Chapter 1. Functions, Graphs, and Limits

Transcription

1 Chapter 1 Functions, Graphs, and Limits MA1103 Business Mathematics I Semester I Year 016/017 SBM International Class Lecturer: Dr. Rinovia Simanjuntak

2 1.1 Functions

3 Function A function is a rule that assigns to each object in a set A eactly one object in a set B. The set A is called the domain of the function, and the set of assigned objects in B is called the range. 3

4 Which One is a Function? f A B A f B A f B 4

5 We represent a functional relationship by an equation y f () and y are called variables: y is the dependent variable and is the independent variable. Eample. y f ( ) 4 Note that and y can be substituted by other letters. For eample, the above function can be represented by s t 4 5

6 Function which is Described as a Tabular Data Table 1.1 Average Tuition and Fees for 4-Year Private Colleges Academic Year Tuition and Ending in Period n Fees $1, $, $4, $7, $10, $13, $18,73 6

7 We can describe this data as a function f defined by the rule f ( n) average tuition and fees at the beginning of the nth 5- year period Thus, f ( 1) 1,898, f (),700,, f (7) 18,73 Noted that the domain of f is the set of integers A {1,,...,7} 7

8 Piecewise-defined function A piecewise-defined function is a function that is often defined using more than one formula, where each individual formula describes the function on a subset of the domain. Eample. f ( ) if if 1 1 Find f(-1/), f(1), and f(). 8

9 Natural Domain The natural domain of f is the domain of f to be the set of all real numbers for which f() is defined. There are two situations often need to be considered: 1) division by 0 ) the even root of a negative number Eamples. Find the domain and range of each of these functions f ( ) 1 4. g( u) u 9

10 Functions Used in Economics A demand function p=d() is a function that relates the unit price p for a particular commodity to the number of units demanded by consumers at that price. The total revenue is given by the product R()=(number of items sold)(price per item) =p=d() If C() is the total cost of producing the units, then the profit is given by the function P()=R()-C()=D()-C() 10

11 Eample Market research indicates that consumers will buy thousand units of a particular kind of coffee maker when the unit price is p dollars. The cost of producing the thousand units is C( ) thousand dollars a. What are the revenue and profit functions, R() and P(), for this production process? b. For what values of is production of the coffee makers profitable? 11

12 a. The demand function is, so the revenue is D( ) R( ) D( ) thousand dollars, and the profit is (thousand dollars) P( ) R( ) C( ) ( ) b. Production is profitable when P()>0. We find that P( ).5.5( ).5( )( 17) 0 Thus, production is profitable for <<17. 1

13 Composition of Functions Given functions f(u) and g(), the composition f(g()) is the function of formed by substituting u=g() for u in the formula for f(u). Eample. Find the composition function f(g()), where g( ) 1 Solution. Replace u by +1 in the formula for f(u) to get f ( g( )) ( 1) f ( u) u Question: How about g(f())? Note: In general, f(g()) and g(f()) are not the same. and 13

14 1. The Graph of a Function 14

15 Graph The graph of a function f consists of all points (,y) where is in the domain of f and y=f(), that is, all points of the form (,f()). Rectangular coordinate system, horizontal ais, vertical ais. f ( ) f()

16 Intercepts intercept: points where a graph crosses the ais. y intercept: a point where the graph crosses the y ais. How to find the and y intercepts: The only possible y intercept for a function is y0 f (0), to find any intercept of y=f(), set y=0 and solve for. Note: Sometimes finding intercepts may be difficult. Following aforementioned eample, the y intercept is f(0)=. To find the intercepts, solve the equation f()=0, we have =-1 and. Thus, the intercepts are (-1,0) and (,0). 16

17 Parabolas Parabolas: The graph of y A B C as long as A 0. All parabolas have a U shape and the parabola opens up if A>0 and down if A<0. The peak or valley of the parabola is called its verte, and it always occurs where B A 17

