1. 1 CALCULUS For Business, Economics, and the Social and life Sciences Hoffmann, L.D. & Bradley, G.L. Instructor: Dr. Ken Tsang Room E409-R9 Email: kentsang@uic.edu.hk@uic.edu.hk

2. 2 TA information Mr Zeng Xiaolong 曾晓龙 Room E409 Tel: 3620631 brucezeng@uic.edu.hk Math1050: Mathematics for Science (for Non- STAT students only) (2)

3. 3 Web-page for this class Watch for announcements about this class and download lecture notes from http://www.uic.edu.hk/~kentsang/calculus/cal cu.htm http://www.uic.edu.hk/~kentsang/calculus/cal cu.htm Or from this page: http://www.uic.edu.hk/~kentsang/

4. 4 Tutorials One hour each week Time & place to be announced later (we need your input) More explanations More examples More exercises

5. 5 What can I do to maximize my chances for success? Work hard, more importantly, work smart: 1. Understand, don't memorize. 2. Ask why, not how. 3. See every problem as a challenge. 4. Learn techniques, not results. 5. Make sure you understand each topic before going on to the next.

6. 6 How is my final grade determined?

7. 7 Some requirements on this Course  Assignments must be handed in before the deadline. After the deadline, we refuse to accept your assignments!  We will tell you your scores for the mid-term exam and quizzes so that you know your progress. However, for the final examination, we cannot tell you the score before the AR release the official results.

8. 8 UIC Score System

9. 9 Grade Distribution Guidelines

10. Mathematics is about  numbers, space, structures, … Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions. Most important, it teaches us how to analysis problem in an abstract form, with logical thinking. 10 Mathematics again? Why?

11. What is mathematics? “Mathematics sets the standard of objective truth for all intellectual endeavors, science and technology bear witness to its practical usefulness. Besides language and music, it is one of the primary manifestations of the free creative power of the human mind, and it is the universal organ for world-understanding through theoretical construction. Mathematics must therefore remain an essential element of the knowledge and abilities which we have to teach, of the culture we have to transmit, to the next generation.” 11 --Hermann Weyl

12. 12 They invented Calculus! Sir Isaac Newton (1642-1727) Gottfriend Wilhelm von Leibniz (1646-1716)

13. 13 What is Calculus all about?  Calculus is the study of change, or to be more precise, changing quantities.  The two key areas of Calculus are Differential Calculus and Integral Calculus.  The big surprise is that these two seemingly unrelated areas are actually connected via the Fundamental Theorem of Calculus.

14. 14 Calculus has practical applications, such as understanding the true meaning of the infinitesimals. (Image concept by Dr. Lachowska.)

15. 15 Isaac Newton (4 January 1643 – 31 March 1727) English physicist, mathematician, astronomer, natural philosopher and theologian, one of the most influential men in human history. Newton in a 1702 portrait by Godfrey Kneller

16. 16 Newton ’ s contributions Newton described universal gravitation and the three laws of motion, laying the groundwork for classical mechanics, which dominated the scientific view of the physical Universe for the next three centuries and is the basis for modern engineering. Newton showed that the motions of objects on Earth and of celestial bodies are governed by the same set of natural laws.

17. 17 Cosmos1 Newton's own first edition copy of his Philosophiae Naturalis Principia Mathematica with his handwritten corrections for the second edition. The book can be seen in the Wren Library of Trinity College, Cambridge.

18. 18 Newton's 2nd law of motion Newton's Second Law states that an applied force, on an object equals the rate of change of its momentum, with time. For a system with constant mass, the equation can be written in the iconic form:F= ma, where a is the acceleration of an object. This can be rewritten as a differential equation. Most laws of nature can be expressed as differential equations or partial differential equations (PDE).

