Mathematics for Business (Finance)

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

2. Chapter 6: Calculus of two variables

3. In this Chapter: Functions of 2 Variables Limits and Continuity Partial Derivatives Tangent Planes and Linear Approximations The Chain Rule Maximum and Minimum Values Double integrals and volume evaluation

4. DEFINITION: A function f of two variables is a rule that assigns to each ordered pair of real numbers (x, y) in a set D a unique real number denoted by f (x, y). The set D is the domain of f and its range is the set of values that f takes on, that is, . We write z=f (x, y) to make explicit the value taken on by f at the general point (x, y) . The variables x and y are independent variables and z is the dependent variable.

6. Domain of

7. Domain of

8. Domain of

9. Graph of z=f(x,y)

10. Graph of

11. DEFINITION: The level curves of a function f of two variables are the curves with equations f (x, y)=k, where k is a constant (in the range of f).

13. Contour map of

14. The graph of h(x, y)=4x2+y2 is formed by lifting the level curves.

15. DEFINITION • Let f be a function of two variables whose domain D includes points arbitrarily close to • (a, b). Then we say that the limit of f (x, y) as (x, y) approaches (a ,b) is L and we write • if for every number ε> 0 there is a corresponding number δ> 0 such that • If and • then

17. DEFINITION A function f of two variables is called continuous at (a, b) if We say f is continuous on D if f is continuous at every point (a, b) in D.

18. If f is a function of two variables, its partial derivatives are the functions fx and fy defined by

19. NOTATIONS FOR PARTIAL DERIVATIVES If z=f (x, y) , we write

20. RULE FOR FINDING PARTIAL DERIVATIVES OF z=f (x, y) • To find fx, regard y as a constant and differentiate f (x, y) with respect to x. • 2.To find fy, regard x as a constant and differentiate f (x, y) with respect to y.

21. The partial derivatives of f at (a, b) are the slopes of the tangents to C1 and C2.

24. The second partial derivatives of f. If z=f (x, y), we use the following notation:

25. CLAIRAUT’S THEOREM Suppose f is defined on a disk D that contains the point (a, b) . If the functions fxy and fyx are both continuous on D, then

26. The tangent plane contains the tangent lines T1 and T2

27. Suppose f has continuous partial derivatives. An equation of the tangent plane to the surface z=f (x, y) at the point P(xo ,yo ,zo) is

28. The differential of x is dx=△x, if y=f(x), then dy=f’(x)dx is the differential of y.

29. For a differentiable function of two variables, z= f (x ,y), we define the differentials dx and dy (i.e. small increments in x & y directions). Then the differential dz (total differential), is defined by

31. For such functions the linear approximation is:

32. THE CHAIN RULE Suppose that z=f (x, y) is a differentiable function of x and y, where x=g (t) and y=h (t) and are both differentiable functions of t. Then z is a differentiable function of t and

33. THE CHAIN RULE (GENERAL VERSION) Suppose that u is a differentiable function of the n variables x1, x2,‧‧‧,xn and each xj is a differentiable function of the m variables t1, t2,‧‧‧,tm Then u is a function of t1, t2,‧‧‧, tm and for each i=1,2,‧‧‧,m.

34. DEFINITION A function of two variables has a local maximum at (a, b) if f (x, y) ≤ f (a, b) when (x, y) is near (a, b). [This means that f (x, y) ≤ f (a, b) for all points (x, y) in some disk with center (a, b).] The number f (a, b) is called a local maximum value. If f (x, y) ≥ f (a, b) when (x, y) is near (a, b), then f (a, b) is a local minimum value.

36. THEOREM If f has a local maximum or minimum at (a, b) and the first order partial derivatives of f exist there, then fx(a, b)=0 and fy(a, b)=0.

37. A point (a, b) is called a critical point (or stationary point) of f if fx (a, b)=0 and fy (a, b)=0, or if one of these partial derivatives does not exist.

38. SECOND DERIVATIVES TEST Suppose the second partial derivatives of f are continuous on a disk with center (a, b) , and suppose that both fx (a, b) and fy (a, b)=0 [that is, (a, b) is a critical point of f]. Let • If D>0 and fxx (a, b)>0 , then f (a, b) is a local minimum. • (b)If D>0 and fxx (a, b)<0, then f (a, b) is a local maximum. • (c) If D<0, then f (a, b) is not a local maximum or minimum.

39. NOTE 1 In case (c) the point (a, b) is called a saddle point of f and the graph of f crosses its tangent plane at (a, b). NOTE 2 If D=0, the test gives no information: f could have a local maximum or local minimum at (a, b), or (a, b) could be a saddle point of f. NOTE 3 To remember the formula for D it’s helpful to write it as a determinant:

43. EXTREME VALUE THEOREM FOR FUNCTIONS OF TWO VARIABLES If f is continuous on a closed, bounded set D in R2, then f attains an absolute maximum value f(x1,y1) and an absolute minimum value f(x2,y2) at some points (x1,y1) and (x2,y2) in D.

44. EXTREME VALUE THEOREM FOR FUNCTIONS OF TWO VARIABLES If f is continuous on a closed, bounded set D in R2, then f attains an absolute maximum value f(x1,y1) and an absolute minimum value f(x2,y2) at some points (x1,y1) and (x2,y2) in D.

45. Double integrals over rectangles Suppose f(x) is defined on a interval [a,b]. Recall the definition of definite integrals of functions of a single variable

46. Taking a partition P of [a, b] into subintervals: Using the areas of the small rectangles to approximate the areas of the curve sided echelons

47. and summing them, we have (1) (2)

48. Volume and Double Integral z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d] S={(x,y,z) in R3 | 0 ≤ z ≤ f(x,y), (x,y) in R} Volume of S = ?

49. Double integral of a function of two variables defined on a closed rectangle like the following Taking a partition of the rectangle