18 Eample 6 A manufacturer determines that when hundred units of a particular commodity are produced, they can all be sold for a unit price given by the demand function p=60- dollars. At what level of production is revenue maimized? What is the maimum revenue? Solution: The revenue function R()=(60-) hundred dollars. Note that R() 0 only for The revenue function can be rewritten as R( ) 60 which is a parabola that opens downward (Since A=-1<0) and has its high point (verte) at B A ( 1) Thus, revenue is maimized when =30 hundred units are produced, and the corresponding maimum revenue is R(30)=900 hundred dollars. 18

19 Intersections of Graphs Sometimes it is necessary to determine when two functions are equal. For eample, an economist may wish to compute the market price at which the consumer demand for a commodity will be equal to supply. 19

20 Power, Polynomial, and Rational Functions A power function: A function of the form f number. A polynomial function: A function of the form n n1 p( ) a a a a If where n is a nonnegative integer and a n 0, where n is a real are constants., the integer n is called the degree of the polynomial. A rational function: A quotient n n1 p( ) q( ) ( ) 1, a, 0 1 a, n 0 a n of two polynomials p() and q(). 0

21 The Vertical Line Test A curve is the graph of a function if and only if no vertical line intersects the curve more than once. 1

22 1.3 Linear Functions

23 Linear Functions A linear function is a function that changes at a constant rate with respect to its independent variable. The graph of a linear function is a straight line. The equation of a linear function can be written in the form y m b where m and b are constants. 3

24 The Slope of a Line The slope of the non-vertical line passing through the points 1, y ) and (, y) is given by the formula ( 1 Slope change in y change in y y y 1 1 4

25 Equation of a Line The slope-intercept form: The equation y m b is the equation of a line whose slope is m and whose y intercept is (0,b). The point-slope form: The equation y y ( ) 0 m 0 is an equation of the line that passes through the point (, y 0 0) and has slope equal to m. m (0 0.5) ( 1.5 0) 1 3 The slope-intercept form is y The point-slope form that passes through the point (-1.5,0) is 1 y 0 ( 1.5) 3 5

26 Table 1. lists the percentage of the labour force that was unemployed during the decade Plot a graph with the time (years after 1991) on the ais and percentage of unemployment on the y ais. Do the points follow a clear pattern? Based on these data, what would you epect the percentage of unemployment to be in the year 005? Table 1. Percentage of Civilian Unemployment Number of Years Percentage of Year from 1991 Unemployed

27 Parallel and Perpendicular Lines m 1 m Let and be the slope of the non-vertical lines and L. Then L1 and L are parallel if and only if m1 m and are perpendicular if and only if m L L1 L 1 1 m 1 7

28 Let L be the line 4+3y=3 L 1 a. Find the equation of a line parallel to L through P(-1,4). b. Find the equation of a line perpendicular to L through Q(,-3). Solution: L By rewriting the equation 4+3y=3 in the slope-intercept form 4 4 y 1, we see that L has slope m 3 L 3 a. Any line parallel to L must also have slope -4/3. The required line L 1 contains P(-1,4), we have 4 4 y 4 ( 1) y 3 3 b. A line perpendicular to L must have slope m=3/4. Since the 3 required line contains Q(,-3), we have y 3 ( L y )

29 1.4 Functional Models 9

30 Functional Models To analyze a real world problem, a common procedure is to make assumptions about the problem that simplify it enough to allow a mathematical description. This process is called mathematical modelling and the modified problem based on the simplifying assumptions is called a mathematical model. adjustments Testing Prediction Real-world problem Interpretation Formulation Mathematical model Analysis 30

31 Eination of Variables In net eample, the quantity you are seeking is epressed most naturally in term of two variables. We will have to einate one of these variables before you can write the quantity as a function of a single variable. Eample The highway department is planning to build a picnic area for motorists along a major highway. It is to be rectangular with an area of 5,000 square yards and is to be fenced off on the three sides not adjacent to the highway. Epress the number of yards of fencing required as a function of the length of the unfenced side. 31

32 Solution: We denote and y as the lengths of the sides of the picnic area. Epressing the number of yards F of required fencing in terms of these two variables, we get F y. Using the fact that the area 5000 is to be 5,000 square yards that is y 5,000 y and substitute the resulting epression for y into the formula for F to get F( ) 3

