Functions of One Variable Basics

Transcription

1

2 Functions of One Variable Basics Function Notation: = f() Variables and are placeholders for unknowns» We will cover functions of man variables during one of the in-residence QSW lectures. = independent or eogenous variable = dependent or endogenous variable 3 Functions of One Variable Business Applications Economics: Quantit Demanded = f(price) Marketing: : Customer Satisfaction = f(qualit) Finance: : Portfolio Return = f(risk) The Regents of The Universit of Michigan Page 2

3 Review of X and Y Aes Cartesian Coordinate Sstem Since we will be working with functions of two variables and graphing these functions, we need to represent the two variables in two dimensions. In a previous module we used a horizontal number line. This will be the -ais. On top of this we draw a vertical number line, the -ais. This is called the Cartesian Coordinate Sstem. We will refer to a point as having coordinates. Listed first is the -coordinate, followed b the -coordinate. For eample, if a point has coordinates (2, 10) called an ordered pair of numbers then the value of is 2 and the value of is Review of X and Y Aes Finding Points (, ) Origin The Regents of The Universit of Michigan Page 3

4 Review of X and Y Aes To get to the point (-4,2), ou start at the origin and move 4 units to the left along the -ais and 2 units up along the -ais..(-4,2).(-4,-2) To get to the point (-4,-2), move 4 units to the left and 2 units down. Finding Points (, ) To get to the point (4,2), ou start at the origin and move 4 units to the right along the -ais and 2 units up along the -ais..(4,2) (4,-2) To get to the point (4,-2), move 4 units to the right and 2 units down. 7 LINEAR FUNCTIONS General Case = m + b The Regents of The Universit of Michigan Page 4

5 LINEAR FUNCTIONS General Case Revenue = m + b Quantit Application: Revenue is a linear function of quantit sold. Note that the line is drawn starting from the origin. In man business contets the onl relevant part of the coordinate sstem is the upper right-hand quadrant, where and can be either zero or positive, but not negative. 9 LINEAR FUNCTIONS Special Case Constant Function = b b The Regents of The Universit of Michigan Page 5

6 LINEAR FUNCTIONS Special Case Constant Function Overhead Cost = b b Quantit Application: Overhead costs, such as monthl lease paments, are constant. The do not change as quantit sold changes (i.e., the same rent pament is due whether it is a good sales month or a bad one). 11 Functions: Basics & Lines Eercises 1. How much a household spends on water each month depends on a number of factors. The most important factors are the price of water, how much water is consumed, the temperature, and the amount of rainfall (if there is a garden that needs water). Which variables are eogenous and which are endogenous to the household s consumption decision? 2. Draw the line = 10. Draw the line = Plot the following pairs (,): The Regents of The Universit of Michigan Page 6

7 Functions: Basics & Lines Eercises: **Answers** 1. The eogenous variables are the ones that ou cannot control or choose. In this case, the would be the price of water, the temperature, and the amount of rainfall. The endogenous variables are the choice variables. Your onl choice in this eample is the amount of water ou consume. (Note that if this were commercial consumption, rather than residential, the compan ma be able to negotiate over the price. This would make the price variable not strictl eogenous.) 13 Functions: Basics & Lines Eercises: **Answers** 2. Draw the line = 10. Draw the line = The Regents of The Universit of Michigan Page 7

8 Functions: Basics & Lines 3. Plot the following pairs: These points lie along a straight line. In a later CD-ROM module, we will focus eclusivel on working with lines and graphing linear equations (-4, 4,-1) Eercises: **Answers** -2-3 (0,1) (2,2) 15 Quadratic (Parabola) = a 2 + b + c The Regents of The Universit of Michigan Page 8

9 Quadratic (Parabola) = a 2 + b + c Revenue Quantit Application: In some industries, revenue is not related linearl to quantit sold, but is better described as a quadratic function. Revenue increases initiall as sales grow. In order to greatl increase quantit sold, however, ou must lower the price significantl. As ou move further out to the right along the -ais (the quantit-ais), revenues fall. 17 Quadratic (Parabola) Parabolas can point in an direction, but the most common ones that ou will run across are hill- shaped and U-shaped parabolas The Regents of The Universit of Michigan Page 9

