Section 1.3 - Linear Functions and Math Models Example 1: Questions we d like to answer: 1. What is the slope of the line? 2. What is the equation of the line? 3. What is the y-intercept? 4. What is the x-intercept? Definition: If (x 1,y 1 ) and (x 2,y 2 ) are two distinct points on a line L, then the slope (m) of L is (i.e. the rate of change of y with respect to x) Answer to Question 1: 1
Definition: (Point-Slope Form) The equation of the line that passes through the point (x 1,y 1 ) and has slope m is given by: Definition (Slope-Intercept Form) The equation of the line that has slope m and intersects the y-axis at the point (0,b) is given by: Answer to Question 2: Answer to Question 3: Answer to Question 4: Equations of Special Lines: Horizontal Line: Vertical Line: 2
Definition: A function is a rule that assigns to each value of x one and only one value of y. We refer to x as the variable and to y as the variable. The set of all possible values that x can assume is the and the set of all possible values that y can assume is the. We will concentrate on linear functions (those that can be represented graphically by a line) Applications: 1. Linear Depreciation Example 2: In 2009 Texas A&M installed a new machine in one of its departments at a cost of $250,000. The machine is depreciated linearly over 10 years with a scrap value of $10,000. (a) Find an expression for the machine s book value in the tth year of use. (b) Sketch the graph of the function in part (a). (c) Find the machine s book value in 2013. (d) Find the rate at which the machine is being depreciated. 3
2. Linear Demand and Supply Curves Example 3: At a unit price of $55, the quantity demanded of a certain commodity is 1000 units. At a unit price of $85, the demand drops to 600 units. (a) Given that it is linear, find the demand equation. (b) Sketch the demand curve. (c) What quantity would be demanded if the commodity were free? Example 4: Producers will make 2000 cell phones available when the unit price is $100. At a unit price of $200, 6000 cell phones will be marketed. (a) Given that it is linear, find the supply equation. (b) Sketch the supply curve. (c) What is the unit price when 3,000 cell phones are supplied? 4
3. Cost, Revenue, and Profit Functions Cost Function (cost of manufacturing x units of a product) where c is the cost per unit and F is the fixed cost. Revenue Function (revenue realized from selling x units of the product) where s is the selling price. Profit Function Example 5: A manufacturer has a monthly fixed cost of $100,000 and a production cost of $14 for each unit produced. The product sells for $20/unit. (a) What is the cost function? (b) What is the revenue function? (c) What is the profit function? (d) Compute the profit (loss) corresponding to production levels of 12,000 and 20,000 units. Section 1.3 Highly Suggested Homework Problems: 11, 13, 15, 21, 23, 33, 37, 45 5
Section 1.4 Intersection of Straight Lines Calculator Steps: Graph the two functions by entering the slope-intercept form of the lines into Y1 and Y2 (These are located under the Y= button). After chosing the appropriate Window, hit GRAPH. Go to CALC which is found by pressing 2nd TRACE. Select option 5: intersect The calculator will prompt you for the first and second curves you want to intersect. The cursor will blink on a function and show the function name in the top left corner of the screen. When it blinks on Y1, press enter and when it blinks on Y2 press ENTER. The calculator will then ask you to guess where the intersection point is located. Use your right and left arrow keys to move to the intersection point and press ENTER. The intersection point will appear at the bottom of the screen. Example 1: Find the point of intersection of the lines y = x + 1 and y = 2x + 4 a) Find the point of intersection algebraically. b) Find the point of intersection using the calculator. 6
Applications: 1. Market Equilibrium: The point at which the consumer and supplier agree upon (i.e. the point of intersection of the supply and demand curves) Example 2: If the supply equation is 5x 6p + 14 = 0 and the demand equation is 4x + 3p 59 = 0 where x represents the quantity demanded in units of 1000 and p is the unit price in dollars, find the equilibrium quantity and the equilibrium price. 2. Break-Even Point: The point at which the company suffers neither a loss or gain. (i.e. the point of intersection of the revenue and cost functions) Example 3: A manufacturer of garbage disposals, has a monthly fixed cost of $10,000 and a production cost of $20 for each garbage disposal manufactured. The units sell for $50 each. (a) What is the cost function? (b) What is the revenue function? (c) What is the profit function? 7
(d) Sketch the graphs of the cost function and the revenue function and hence find the break-even point graphically. (e) Find the break-even point algebraically. (f) Sketch the graph of the profit function. (g) At what point does the graph of the profit function cross the x-axis? Interpret your result. Section 1.4 Highly Suggested Homework Problems: 3, 9, 13, 21, 25, 27 8
Section 1.5 - The Method of Least Squares Example 1: The following table gives the actual high temperature in College Station for a few days in August where x represents the day of the month of August and y represents the temperature in F. x y 1 99 2 102 4 105 Use the above model to predict the temperature on August 10th. 9
Calculator Steps Enter your x and y values into lists. To do this, hit STAT and select 1: Edit... If you have anything in L1 and L2, cursor up to the name of the list, hit CLEAR and ENTER. Now just enter in your values one at a time by pressing ENTER after each number. To find the regression equation, first hit STAT, cursor right to CALC and select option 4: LinReg(ax+b). To have the calculator automatically store the equation into Y1 press VARS, arrow to Y-Vars, select 1:Function and then select Y1. Note: Your homescreen should say LinReg(ax+b) Y1. Now press ENTER To graph the regression line with data points, first enter the equation of the line into Y1(if you didn t automatically store it there). Then make sure your stat plot is turned on. To do this, hit 2nd Y= and select 1...by pressing ENTER. You can then highlight any of the options you would like. Now press ZOOM and select 9: ZoomStat. You will have a plot of the original data points and the least squares line. Example 2: A manufacturer of electric motors, submitted the accompanying data. The table shows the net sales (in millions of dollars) during a 5 year period: Year Net Sales 1999 426 2000 437 2001 460 2002 473 2003 477 a) Determine the equation of the least-squares line for these data. (Let x represent the number of years since 1999.) b) Draw a scatter diagram and the least-squares line for these data. c) Use the result obtained in part (a) to predict the net sales for 2010. d) When will the net sales reach $500 million? Section 1.5 Highly Suggested Homework Problems: 3, 7, 11 10