Functions of 2 Variables

Transcription

1 Functions of 2 Variables Functions and Graphs In the last chapter, we extended di erential calculus to vector-valued functions. In this chapter, we extend calculus primarily to functions of two variables, which are functions like f (x; y) = x 2 + y 2 and g (x; y) = sin (x) cos (y) : Intuitively, a function of 2 variables maps points (x; y) in the xy-plane to numbers z on the z-axis (rigorous de nition in part 2). We often denote functions of 2 variables by f (x; y) ; which means the output from an input of (x; y) ; and we often de ne these functions in the form f (x; y) = \expression in x and y Equivalently, we can consider f (x; y) to be the assignment of a real number to a point (x; y) in the xy-plane. EXAMPLE 1 Evaluate f (1; 2) and f (2; 5) if f (x; y) = x 2 + 2xy Solution: To begin with, f (1; 2) = = 5; which is to say that f (x; y) = x 2 + 2xy maps the point (1; 2) to the number 5: Likewise, f (2; 5) = = 24: The graph of f (x; y) is the set of points in R 3 that satisfy z = f (x; y) : That is, the graph of f (x; y) is the surface z = f (x; y) and the output z is the height of the surface at the point (x; y) : 1

2 A graphing calculator or computer algebra system is often used to produce an approximation of the graph of a function. EXAMPLE 2 Use a computer algebra system to graph the function f (x; y) = x 2 + y 2 for x and y in [ 1; 1] : Solution: The graph of f (x; y) = x 2 +y 2 is by de nition the set of all points (x; y; z) with x in [ 1; 1], y in [ 1; 1], and z = x 2 +y 2 : This set of points forms the surface which is shown in the four di erent types of plots below.a patch plot shows only the surface, while a patch and grid plot shows the graph along with a grid of curves on the surface. A contour plot shows the surface along with horizontal cross-sections at various heights. A wireframe plot shows only a grid of curves on a surface. 2

3 We often let x = hx; yi be the position vector of a point in the xy-plane, and then we write f (x) = f (x; y) This allows us to use vectors to de ne functions of 2 variables. EXAMPLE 3 What function f (x; y) is represented by f (x) = jjxjj 2 Solution: Since x = hx; yi ; we have f (x; y) = jjhx; yijj 2 = x 2 + y 2 Note: We will often use the bolded rst letter of a vector of variables to denote the vector. For example, we write x = hx; yi ; p = hp; qi ; v = hv 1 ; v 2 ; v 3 i ; and so on Check your Reading: How is z = x 2 + y 2 related to examples 2 and 3? De nition and Domains In particular, a function of 2 variables is a function whose inputs are points (x; y) in the xy-plane and whose outputs real numbers. De nition 1.1 A function of 2 variables f (x; y) is a relation which maps each point (x; y) in a set D in the xy-plane to at most one real number z. The set D is called the domain of the function, which is denoted dom (f) : z y f (x,y) D x f(x,y) 3

4 Domains are often written in set notation, where fg represents the phrase is the set of and a vertical bar j represents the phrase such that. EXAMPLE 4 Determine the domain of f (x; y) = ln (y 2x) Solution: Since the argument of ln () must be positive, the domain of f is the set of points (x; y) for which y 2x is positive. However, y 2x > 0 means that y > 2x In set notation this is written as dom (f) = f(x; y) j y > 2xg : In this text, most of the sets in the xy-plane we encounter will be bounded by a closed curve. The set of all points inside of but not including a closed curve is said to be an open region, while the set of all points inside of and including a closed curve is said to be a closed region. Similar de nitions hold for a nite union of closed curves. More generally, a point (p; q) is a boundary point of a set R is any circle centered at (p; q) contains both points inside of and points outside of R: A set R is open if it contains none of its boundary points and is closed if it contains all of its boundary points (if it contains some but not all of its boundary points, then it is neither open or closed). In addition, we say that the domain of a function is bounded if there is a number R > 0 such that the domain is inside of the circle centered at (0; 0) 4