19. 19 If you are a finance major Finance is a quantitative discipline.  How to calculate the return of your investment?  Asset valuation  Portfolio theory  Derivatives  Risk management

20. 20 If your major is finance, you will know this: Fischer Black and Myron Scholes first articulated the Black-Scholes formula in their 1973 paper, "The Pricing of Options and Corporate Liabilities." Robert C. Merton was the first to publish a paper expanding the mathematical understanding of the options pricing model and coined the term "Black- Scholes" options pricing model. Merton and Scholes received the 1997 Prize in Economic Sciences in Memory of Alfred Nobel for this and related work.

21. 21 The Black-Scholes model In the Black-Scholes model, we assume that the underlying security (typically the stock) follows a geometric Brownian motion. That is, where S is the price of the stock at time t, μis the drift rate of S, annualized, σis the volatility of the stock, the dW term here stands in for any and all sources of uncertainty in the price history of a stock.

22. 22 If you are a science major Science is  Quantitative  Logical

23. 23 Ecology: Population dynamics The basic accounting relation for population dynamics is: N 1 = N 0 + B − D + I − E where N 1 is the number of individuals at time 1, N 0 is the number of individuals at time 0, B is the number of individuals born, D the number that died, I the number that immigrated, and E the number that emigrated between time 0 and time 1.

24. 24 The Lotka – Volterra (predator – prey) equations are a pair of first-order, non-linear, differential equations frequently used to describe the dynamics of biological systems in which two species interact, one (y) a predator and the other (x) its prey.

25. 25 where * y is the number of some predator (for example, wolves); * x is the number of its prey (for example, rabbits); * dy/dt and dx/dt represents the growth of the two populations against time; * t represents the time; and * α, β, γ and δ are parameters representing the interaction of the two species.

26. 26 Suppose there are two species of animals, a baboon (prey) and a cheetah (predator). If the initial conditions are 80 baboons and 40 cheetahs, one can plot the progression of the two species over time.

27. 27 OK, any question? Enough for introduction. Let’s begin the real thing!

28. 28 Chapter 1 Functions, Graphs and Limits In this Chapter, we will encounter some important concepts.  Functions ( 函数 )  Limits （极限）  One-sided Limits （单边极限） and Continuity （连续）

29. 29 Section 1.1 Functions （函数）  A function is a rule that assigns to each object in a set A exactly one object in a set B.  The set A is called the domain （定义域） of the function, and the set of assigned objects in B is called the range. （值域）

30. 30 Function, or not? YES NO NO

31. 31  To be convenient, we represent a functional relationship by an equation  In this context, x and y are called variables, furthermore, we refer to y as the dependent variable ( 因变量 ) and to x as the independent variable （自变量）. For instant, the function representation  Noted that x and y can be substituted by other letters. For example, the above function can be represented by

32. 32 Function that describes tabular data Academic Year Tuition and Ending in Period n Fees 1973 1 $1,898 1978 2 $2,700 1983 3 $4,639 1988 4 $7,048 1993 5 $10,448 1998 6 $13,785 2003 7 $18,273 Table 1.1 Average Tuition and Fees for 4-Year Private Colleges

33. 33  We can describe this data as a function f defined by the rule Thus,  Noted that the domain of f is the set of integers Solution: Solution:

34. 34 Piecewise-defined function （分段函数）  A piecewise-defined function is such a function that is often defined using more than one formula, where each individual formula describes the function on a subset of the domain.  Here is an example of such a function

35. 35 Find f(-1/2), f(1), and f(2), where the piecewise-defined function f(x) is given at the above slide. Since satisfies x<1, use the top part of the formula to find However, x=1 and x=2 satisfy x≥1, so f(1) and f(2) are both found by using the bottom part of the formula: and Example 1 Example 1 Solution: Solution:

36. 36 Domain Convention  We assume the domain of f to be the set of all numbers for which f(x) is defined (as a real number).  We refer to this as the natural domain of f. In general, there are two situations where a number is not in the domain of a function: 1) division by 0 2) The even number root of a negative number

37. 37 Example 2 Find the domain and range of each of these functions a. b. Solution: Solution: a.Since division by any number other than 0 is possible, the domain of f is the set of all numbers except -1 and 1. The range of f is the set of all numbers y except 0. b.Since negative numbers do not have real fourth roots, so the domain of g is the set of all numbers u such as u≥-2. The range of g is the set of all nonnegative numbers.

