Mathematics for Business

1. Mathematics for Business Instructor: Prof. Ken Tsang Room E409-R11 Email: kentsang@uic.edu.hk

2. CALCULUSFor Business, Economics, and the Social and life Sciences Hoffmann, L.D. & Bradley, G.L.

3. TA information Mr Zhu Zhibin Room E409 Tel: 3620630 zhibinzhu@uic.edu.hk

4. Web-page for this class • Watch for announcements about this class and • download lecture notes from • http://www.uic.edu.hk/~kentsang/calcu2012/calcu.htm • Or from this page: http://www.uic.edu.hk/~kentsang/ Or from Ispace

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

6. How is my final grade determined? • Quizzes 20% • Mid-term exam 20% • Assignments 10% • Final Examination 50%

7. UIC Score System

8. Grade Distribution Guidelines

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

10. More info about this Course • Assignments must be handed in before the deadline. • There will be about 3 to 4 quizzes. • We will tell you your scores for the mid-term test 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.

11. Mathematics? Why? • 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.

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

13. What is Calculus all about? • Calculus is the study of changing quantities, or more precisely, the rate of changes: e.g. velocities, interest rate, return on an asset. • 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. Calculus has practical applications, such as understanding the true meaning of the infinitesimals. (Image concept by Dr. Lachowska.)

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. 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. 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. Cosmos1

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. Acceleration is the rate of change in velocity. This can be rewritten as a differential equation. Most laws of nature can be expressed as differential equations or partial differential equations (PDE).

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. A simple example in asset valuation • Suppose we have a riskless asset • r is the constant rate of return

21. 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.

22. 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, μ is the drift rate of S, annualized, σ is 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, modeled by a Brownian motion.

23. If you are a science major • Science is • Quantitative • Logical

24. Ecology: Population dynamics • The basic accounting relation for population dynamics is: N1 = N0 + B − D + I − E where N1 is the number of individuals at time 1, N0 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.

25. 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 a predator and one its prey.

26. 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.

27. 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.

28. OK, any question? • That’s all for introduction. • Let’s begin the real thing!

29. Chapter 1Functions, 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）

30. 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）

31. Function, or not? YES NO NO

32. 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

33. Function that describes tabular data Table 1.1 Average Tuition and Fees for 4-Year Private Colleges

34. Solution: • 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

35. 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

36. Example 1 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 Solution:

37. 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

38. Find the domain and range of each of these functions Example 2 a. b. Solution: • 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. • 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 ofg is the set of all nonnegative numbers.

39. 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)

40. 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?

41. Solution: • The demand function is , so the revenue is • 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.

42. 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: 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.

43. 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. • Express the level of carbon monoxide in the air as a function of time. • When will the carbon monoxide level reach 6.8 parts per million?

44. 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 That is, 4 years from now the level of carbon monoxide will be 6.8 parts per million.

45. Section 1.2 The Graph of a Function • The graph of a function fconsists 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.

46. 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).

47. 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

48. 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: 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.

49. 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.

50. 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).