Section 2.5. Functions and Surfaces Brief review for one variable functions and curves: ² A (one variable) function is rule that assigns to each member x in a subset D in R1 a unique real number denoted by f (x) . The set D is called the domain of f (x) , and the set R of all values f (x) that f takes on, i.e., R = f (x) x D f j 2 g is called the range of f (x) . The subset of points in R2 de…ned by G = (x, f (x)) x D f j 2 g is called the graph of f. The graph of any one variable function is usually a curve in the common sense. Some times, a curve can be used to de…ne a function provided that if satis…es "vertical line test". A table may also de…ne a function. – Determine domains of some commonly used functions: ² D (loga (x)) = x x > 0 D x1/n = x f xj 0 g(n = even integer) D (sin x, cosfx, aj x, x¸m)g= R1 (a > 0, m = positive integer) ¡ ¢ f D = D (f) D (g) g = 0 = x both f (x) and g (x) are defined, AND g (x) = 0 g \ \f 6 g f j 6 g µ ¶ All these concepts and principles extend to two-variable functions. Two-variable functions ² De…nition. A function of two variables is rule that assigns to each point (x, y)in a subset D in R2 a unique real number denoted by f (x, y) . The set D is called the domain of f, and the set R of all values f (x, y) that f takes on, i.e., R = f (x, y) (x, y) D R1 f j 2 g ½ 1 is called the range of f. The subset of points in R3 de…ned by G = (x, y, z) z = f (x, y) , (x, y) D = (x, y, f (x, y)) (x, y) D f j 2 g f j 2 g is called the graph of f. The graph of any two-variable function is usually a SURFACE in the common sense. A surface may be used to de…ne a function as long as it passes the "vertical line test" that each line perpendicular to xy plane intersects the surface at most once. ¡ A function of two variables may also be de…ned through a 2D table. Example 5.1. Determine domains and discuss ranges for the following functions: (a) f (x, y) = 4x2 + y2, px + y + 1 (b) g (x, y) = , x 1 (c) h (x, y) = x ln (y¡2 x) . Solution: (a) f (x, y) i¡s de…ned for all (x, y) . Its graph is called paraboloid. (b) The function g (x, y) is unde…ned when either x = 1 (denominator = zero) or x + y + 1 < 0. So D (g) = (x, y) x + y + 1 0, x = 1 . f j ¸ 6 g In xy plane, both x + y + 1 = 0 and x = 1 are straight lines. ¡ 2 So D (g) consists of all points on one-side of the line x + y + 1 = 0 that containing (0, 0) , including the line x + y + 1 = 0, but excluding those on vertical line x = 1. Domain of g (x, y) Graph of g (x, y) 3 (c) h (x, y) = x ln (y2 x) . This function is de…ned as long as ¡ y2 x > 0, or x < y2. ¡ Domain of h : Shaded area, excluding the parabola Graph of h (x, y) 4 Graphs of Two-variable Functions ² The graph of z = f (x, y) , G = (x, y, f (x, y)) (x, y) D f j 2 g may be understood as a surface formed by two families of cross-section curves (or trace) as follows. For any …xed y = b, the one-variable function z = f (x, b) represents a curve. With various choices for b, for instance, b = 0, 0.1, 0.2, ..., there is a family of such curves z = f (x, 0) , z = f (x, 0.1) , z = f (x, 0.2) , ... In the same 3D coordinate system, each curve is the intersection of G and a coordinate plane (parallel to zx-plane) y = b, i.e., it is the solution of the system z = f (x, y) y = b. On the other hand, if we …x x = a,the one-variable function z = f (a, y) also represents a curve. With various choices for a, for instance, a = 0, 0.1, 0.2, ..., there is a family of such curves z = f (0, y) , z = f (0.1, y) , z = f (0.2, y) , ... In the same 3D coordinate system, each curve is the intersection of G and a coordinate plane (parallel to yz plane) x = a, i.e., it is the solution of the system ¡ z = f (x, y) x = a. Another way to study two-variable functions, or surfaces are often through one-variable functions, or curves. Consider, for example, z = T (x, y) is the temperature function of Dayton area in a certain time. Then, for each …xed number T0 = 50, for instance, the set (x, y) T (x, y) = 50 f j g 5 de…nes a one variable function. For each x = a, y is the solution of T (a, y) = 50. The graph of this one variable function is a curve called contour. It represents the path along which the temperature maintains at 50 degree. When f (x, y) is a polynomial of degree one or two, the function is called a quadratic function. Graphs of some two-variable functions ² Example 5.2. (a) The graph of z = 6 3x 2y ¡ ¡ is the plane 3x + 2y + z 6 = 0, ¡ passing through P0 (2, 0, 0) (by setting y = z = 0, and then solving for x = 2) with a normal 2, 3, 1 . h i One way to graph a plane is to …nd all three intercepts: intersection of the plane and coordinate axis: x intercept is x = 2 on x axis, y intercept is y = 3 on y axis, and z ¡intercept is z = 6 on z ¡axis. ¡ (b) The gr¡aph of ¡ ¡ z = 1 x2 y2 ¡ ¡ p 6 is the upper-half unit sphere. (c) The graph of z = 1 x2 y2 ¡ ¡ ¡ p is the other half of the unit sphere. Example 5.3. Sketch z = x2 7 parabolic cylinder We …rst view this as a one-variable function whose graph is a curve on xz plane. ¡ Now as a two-variable function, since z = x2 is independent of y, if a point P (x0, y0, z0) is on the surface, so is the entire line (x0, y, z0), passing through 8 P (x0, y0, z0) and parallel to y axis is on the surface. So it is a cylinder with cross section being the parabo¡la. One can also view this surface is generated by moving a line parallel to y axis parallel along above parabolic curve. In general, if one varia¡ble, for instance, yis missing, then the graph z = f (x)is a cylinder with generating lines parallel to y axis. ¡ Example 5.4. z = 2x2 + y2 (elliptic paraboloid) We now try to use trace method to analyze the above graph. Consider horizontal cross-section z = c,i.e., the intersection with a coordinate plane; z = 2x2 + y2, z = c. The cross-section, or trace, is the curve ellipse if c > 0 c = 2x2 + y2 : empty if c < 0 ½ on the plane z = c that is parallel to xy plane. When c > 0,the standard form is ¡ x2 y2 2 + 2 = 1, horizontal half axis = c/2 , vertical Half axis = pc. c/2 (pc) p ³p ´ 9 So as c increases (i.e., moving parallel to xy plane upward) starting at c = 0, the trace, which is ellipse, getting larger¡and larger. We next look at cross-sections parallel to yz plane : x = a, or ¡ z = 2x2 + y2, x = a. The trace is a curve z = 2a2 + y2 a = 0 (solid), a = 1 (dash), a = 2 (dot) on yz plane, which is a parabola with vertex y = 0, z = 2a2. Sim¡ilarly, along zx plane direction, the cross-section with y = b is a parabola ¡ z = 2x2 + b2. In summary, cross-sections are either ellipse or parabola. Example 5.5. z = y2 x2 (hyperbolic paraboloid) ¡ 10 We again try to use trace method to analyze this graph. Consider horizontal cross-section z = c,i.e., the intersection with a coordinate plane; z = y2 x2, ¡ z = c. The cross-section, or trace, is the curve hyperbola (opening along y axis) if c > 0 c = y2 x2 : ¡ ¡ hyperbola (opening along x axis) if c < 0 ½ ¡ on the plane z = c that is parallel to xy plane. The standard forms are ¡ y2 x2 y2 x2 = 1 (c > 0), or = 1 (c < 0). (pc)2 ¡ (pc)2 p c 2 ¡ p c 2 ¡ ¡ ¡ So as c increases (moving upward) s¡tartin¢g at ¡c = ¢0, the trace becomes vertical hyperbola with increasing half axis pc. However, when c decreases (moving downward) starting at c = 0, the trace becomes horizontal hyperbola y2 x2 with increasing half axis c . = 1 j j (pc)2 ¡ (pc)2 p 11 c = 1, 3 (solid), c = 1, 3 (dash) ¡ ¡ We next look at cross-sections parallel to yz plane : x = a, or ¡ z = y2 x2, ¡ x = a. The trace is a curve z = y2 a2 ¡ on yz plane, which is a parabola with vertex y = 0, z = a2. ¡ ¡ 12 a = 0 (solid), a = 1 (dash), a = 2 (dot) Similarly, along zx plane direction, the cross-section with y = b is a parabola ¡ z = b2 x2 ¡ opening opposite to z axis : ¡ b = 0 (solid), b = 1 (dash), b = 2 (dot) 13 In summary, cross-sections are either ellipse or hyperbola.