38. 38 Functions Used in Economics  A demand function ( 需求函数 ) p=D(x) is a function that relates the unit price p for a particular commodity to the number of units x demanded by consumers at that price.  The total revenue （总收入） is given by the product R(x)=(number of items sold)(price per item) =xp=xD(x)  If C(x) is the total cost （总成本） of producing the x units, then the profit （利润） is given by the function P(x)=R(x)-C(x)=xD(x)-C(x)

39. 39 Example 3 Market research indicates that consumers will buy x thousand units of a particular kind of coffee maker when the unit price is dollars. The cost of producing the x thousand units is thousand dollars a. What are the revenue and profit functions, R(x) and P(x), for this production process? b. For what values of x is production of the coffee makers profitable?

40. 40 Solution: Solution: a.The demand function is, so the revenue is thousand dollars, and the profit is (thousand dollars) thousand dollars, and the profit is (thousand dollars) b. Production is profitable when P(x)>0. We find that Thus, production is profitable for 2<x<17.

41. 41 Composition of Functions （复合函数）  Composition of Functions: Given functions f(u) and g(x), the composition f(g(x)) is the function of x formed by substituting u=g(x) for u in the formula for f(u). Example 4 Find the composition function f(g(x)), where and Solution: Solution: Replace u by x+1 in the formula for f(u) to get Question: How about g(f(x))? Note: In general, f(g(x)) and g(f(x)) will not be the same.

42. 42 Example 5 An environmental study of a certain community suggests that the average daily level of carbon monoxide in the air will be parts per million when the population is p thousand. It is estimated that t years from now the population of the community will be thousand. a.Express the level of carbon monoxide in the air as a function of time. b.When will the carbon monoxide level reach 6.8 parts per million?

43. 43 Solution: Solution: a. Since the level of carbon monoxide is related to the variable p by the equation, and the variable p is related to the variable t by the equation It follows that the composite function expresses the level of carbon monoxide in the air as a function of the variable t. b. Set c(p(t)) equal to 6.8 and solve for t to get b. Set c(p(t)) equal to 6.8 and solve for t to get That is, 4 years from now the level of carbon monoxide will be 6.8 parts per million.

44. 44 Section 1.2 The Graph of a Function  The graph of a function f consists of all points (x,y) where x is in the domain of f and y=f(x), that is, all points of the form (x,f(x)).  Rectangular coordinate system （平面直角坐标系）, Horizontal axis （横坐标）, vertical axis （纵坐标）.  The below example shows that the function can be sketched by plotting a few points. x -3-201234 f(x)-10-40220 -10

45. 45 Intercepts  x intercepts: The points where a graph crosses the x axis.  A y intercept: A point where the graph crosses the y axis.  How to find the x and y intercepts: The only possible y intercept for a function is, to find any x intercept of y=f(x), set y=0 and solve for x.  Note: Sometimes finding x intercepts may be difficult.  Following above example, the y intercept is f(0)=2. To find the x intercepts, solve the equation f(x)=0, we have x=-1 and 2. Thus, the x intercepts are (-1,0) and (2,0).

46. 46 Parabolas （抛物线）  Parabolas: The graph of 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 vertex （顶点）, and it always occurs where

47. 47 Example 6 A manufacturer determines that when x hundred units of a particular commodity are produced, they can all be sold for a unit price given by the demand function p=60-x dollars. At what level of production is revenue maximized? What is the maximum revenue? Solution: Solution: The revenue function R(x)=x(60-x) hundred dollars. Note that R(x) ≥0 only for 0≤x≤60. The revenue function can be rewritten as which is a parabola that opens downward (Since A=-1<0) and has its high point (vertex) at Thus, revenue is maximized when x=30 hundred units are produced, and the corresponding maximum revenue is R(30)=900 hundred dollars.

48. 48 Intersections of Graphs  Sometimes it is necessary to determine when two functions are equal.  For example, an economist may wish to compute the market price at which the consumer demand for a commodity will be equal to supply.