10 Quadratic (Parabola) Parabolas can point in an direction, but the most common ones that ou will run across are hill- shaped and U-shaped parabolas. Cost per unit Quantit Application: Unit or average cost curves are often U-shaped. On the one hand, as production increases the firm s average overhead ependiture declines. On the other hand, the firm s average variable ependiture ma eventuall rise if ou have to hire more workers or pa overtime, or if more production errors occur as ou push the limits of the sstem. 19 Cubic = a 3 + b 2 + c + d The Regents of The Universit of Michigan Page 10

11 Cubic Total Cost = a 3 + b 2 + c + d Quantit Application: Total cost rises in the short run as output increases, first at a decreasing rate, then at an increasing rate as ou begin to hit bottlenecks in production and capacit constraints. 21 Cubic Cumulative # Adopters = a 3 + b 2 + c + d Time Another Application: The tpical pattern of product diffusion, the rate at which new users bu a product, looks like an S-curve. Product sales start slowl, increase at an increasing rate, and then plateau. The steepness of the curve depends on such factors as price, product attributes, advertising, and so on The Regents of The Universit of Michigan Page 11

12 Rational A rational function is a ratio of functions. A reciprocal function, which we cover here, is a special case, where the in the denominator is raised to the power of one. a = 23 Rational = a Special Terminolog: Since ou cannot divide b zero, these graphs have a vertical asmptote and a horizontal asmptote, where the curve gets ver close to a line but never touches it. In this graph the vertical asmptote is the -ais and the horizontal asmptote is the -ais The Regents of The Universit of Michigan Page 12

13 Rational = a Price Quantit Application: If a consumer alwas spends $100/month on lotter tickets, regardless of the ticket price, then P Q = 100 or P = 100/Q 25 Graphing Nonlinear Functions Select a few values for Solve for = f() Plot the pairs (,) Connect the points In most cases it is not necessar to plot the function precisel. It usuall suffices to draw a rough sketch that is accurate enough to show the shape and approimate placement of the curve The Regents of The Universit of Michigan Page 13

14 Graphing Nonlinear Functions Graph = Graphing Nonlinear Functions Graph = 100/, onl for values of The Regents of The Universit of Michigan Page 14

15 Graphing Nonlinear Functions Eercises 1. Graph = Graph = 100/ + 20, onl for values of 0 29 Graphing Nonlinear Functions Eercises: **Answers** 1. Graph = The Regents of The Universit of Michigan Page 15

16 Graphing Nonlinear Functions Eercises: **Answers** 2. Graph = 100/ + 20, onl for values of Application: Learning Curve As a firm gains eperience in production, it is often the case that unit costs decline, particularl for comple products: workers become more efficient, technical knowledge increases, and product design improvements are made. A learning curve describes the relationship between production costs and cumulative output. Learning curves (or eperience curves) have been estimated for a number of industries. A common eperience curve is an 80 percent curve. This means that unit production costs drop b 20 percent with each doubling of cumulative output. A 90 percent curve would mean that costs drop b 10 percent as cumulative output doubles The Regents of The Universit of Michigan Page 16

17 Application: Learning Curve An eample of an 85 percent learning curve is shown below, using hpothetical data. The learning curve is clearl nonlinear. You can see how quickl labor hours (and therefore labor costs) drop as cumulative output increases. LEARNING CURVE Labor Hours per Unit Cumulative Units 33 Application: Learning Curve An 85 percent eperience curve just like this was assumed b Douglas Aircraft when the set their pricing strateg for the DC-9. The fied prices based on an epectation that there would be an 85% eperience curve. In other words, the prices the set were below current cost, but the epected costs to fall quickl with time and eperience. Unfortunatel, the epected cost savings did not occur. Douglas lost mone and was later acquired b the McDonnell Compan.* * See Pankaj Ghemawat Building Strateg on the Eperience Curve, Harvard Business Review,, March-April The Regents of The Universit of Michigan Page 17