linear function is a function defined by y = f(x) = mx + b for real numbers m and b.
L3-2 ex. Express the given linear equation as a linear function, and then find f( 2). 1) 3x + 4y 6 = 0 2) 2y 5 = 0 ex. 2x + 3 = 0 NOTE: If B = 0 in the equation Ax + By = C, we do not have a function.
L3-3 Supply and Demand An important idea in economics is the relationship between the price of an item and the number of items sold or produced. The relationship between the price of an item p and quantity q can often be represented as a linear function, at least approximately. A demand function expresses the relationship between price p and q, the number of items that can be sold. For most examples, we let q be the independent variable, so p is the price at which q items will sell. ex. When a graphing calculator is priced at $90, a district manager at Office Depot observes that monthly sales average 480 units in his stores. When the price is reduced periodically for a special sale, his average monthly sales increase by 40 calculators per $4 price decrease. 1) The relationship between price p and quantity q appears to be linear. Why?
2) Find the linear demand function denoted p = D(q) which expresses this relationship. L3-4 Why is slope negative? 3) Find the demand for calculators when the price is set at $100.
L3-5 A supply function expresses the price p at which a producer is willing to supply q items. ex. Suppose that the price p at which the producer will supply q calculators is given by the supply function p = S(q) = 1 5 q + 12. 1) Why is slope positive? 2) If the price is set at $100, how many calculators will be supplied to the Office Depot stores? Looking at both supply and demand at $100, what should happen to the price of a calculator?
L3-6 For our example, find the equilibrium quantity at which the supply equals demand. What is the equilibrium price?
L3-7 Break Even Analysis ex. An inventor would like to market his latest gadget. Each item costs $2.50 to produce, and his fixed costs of production are $1200. He plans to sell his gadget for $10. If the inventor thinks he can sell 150 items, should he put the gadget on the market? To answer the question, we consider the following ideas: 1) Find C(x), which gives the total cost of producing x items. Marginal Cost 2) Find the revenue function R(x).
L3-8 3) Find the break-even point. Should the inventor move ahead with production of his gadget? 4) Find the profit function P (x). What is the marginal profit?
L3-9 Additional Topic: Section 1.3 Least Squares Line and Linear Regression Consider the following data relating food availability in daily calorie supply and life expectancy for various countries: Country Calories(x) Life Expectancy (y) Belize 2818 75.4 Cambodia 2155 59.4 France 3602 80.4 India 2305 62.7 Mexico 3265 75.5 New Zealand 3235 79.8 Peru 2450 72.5 Sweden 3120 80.5 Tanzania 2010 53.7 United States 3826 78.7 Look at a scatter plot of this data. Does it appear to be linear? 80 70 60 2 2.5 3 3.5 4
L3-10 To find the linear model which best fits the set of data, we use the least squares line. This line is based on the vertical distance between an actual data point and the line; those distances for each point are summed and then minimized using calculus (you will learn how to do this in MAC 2234). If the least squares line has the form y = ax + b, the formulas for finding a and b are listed in your text, on page 27. A calculator or Excel spreadsheet can compute these values for a given set of data. In our case, the line of best fit has the formula y = 0.01367x + 32.503235. ex. Use your linear model to find the life expectancy of a citizen of the United Kingdom, with a daily calorie supply of 3426. (The actual life expectancy is 79.0 years). The Correlation Coefficient r, whose formula is given on page 30, is a measure of the accuracy of the line of best fit to the actual data. It lies between the numbers 1 and 1, and the closer r is to 1, the more linear the data and the better the fit. It can be computed on a calculator or spreadsheet.
L3-11 ex. For our example, r = 0.88. 90 80 70 60 2 2.5 3 3.5 4
L3-12 Now you try it! 1. Suppose that Avis leases a Ford Focus for $32 per day plus 45 cents per mile, while Enterprise leases the same car for $23 per day plus 60 cents per mile. If a customer plans to rent the car for a one day trip of 100 miles, which company should she use to minimize her cost? Find the number of miles for which the charge is the same for the two companies. 2. According to the US Department of Agriculture, the per capita consumption of beef in the United States decreased from 115.7 lb. in 1974 to 92.9 lb in 2007. Assuming that the consumption is decreasing linearly, write the function C(t) which expresses yearly consumption C in pounds as a function of t, the number of years since 1950 (so t = 0 in 1950). Round the slope to 2 decimal places. Using this model and rounding to the nearest whole number, in what year should the per capita consumption drop to 100 pounds? 3. For a certain product, the cost function is linear with fixed costs of $480. If the product will sell for $25 per unit and the break even quantity is 80 units, find the marginal cost and marginal profit for the product. Remember that marginal refers to the rate of change of the function if one more item is sold. How is it found? 4. Suppose the supply and demand functions for a given product are S(q) = 3.6q + 86 and D(q) = 2.8q + 246. (a) How many items will the supplier provide for a price of $86? (b) Find the equilibrium quantity and price. 5. A marketing firm wants to test a new weight loss product. (a) Find the linear supply function if the manufacturer will provide 60 10-pill samples at a unit price of $40, and 100 samples if the price is $60 each. (b) The marketing firm has determined that demand for the 10-pill sample can be modeled by the demand function p = 1.3q + 154. Find the equilibrium quantity and price. 6. Projected spending on home care and medical equipment (in billions of dollars) from 2004 to 2016 is given in the following table. Year x 2004 2006 2008 2010 2012 2014 2016 Spending y 60 74 90 106 118 128 150 (a) Find the least squares line and the correlation coefficient. (b) Use the regression line to find the projected spending in 2018. (c) If you have graphing capability, draw a scatterplot and add the least squares line.