49. 49 Power Functions, Polynomials, and Rational Functions  A Power Function （幂函数） : A function of the form, where n is a real number.  A Polynomial Function （多项式） : A function of the form where n is a nonnegative integer and are constants. If, the integer n is called the degree （阶） of the polynomial.  A Rational Function （有理函数） : A quotient of two polynomials p(x) and q(x).

50. 50 The Vertical Line Test 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.

51. 51 Section 1.3 Linear Functions A linear function （线性函数） 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 where m and b are constants.

52. 52 The Slope of a Line （斜率）  The Slope of a Line: The slope of the non-vertical line passing through the points and is given by the formula  Sign and magnitude of slope

53. 53 Forms of the equation of a line  The Slope-Intercept Form: The equation is the equation of a line whose slope is m and whose y intercept is (0,b).  The Point-Slope Form: The equation is an equation of the line that passes through the point and that has slope equal to m. The slope-Intercept form is The point-slope form that passes through the point (-1.5,0) is

54. 54 Example 7 Table 1.2 lists the percentage of the labour force that was unemployed during the decade 1991-2000. Plot a graph with the time (years after 1991) on the x axis and percentage of unemployment on the y axis. Do the points follow a clear pattern? Based on these data, what would you expect the percentage of unemployment to be in the year 2005? Number of Years Percentage of Year from 1991 Unemployed 1991 0 6.8 1992 1 7.5 1993 2 6.9 1994 3 6.1 1995 4 5.6 1996 5 5.4 1997 6 4.9 1998 7 4.5 1999 8 4.2 2000 9 4.0 Table 1.2 Percentage of Civilian Unemployment

55. 55 Solution: Solution: The pattern does suggest that we may be able to get useful information by finding a line that “best fits” the data in some meaningful way. One such procedure, called “least- squares approximation”, require the approximating line to be positioned so that the sum of squares of vertical distances from the data points to the line is minimized. It produces the “best-fitting line”. Based on this formula, we can attempt a prediction of the unemployment rate in the year 2005: Note: Care must be taken when making predictions by extrapolating from known data, especially when the data set is as small as the one in this example.

56. 56 Parallel （平行） and Perpendicular （垂直） Lines  Let and be the slope of the non-vertical lines and. Then and are parallel if and only if and are perpendicular if and only if

57. 57 Example 8 Let L be the line 4x+3y=3 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(2,-3). Solution: Solution: By rewriting the equation 4x+3y=3 in the slope-intercept form, we see that L has slope, we see that L has slope a.Any line parallel to L must also have slope -4/3. The required line contains P(-1,4), we have contains P(-1,4), we have b.A line perpendicular to L must have slope m=3/4.Since the required line contains Q(2,-3), we have

58. 58 Section 1.4 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. Real-world problem Testing Interpretation Mathematical model adjustments Prediction Analysis Formulation

59. 59 Elimination of Variables  In next example, the quantity you are seeking is expressed most naturally in term of two variables. We will have to eliminate one of these variables before you can write the quantity as a function of a single variable. Example 9 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. Express the number of yards of fencing required as a function of the length of the unfenced side.

60. 60 Solution: Solution: We denote x and y as the lengths of the sides of the picnic area. Expressing the number of yards F of required fencing in terms of these two variables, we get. Using the fact that the area is to be 5,000 square yards that is and substitute the resulting expression for y into the formula for F to get

61. 61 Modelling in Business and Economics Example 10 A manufacturer can produce blank videotapes at a cost of $2 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: Express 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 maximum profit? What is the maximum profit?