5 with radius R: For example, the region in example 2 is bounded. A region is unbounded if it cannot be contained in any circle centered at the origin. Also, a region R is connected if any two points in R can be joined by a curve which is contained in R: 10 1 con It is important to identify if the domain of a function has these properties. EXAMPLE 5 open or closed Determine if the domain of the following function is f (x; y) = p 9 x 2 y 2 Solution: To begin with, the quantity 9 x 2 y 2 cannot be negative since it is under the square root. Thus, the domain of f is the set of points that satisfy 9 x 2 y 2 0 or 9 x 2 + y 2 That is, the domain is the set of points (x; y)inside the circle of radius 3 centered at the origin, which we write as dom (f) = (x; y) j x 2 + y 2 9 Since dom (f) contains it boundary i.e., the circle of radius 3 it is a closed region of the xy-plane. 5

6 Check your Reading: Is the domain in example 4 open or closed? Boundedness and Connectedness In addition, we say that the domain of a function is bounded if there is a number R > 0 such that the domain is inside of the circle centered at (0,0) with radius R. For example, the domain in example 5 is bounded because it is itself a circle centered at the origin. Conversely, a set is unbounded if it cannot be contained in any circle centered at the origin. For example, set f(x; y) j y > 2xg is unbounded since no circle centered at the origin can contain all the points (x; y) for whichy > 2x. blueexample 6 blackdetermine if the domain of f(x; y) = ln(xy+ 1) is bounded or unbounded. Solution: To begin with, we must have xy + 1 > 0, which requires that xy > 1. If x < 0, then xy > 1 is the same as y < 1=x. If x 0, then xy > 1 is the same as y > 1=x. We can combine the two cases by using the symbol for a union of sets, which is [: That is, dom (f) = (x; y) j x < 0 and y < 1 [ (x; y) j x 0 and y > 1 x x The domain of the function is shown below: Clearly, it cannot be contained in any circle centered at the origin. Therefore, dom (f) is unbounded. In addition, we say that a set R is connected if any two points in R can be joined by a curve which is contained in R: blueexample 7 blackdetermine if the domain of the following function is connected: f (x; y) = y p x 2 1 6

7 Solution: The domain of f is the set of points (x; y) for which x 2 1 0; which is where x 2 1: This implies that either x 1 or x 1; so that the domain is the two strips shown below: Clearly, the domain is not connected. Check your Reading: Is the domain in example 6 bounded or unbounded? Functions of Space and Time A function of the form u (x; t) is often interpreted to be a function of x at a given point in time. For example, let s place an xy-coordinate system on a violin whose strings have a length of l; as shown in the gure below: If u (x; t) is de ned to be the displacement of the string above or below a point x on the x-axis at a time t; then y = u (x; t) is the shape of the string at a xed 7

8 time t: Thus, u (x; t) models the motion of the string as t increases. EXAMPLE 8 The displacement of a1 foot long violin string at a distance x from the bridge and at time t in seconds is given by u (x; t) = 60 sin (x) cos (880t) Describe the shape of the string at times t = 0; 0:5; 1; and 1:5 milliseconds. Solution: At time t = 0; we have the curve u (x; 0) = 60 sin (x) ; which is a sine curve beginning at the origin and intersecting the 8

9 x-axis again at x = 1: At time t = 0:5 milliseconds, which is t = 0:00005 seconds, we have the curve u (x; 0:00005) = 60 sin (x) cos (0:44) = 11:243 sin (x) At time t = 1 millisecond, which is t = 0:001 seconds, we have u (x; 0:001) = 60 sin (x) cos (0:88) = and at t = 1:5 milliseconds, we have u (x; 0:0015) = 60 sin (x) cos (1:32) = 55:787 sin (x) 32:15 sin (x) These curves are shown in sequence in the "slide show" below: Likewise, u (x; t) might represent evolution of the temperature distribution of a thin rod, in that u might represent the temperature at time t at a distance x from one end. The function u (x; t) then models the evolution of the temperature distribution over time. In general, any process which can be modeled as the evolution in time of the graph of a function can be modeled by a function of 2 variables. 9