33 Modelling in Business and Economics A manufacturer can produce blank videotapes at a cost of $ per cassette. The cassettes have been selling for $5 a piece. Consumers have been buying 4000 cassettes a month. The manufacturer is planning to raise the price of the cassettes and estimates that for each $1 increase in the price, 400 fewer cassettes will be sold each month. a. Epress the manufacturer s monthly profit as a function of the price at which the cassettes are sold. b. Sketch the graph of the profit function. What price corresponds to maimum profit? What is the maimum profit? 33

34 Solution: a. As we know, Profit=(number of cassettes sold)(profit per cassette) Let p denote the price at which each cassette will be sold and let P(p) be the corresponding monthly profit. Number of cassettes sold Profit per cassette=p- The total profit is P( p) = (number of $1 increases) = (p-5)= p ( p)( p ) 400 p 6800 p

35 b. The graph of P(p) is the downward opening parabola shown in the bottom figure. Profit is maimized at the value of p that corresponds to the verte of the parabola. We know B 6800 p A ( 400) 8.5 Thus, profit is maimized when the manufacturer charges $8.50 for each cassette, and the maimum monthly profit is P P(8.5) 400(8.5) 6800(8.5) 1000 $16900 ma 35

36 Market Equilibrium The law of supply and demand: In a competitive market environment, supply tends to equal demand, and when this occurs, the market is said to be in equilibrium. The demand function: p=d() The supply function: p=s() The equilibrium price: p e D( e) S( e) Shortage: D()>S() Surplus: S()>D()

37 Market research indicates that manufacturers will supply units of a particular commodity to the marketplace when the price is p=s() dollars per unit and that the same number of units will be demanded by consumers when the price is p=d() dollars per unit, where the supply and demand functions are given by S( ) 14 D( ) a. At what level of production and unit price p is market equilibrium achieved? b. Sketch the supply and demand curves, p=s() and p=d(), on the same graph and interpret. 37

38 Solution: a. Market equilibrium occurs when S()=D(), we have ( 10)( ) 0 10 or 16 Only positive values are meaningful, D( 10) 174 6(10) 114 p e 38

39 Break-Even Analysis At low levels of production, the manufacturer suffers a loss. At higher levels of production, however, the total revenue curve is the higher one and the manufacturer realizes a profit. Break-even point: The total revenue equals total cost. 39

40 A manufacturer can sell a certain product for $110 per unit. Total cost consists of a fied overhead of $7500 plus production costs of $60 per unit. a. How many units must the manufacturer sell to break even? b.what is the manufacturer s profit or loss if 100 units are sold? c.how many units must be sold for the manufacturer to realize a profit of $150? Solution: If is the number of units manufactured and sold, the total revenue is given by R()=110 and the total cost by C()=

41 a. To find the break-even point, set R() equal to C() and solve 110= , so that =150. It follows that the manufacturer will have to sell 150 units to break even. b. The profit P() is revenue minus cost. Hence, P()=R()-C()=110-( )= The profit from the sale of 100 units is P(100)=-500 It follows that the manufacturer will lose $500 if 100 units are sold. c. We set the formula for profit P() equal to 150 and solve for, we have P()=150, =175. That is 175 units must be sold to generate the desired profit. 41

42 A certain car rental agency charges $5 plus 60 cents per mile. A second agency charge $30 plus 50 cents per mile. Which agency offers the better deal? Solution: Suppose a car is to be driven miles, then the first agency will charge C ( ) dollars and the second will charge C ( ) So that =50. For shorter distances, the first agency offers the better deal, and for longer distances, the second agency is better. 4

43 1.5 Limits 43

44 Illustration of Limit The it process involves eamining the behaviour of a function f() as approaches a number c that may or may not be in the domain of f. Illustration. Consider a manager who determines that when percent of her company s plant capacity is being used, the total cost is 8 C( ) hundred thousand dollars. The company has a policy of rotating maintenance in such a way that no more than 80% of capacity is ever in use at any one time. What cost should the manager epect when the plant is operating at full permissible capacity? 44

45 It may seem that we can answer this question by simply evaluating C(80), but attempting this evaluation results in the meaningless fraction 0/0. However, it is still possible to evaluate C() for values of that approach 80 from the left (<80) and the right (>80), as indicated in this table: approaches 80 from the left approaches 80 from the right C() The values of C() displayed on the lower line of this table suggest that C() approaches the number 7 as gets closer and closer to 80. The functional behavior in this eample can be describe by C( )