62. 62 Solution: 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. 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 Number of cassettes sold =4000-400(number of $1 increases) =4000-400(number of $1 increases) =4000-400(p-5)=6000-400p =4000-400(p-5)=6000-400p Profit per cassette=p-2 Profit per cassette=p-2 The total profit is The total profit is

63. 63 b.The graph of P(p) is the downward opening parabola shown in the bottom figure. Profit is maximized at the value of p that corresponds to the vertex of the parabola. We know Thus, profit is maximized when the manufacturer charges $8.50 for each cassette, and the maximum monthly profit is Thus, profit is maximized when the manufacturer charges $8.50 for each cassette, and the maximum monthly profit is

64. 64 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(x) The supply function: p=S(x) The equilibrium price: Shortage: D(x)>S(x) Surplus: S(x)>D(x)

65. 65 Example 11 Market research indicates that manufacturers will supply x units of a particular commodity to the marketplace when the price is p=S(x) dollars per unit and that the same number of units will be demanded by consumers when the price is p=D(x) dollars per unit, where the supply and demand functions are given by a.At what level of production x and unit price p is market equilibrium achieved? b.Sketch the supply and demand curves, p=S(x) and p=D(x), on the same graph and interpret.

66. 66 Solution: Solution: a. Market equilibrium occurs when S(x)=D(x), we have Only positive values are meaningful,

67. 67 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.

68. 68 Example 12 A manufacturer can sell a certain product for $110 per unit. Total cost consists of a fixed 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 $1250? Solution: Solution: If x is the number of units manufactured and sold, the total revenue is given by R(x)=110x and the total cost by C(x)=7500+60x

69. 69 a.To find the break-even point, set R(x) equal to C(x) and solve 110x=7500+60x, so that x=150. It follows that the manufacturer will have to sell 150 units to break even. It follows that the manufacturer will have to sell 150 units to break even. b.The profit P(x) is revenue minus cost. Hence, P(x)=R(x)-C(x)=110x-(7500+60x)=50x-7500 P(x)=R(x)-C(x)=110x-(7500+60x)=50x-7500 The profit from the sale of 100 units is P(100)=-2500 The profit from the sale of 100 units is P(100)=-2500 It follows that the manufacturer will lose $2500 if 100 units are sold. It follows that the manufacturer will lose $2500 if 100 units are sold. c.We set the formula for profit P(x) equal to 1250 and solve for x, we have P(x)=1250, x=175. That is 175 units must be sold to generate the desired profit.

70. 70 Example 13 A certain car rental agency charges $25 plus 60 cents per mile. A second agency charge $30 plus 50 cents per mile. Which agency offers the better deal? Solution: Solution: Suppose a car is to be driven x miles, then the first agency will charge dollars and the second will charge. So that x=50.. So that x=50. For shorter distances, the first agency offers the better deal, and for longer distances, the second agency is better.

71. 71 Section 1.5 Limits （极限）  Roughly speaking, the limit process involves examining the behavior of a function f(x) as x approaches a number c that may or may not be in the domain of f.  To illustrate the limit process, consider a manager who determines that when x percent of her company’s plant capacity is being used, the total cost is 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 expect when the plant is operating at full permissible capacity?

72. 72 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(x) for values of x that approach 80 from the left (x 80), as indicated in this table: x approaches 80 from the left → ← x approaches 80 from the right x79.879.9979.9998080.000180.00180.04 C(x)6.997826.999896.999997.0000017.000017.00043 The values of C(x) displayed on the lower line of this table suggest that C(x) approaches the number 7 as x gets closer and closer to 80. The functional behavior in this example can be describe by

73. 73 Limit: If f(x) gets closer and closer to a number L as x gets closer and closer to c from both sides, then L is the limit of f(x) as x approaches c. The behavior is expressed by writing

74. 74 Example 14 Solution: Solution: Use a table to estimate the limit Let and compute f(x) for a succession of values of x approaching 1 from the left and from the right. x → 1 ← x x0.990.9990.9999 11.000011.00011.001 f(x)0.501260.500130.500010.4999990.499990.49988 The numbers on the bottom line of the table suggest that f(x) approaches 0.5 as x approaches 1. That is

75. 75 Three functions for which It is important to remember that limits describe the behavior of a function near a particular point, not necessarily at the point itself.

76. 76 The figure below shows that the graph of two functions that do not have a limit as x approaches 2. Figure (a): The limit does not exist; Figure (b): The function has no finite limit as x approaches 2. Such so- called infinite limits will be discussed later.