10 Exercises Find the domains of the following functions. Sketch a graph of the domain, and then determine whether is it open or closed, bounded or unbounded, connected or not connected. 1. f (x; y) = (1 2x + y) 1=2 2. f (x; y) = (1 3x + 2y) 1=2 3. f (x; y) = p x + p y 4. f (x; y) = p xy 1 5. f (x; y) = x x 2 +y 6. f (x; y) = x y 2 x+y 7. f (x; y) = x2 y 1 x 2 y 2 8. f (x; y) = x 2 y 1 x 2 y 2 9. f (x; y) = y x 10. f (x; y) = y tan(x) 11. f (x; y) = p x ln (1 y) 12. f (x; y) = ln 1 x 2 y f (x; y) = ln x 2 y f (x; y) = ln jx xyj 15. f (x; y) = sin 1 (x y) 16. f (x; y) = sin 1 (xy) Use a computer algebra system to sketch the graph of the given function. 17. f (x; y) = y 18. f (x; y) = 3x + 2y f (x; y) = 9 x 2 y f (x; y) = 1 xy 21. f (x; y) = sin x 2 + y f (x; y) = sin (x) cos (y) sin (x) 23. f (x; y) = 1 + x f (x; y) = e y 1 + y 2 x 25. f (x; y) = x 2 + y f (x; y) = x + y x 2 + y 2 In 27-30, u (x; t) is the displacement of a vibrating string at a point x in [0; ] and at time t: Sketch the graph of the string at each of the given times. 27. u (x; t) = cos 6 t sin (x) 28. u (x; t) = cos (t) cos (2x) t = 0; 1; 2; 3 t = 0; 1; 2; u (x; t) = sin 4 t cos (2x) 30. u (x; t) = e t=5 sin 6 t sin (x) t = 0; 1; 2; 3 t = 0; 1; 2; Explain why the set f(x; y) j y 6= 2xg is the same as f(x; y) j y < 2xg [ f(x; y) j y > 2xg 32. What is the domain of the function f (x; y) = p 1 2 cosh (xy) (1) 10

11 Does it contain any points at all? Can (1) even be considered a function? 33. The function u (x; t) = sin (x t)+sin (x + t) models the shape of a vibrating string which is xed at x = 0 and x = : Graph u (x; t) over x in [0; ] for several di erent values of t (e.g., t = 0; 1; 2, etc.). What does the vibration of the string look like? How long does it take until it returns to its original shape? 34. The function u (x; t) = e t sin 2 (x) + 32 models the temperature in F of a 1 foot long thin rod in which both ends are held at the freezing point at all times t. Graph u (x; t) over x in [0; 1] for several di erent values of t (e.g., t = 0; 1; 2, etc.). What happens to the temperature of the rod as t approaches 1? 35. The function (x t)2 u (x; t) = (x t) e is an example of a traveling wave. Graph u (x; t) for several di erent values of t (e.g., t = 0; 1; 2, etc.). Why might we call this a traveling wave? 36. The 1-dimensional heat kernel is de ned to be u (x; t) = 1 p 4t e x2 =(4t) Graph u (x; t) for several di erent values of t (e.g., t = 1; 2, 3). What does the graph look like when t is close to 0? 37. Suppose that a function of 1 variable, g (x) ; has a domain of [a; b] : If we de ne a function of 2 variables f (x; y) = g (x) ; then what is dom (f)? Is it bounded? Is it open or closed? 38. Suppose that g (x) has a domain of [a; b] and that h (y) has a domain of [c; d] : Then what is the domain of f (x; y) = g (x) h (y)? Is it bounded? Is it open or closed? 39. Write to Learn: Suppose that f (x; y) is de ned everywhere and that f (x; y) = 0 is a smooth closed curve. What is the domain of h (x; y) = 1 f (x; y) Is dom (h) open or closed? Bounded or unbounded? Connected or disconnected? Report your conclusions in a short essay with supporting diagrams. 40. Write to Learn: Suppose that the domain of f (x; y) is dom (f) and the domain of g (x; y) is dom (g) : Write a short essay which explains why if h (x; y) = f (x; y) + g (x; y) ; then the domain of h (x; y) is dom (h) = dom (f) \ dom (g) 41. Let s suppose that f (x; y) is the shortest distance from (x; y) to the origin along paths that contain only horizontal and vertical line segments (this is known as taxicab distance). What is f (x; y)? What does its graph look like? 42. Let G (x; y) be the gallons of fuel consumed by a taxicab that drives from a point (x; y) to the origin along only horizontal and vertical line segments given that it consumes 0:05 gallons per mile (see question 41). What is G (x; y)? What is its domain? Explain. 11