46 Limits If f() gets closer and closer to a number L as gets closer and closer to c from both sides, then L is the it of f() as approaches c. The behaviour is epressed by writing f ( ) L c 46

47 Eample Use a table to estimate the it Let 1 f ( ) and compute f() for a succession of values of approaching 1 from the left and from the right f() The table suggest that f() approaches 0.5 as approaches 1. That is

48 It is important to remember that its describe the behavior of a function near a particular point, not necessarily at the point itself. Three functions for which 3 f ( ) 4 48

49 The figure below shows that the graph of two functions that do not have a it as approaches. Figure (a): The it does not eist; Figure (b): The function has no finite it as approaches. Such so-called infinite its will be discussed later. 49

50 Properties of Limits then eist, ) ( and ) ( If g f c c ) ( ) ( )] ( ) ( [ g f g f c c c ) ( ) ( )] ( ) ( [ g f g f c c c for any constant ) ( ) ( k f k kf c c 50 )] ( )][ ( [ )] ( ) ( [ g f g f c c c 0 ) ( if ) ( ) ( ] ) ( ) ( [ g g f g f c c c c eists )] ( [ if )] ( [ )] ( [ p c p c p c f f f

51 For any constant k, k k c and c c That is, the it of a constant is the constant itself, and the it of f()= as approaches c is c. 51

52 Eamples Find (a) ( ) (b) a. Apply the properties of its to obtain ( ) ( 1) 4( 1) b. Since ( ) 0, you can use the quotient rule for 0 its to get

53 Limits of Polynomials and Rational Functions If p() and q() are polynomials, then and p( ) c q( ) c p( ) p( c) p( c) if q( c) 0 q( c) Eample. Find 1 The quotient rule for its does not apply in this case since the it of the denominator is 0 and the it of the numerator is 3. 53

54 Indeterminate Form g( ) 0 If f ( ) 0 and, then is said to be c c c ( ) indeterminate. The term indeterminate is used since the it may or may not eist. Eamples. 1 (a) Find 1 3 (b) Find 1 f ( ) g ( 1)( 1) 1 a ( 1)( ) 1 1 b ( 1) 1 1 ( 1)

55 Limits Involving Infinity Limits at Infinity If the value of the function f() approach the number L as increases without bound, we write f ( ) L Similarly, we write f ( ) M when the functional values f() approach the number M as decreases without bound. 55

56 Reciprocal Power Rules A A For constants A and k, with k>0, 0 and 0 k k Eample. Find 1 / 1 1 1/ / / 1/ 1/

57 Procedure for Evaluating a Limit at Infinity of f()=p()/q() Step 1. Divide each term in f() by the highest power k that appears in the denominator polynomial q(). Step. Compute f ( ) or f ( ) using algebraic properties of its and the reciprocal rules. Eample

58 Infinite Limits If f() increases or decreases without bound as c, we have f ( ) or f ( ) c Eample. ( ) c From the figure, we can guest that ( ) 58

59 1.6 One-sided Limits and Continuity 59

60 One-Sided Limits If f() approaches L as tends toward c from the left (<c), we write f ( ) L c c where L is called the it from the left (or left-hand it) Likewise if f() approaches M as tends toward c from the right (>c), then f ( ) M is called the it from the right (or right-hand it). M 60

61 Eample. For the function evaluate the one-sided its and 61 if 1 if 1 ) ( f ) ( f ) ( f Since for <, we have 1 ) ( f 3 ) (1 ) ( f Similarly, f()=+1 if, so 5 1) ( ) ( f

62 Eistence of a Limit The two-sided it f ( ) eists if and only if the two c one-sided its f ( ) and f ( ) eist and are c c equal, and then c f ( ) c f ( ) c f ( ) Recall. Find 1 6

63 1 f ( ) does not eist! Since the left and right hand its are not equal. At =1: 1 f 0 1 f 1 f 1 1 Left-hand it Right-hand it value of the function 63