77. 77 Properties of Limits

78. 78 For any constant k, That is, the limit of a constant is the constant itself, and the limit of f(x)=x as x approaches c is c.

79. 79 Computation of Limits Example 15 Find (a) (b) Solution: Solution: a.Apply the properties of limits to obtain b.Since, you can use the quotient rule for limits to get

80. 80 Limits of Polynomials and Rational Functions: If p(x) and q(x) are polynomials, then and Example 16 Find Solution: Solution: The quotient rule for limits does not apply in this case since the limit of the denominator is 0 and the limit of the numerator is 3. So the limit of the quotient does not exist.

81. 81 If and, then is said to be indeterminate. The term indeterminate is used since the limit may or may not exist. Indeterminate Form （不定形） Example 17 (a) Find (b) Find Solution: Solution: a. a. b. b.

82. 82 Limits Involving Infinity Limits at Infinity: If the value of the function f(x) approach the number L as x increases without bound, we write Similarly, we write when the functional values f(x) approach the number M as x decreases without bound.

83. 83 Reciprocal Power Rules: For constants A and k, with k>0 Example 18 Find Solution: Solution:

84. 84 Procedure for Evaluating a Limit at Infinity of f(x)=p(x)/q(x) Step 1. Divide each term in f(x) by the highest power x k that appears in the denominator polynomial q(x). Step 2. Compute or using algebraic properties of limits and the reciprocal rules. Exercise Example 19 If a crop is planted in soil where the nitrogen level is N, then the crop yield Y can be modeled by the Michaelis- Menten function where A and B are positive constants. What happens to crop yield as the nitrogen level is increased indefinitely? ( Optional Question! )

85. 85 Solution: Solution: We wish to compute Thus, the crop yield tends toward the constant value A as the nitrogen level N increases indefinitely. For this reason, A is called the maximum attainable yield.

86. 86 Infinite Limits （无穷极限） : If f(x) increases or decreases without bound as x → c, we have For example From the figure, we can guest that

87. 87 Section 1.6 One-sided Limits and Continuity If f(x) approaches L as x tends toward c from the left (x<c), we write where L is called the limit from the left (or left- hand limit) （左极限） Likewise if f(x) approaches M as x tends toward c from the right (x>c), then M is called the limit from the right (or right-hand limit.) （右极限）

88. 88 Example 20 For the function evaluate the one-sided limits and Solution: Solution: Since for x<2, we have Similarly, f(x)=2x+1 if x ≥ 2, so

89. 89 Existence of a Limit: The two-sided limit exists if and only if the two one-sided limits and exist and are equal, and then Notice that the limit of the piecewise-define function f(x) in example 20 does not exist, that is does not exist, since does not exist, since

90. 90 At x=1 : Left-hand limitRight-hand limit value of the function does not exist! does not exist! Since the left and right hand limits are not equal.

91. 91 At x=2 : Left-hand limitRight-hand limit value of the function does exist! does exist! Since the left and right hand limits are equal, However, the limit is not equal to the value of function.

92. 92 At x=3 : Left-hand limitRight-hand limit value of the function does exist! does exist! Since the left and right hand limits are equal, and the limit is equal to the value of function.

93. 93 Nonexistent One-sided Limits A simple example is provided by the function As x approaches 0 from either the left or the right, f(x) oscillates between -1 and 1 infinitely often. Thus neither one-sided limit at 0 exists.

94. 94 Continuity （连续性）  A continuous function is one whose graph can be drawn without the “pen” leaving the paper. (no holes or gaps )

95. 95 A “hole “ at x=c

96. 96 A “gap” at x=c

97. 97  So what properties will guarantee that f(x) does not have a “hole” or “gap” at x=c? Continuity: A function f is continuous at c if all three of these conditions are satisfied: a. a. b. b. c. c. If f(x) is not continuous at c, it is said to have a discontinuity there.

98. 98 f(x) is continuous at x=3 because the left and right hand limits exist and equal to f(3). At x=1: At x=2: At x=3: Discontinuous Continuous

99. 99 Continuity Polynomials and Rational Functions  If p(x) and q(x) are polynomials, then A polynomial or a rational function is continuous wherever it is defined

100. 100 Example 21 Show that the rational function is continuous at x=3. Solution: Solution: Note that f(3)=(3+1)/(3-2)=4, since, you will find that as required for f(x) to be continuous at x=3, since the three criteria for continuity are satisfied.

101. 101 Example 22 Determine where the function below is not continuous. Solution: Solution: Rational functions are continuous everywhere except where we have division by zero. The function given will not be continuous at t=-3 and t=5.

102. 102 Exercise Discuss the continuity of each of the following functions

103. 103 Example 23 For what value of the constant A is the following function continuous for all real x? Solution: Solution: Since Ax+5 and are both polynomials, it follows that f(x) will be continuous everywhere except possibly at x=1. According to the three criteria for continuity, we have This means that f is continuous for all x only when A=-3

104. 104 Example 24 Find numbers a and b so that the following function is continuous everywhere. Solution: Solution: Since the “parts” fare polynomials, we only need to choose a and b so that f is continuous at x=-1 and 1. Since the “parts” f are polynomials, we only need to choose a and b so that f is continuous at x=-1 and 1. At x=-1 At x=1 We have a=-1/3 and b=1/3 for f is continuous everywhere

105. 105 Continuity on an Interval  A function f(x) is said to be continuous on an open interval a<x<b if it is continuous  A function f(x) is said to be continuous on an open interval a<x<b if it is continuous at each point x=c in that interval.  f is continuous on closed interval a≤x≤b, if it continuous on the open interval a<x<b and  is continuous on [-1,1]

106. 106 Example 25 Discuss the continuity of the function on the open interval -2<x<3 and on the closed interval -2≤x≤3 Solution: Solution: The rational function f(x)is continuous for all x except x=3. Therefore, it is continuous on the open interval -2<x<3 but not on the closed interval -2≤x≤3,since it is discontinuous at the endpoint 3 (where its denominator is zero). The graph of f is shown in below Figure.

107. 107 The Intermediate Value Property  Suppose that f(x) is continuous on the interval a≤x≤b and L is a number between f(a) and f(b), then there exists a number c between a and b, such that f(c)=L.

108. 108 Corollary If f is continuous on the closed interval [a,b], and f(a) and f(b) have opposite signs, then there exists a number c in (a,b) where f(c)=0.

109. 109 Example 26 Show that the equation has a solution for 1<x<2 Solution: Solution: Let. Then f(1)=-3/2 and f(2)=2/3. Since f(x) is continuous for 1≤x≤2, it follows from the intermediate value property that the graph must cross the x axis somewhere between x=1 and x=2.

110. 110 Summary  Function: domain and range of a function domain and range of a function composition of function f(g(x)) composition of function f(g(x))  Graph of a function: x and y intercepts, x and y intercepts, Piecewise-defined function, power function Piecewise-defined function, power function Polynomial, Rational function, Vertical line test Polynomial, Rational function, Vertical line test  Linear function: Slope, Slope-intercept formula, point-slope formula Slope, Slope-intercept formula, point-slope formula Parallel and perpendicular lines. Parallel and perpendicular lines.

111. 111  Function Models: Market equilibrium: law of supply and demand Market equilibrium: law of supply and demand Shortage and surplus, Break-even analysis Shortage and surplus, Break-even analysis  Limits: Calculation of limits: limits of polynomial and Calculation of limits: limits of polynomial and rational function, limits at infinity: calculation of rational function, limits at infinity: calculation of limits at the infinity (Reciprocal power Rules), limits at the infinity (Reciprocal power Rules), Infinite limit, One sided limit, Existence of limit Infinite limit, One sided limit, Existence of limit Continuity of f(x) at x=c:  Continuity of f(x) at x=c: Continuity on an interval, Continuity of polynomials Continuity on an interval, Continuity of polynomials and rational function, Intermediate value property and rational function, Intermediate value property