64 At =: f 1 f 1 f Since the left and right hand its are equal. However, the it is not equal to the value of function. Left-hand it Right-hand it f ( ) does eist! value of the function 64

65 At =3: 3 f 3 f f 3 3 f ( ) Left-hand it Right-hand it value of the function does eist! Since the left and right hand its are equal, and the it is equal to the value of function. 65

66 Non-eistent One-sided Limits A simple eample is provided by the function f ( ) sin(1/ ) As approaches 0 from either the left or the right, f() oscillates between -1 and 1 infinitely often. Thus neither one-sided it at 0 eists. 66

67 Continuity A continuous function is one whose graph can be drawn without the pen leaving the paper. (no holes or gaps ) 67

68 A hole at =c 68

69 A gap at =c 69

70 What properties will guarantee that f() does not have a hole or gap at =c? A function f is continuous at c if all three of these conditions are satisfied: a. b. c. f (c) c c is f ( ) f ( ) defined eists f ( c) If f() is not continuous at c, it is said to have a discontinuity there. 70

71 f() is continuous at =3 because the left and right hand its eist and equal to f(3). At =1: 1 f ( ) 1 f ( ) Discontinuous At =: f ( ) f ( ) f () Discontinuous At =3: 3 f ( ) 3 f ( ) f (3) Continuous 71

72 7 Continuity of Polynomials and Rational Functions If p() and q() are polynomials, then ) ( ) ( c p p c 0 ) ( if ) ( ) ( ) ( ) ( c q c q c p q p c A polynomial or a rational function is continuous wherever it is defined

73 Eample. Show that the rational function continuous at =3. f ( ) 1 is Note that f(3)=(3+1)/(3-)=4, since ( ) 0, you will find that 3 1 ( 1) 3 f ( ) 3 3 ( ) 3 as required for f() to be continuous at =3, since the three criteria for continuity are satisfied f (3) 73

74 Eample. Determine where the function below is not continuous. Rational functions are continuous everywhere ecept where we have division by zero. The function given will not be continuous at t=-3 and t=5. 74

75 75 Eample. Discuss the continuity of each of the following functions 1 if 1 if 1 ) (. 1 1 ) (. 1 ) (. h c g b f a

76 Eample. For what value of the constant A is the following function continuous for all real? A 5 if 1 f ( ) 3 4 if 1 Since A+5 and 3 4 are both polynomials, it follows that f() will be continuous everywhere ecept possibly at =1. According to the three criteria for continuity, we have 1 f ( ) 1 f ( ) f (1) A 5 f (1) A 3 This means that f is continuous for all only when A=-3 76

77 Eample. Find numbers a and b so that the following function is continuous everywhere. a if -1 f ( ) a b if 1 1 b if 1 Since the parts f are polynomials, we only need to choose a and b so that f is continuous at =-1 and 1. At =-1 f ( ) f ( ) f ( 1) a 1 a b a b At =1 f ( ) f ( ) f (1) 1 a b b a b We have a=-1/3 and b=1/3 for f is continuous everywhere 77

78 Continuity on an Interval A function f() is said to be continuous on an open interval a<<b if it is continuous at each point =c in that interval. f is continuous on closed interval a b, if it continuous on the open interval a<<b, f ( ) f ( a) and b f ( ) f ( b) a f ( ) 1 is continuous on [-1,1] 78

79 Eample. Discuss the continuity of the function f ( ) on the open 3 interval -<<3 and on the closed interval - 3 The rational function f()is continuous for all ecept =3. Therefore, it is continuous on the open interval -<<3 but not on the closed interval - 3, since it is discontinuous at the endpoint 3 (where its denominator is zero). 79

80 Summary Function: Domain and range of a function Composition of function f(g()) Graph of a function: and y intercepts, Piecewise-defined function, power function Polynomial, rational function, vertical line test Linear function: Slope, slope-intercept formula, point-slope formula Parallel and perpendicular lines 80

81 Function Models: Market equilibrium: law of supply and demand Shortage and surplus, break-even analysis Limits: f ( ) L c Calculation of its, its of polynomial and rational function Limits at infinity: its at the infinity (Reciprocal power Rules), infinite it One sided it, eistence of it Continuity of f() at =c: Continuity of polynomials and rational function 81