Section 2.5. Functions and Surfaces

Transcription

1 Section 2.5. Functions and Surfaces ² Brief review for one variable functions and curves: A (one variable) function is rule that assigns to each member x in a subset D in R 1 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 = ff (x) j x 2 Dg is called the range of f (x). The subset of points in R 2 de ned by G = f(x, f (x)) j x 2 D 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. D ² Determine domains of some commonly used functions: D (log a (x)) = fx j x > 0g D x 1/n = fx j x 0g (n = even integer) D (sin x, cos x, a x, x m ) = R 1 (a > 0, m = positive integer) µ f = D (f)\d (g)\fg 6= 0g = fx j both f (x) and g (x) are defined, AND g (x) 6= 0g g All these concepts and principles extend to two-variable functions. ² Two-variable functions De nition. A function of two variables is rule that assigns to each point (x, y)in a subset D in R 2 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 = ff (x, y) j (x, y) 2 Dg ½ R 1 1

2 is called the range of f. The subset of points in R 3 de ned by G = f(x, y, z) j z = f (x, y), (x, y) 2 D g = f(x, y, f (x, y)) j (x, y) 2 D 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) = 4x p 2 + y 2, x + y + 1 (b) g (x, y) =, x 1 (c) h (x, y) = x ln (y 2 x). Solution: (a) f (x, y) is 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) = f(x, y) j x + y + 1 0, x 6= 1g. In xy plane, both x + y + 1 = 0 and x = 1 are straight lines. 2

3 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

4 (c) h (x, y) = x ln (y 2 x). This function is de ned as long as y 2 x > 0, or x < y 2. Domain of h : Shaded area, excluding the parabola Graph of h (x, y) 4

5 ² Graphs of Two-variable Functions The graph of z = f (x, y), G = f(x, y, f (x, y)) j (x, y) 2 D 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 T 0 = 50, for instance, the set f(x, y) j T (x, y) = 50g 5

6 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. ² Graphs of some two-variable functions Example 5.2. (a) The graph of is the plane z = 6 3x 2y 3x + 2y + z 6 = 0, passing through P 0 (2, 0, 0) (by setting y = z = 0, and then solving for x = 2) with a normal h2, 3, 1i. 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 graph of z = p 1 x 2 y 2 6

7 is the upper-half unit sphere. (c) The graph of z = p 1 x 2 y 2 is the other half of the unit sphere. Example 5.3. Sketch z = x 2 7

8 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 = x 2 is independent of y, if a point P (x 0, y 0, z 0 ) is on the surface, so is the entire line (x 0, y, z 0 ), passing through 8

9 P (x 0, y 0, z 0 ) and parallel to y axis is on the surface. So it is a cylinder with cross section being the parabola. 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 variable, for instance, yis missing, then the graph z = f (x)is a cylinder with generating lines parallel to y axis. Example 5.4. z = 2x 2 + y 2 (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 = 2x 2 + y 2, z = c. The cross-section, or trace, is the curve ½ ellipse if c > 0 c = 2x 2 + y 2 : empty if c < 0 on the plane z = c that is parallel to xy plane. When c > 0,the standard form is x 2 ³ pc/2 2 + y2 ( p c) 2 = 1, horizontal half axis = p c/2, vertical Half axis = p c. 9

10 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 = 2x 2 + y 2, x = a. The trace is a curve z = 2a 2 + y 2 a = 0 (solid), a = 1 (dash), a = 2 (dot) on yz plane, which is a parabola with vertex y = 0, z = 2a 2. Similarly, along zx plane direction, the cross-section with y = b is a parabola z = 2x 2 + b 2. In summary, cross-sections are either ellipse or parabola. Example 5.5. z = y 2 x 2 (hyperbolic paraboloid) 10

11 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 = y 2 x 2, z = c. The cross-section, or trace, is the curve ½ hyperbola (opening along y axis) if c > 0 c = y 2 x 2 : hyperbola (opening along x axis) if c < 0 on the plane z = c that is parallel to xy plane. The standard forms are y 2 ( p c) 2 x2 ( p c) 2 = 1 (c > 0), or y 2 p c 2 x2 p c 2 = 1 (c < 0). So as c increases (moving upward) starting at c = 0, the trace becomes vertical hyperbola with increasing half axis p c. However, when c decreases (moving downward) starting at c = 0, the trace becomes horizontal hyperbola with increasing half axis p jcj. y 2 ( p c) 2 x2 ( p c) 2 = 1 11

12 c = 1, 3 (solid), c = 1, 3 (dash) We next look at cross-sections parallel to yz plane : x = a, or z = y 2 x 2, x = a. The trace is a curve z = y 2 a 2 on yz plane, which is a parabola with vertex y = 0, z = a 2. 12

13 a = 0 (solid), a = 1 (dash), a = 2 (dot) Similarly, along zx plane direction, the cross-section with y = b is a parabola z = b 2 x 2 opening opposite to z axis : b = 0 (solid), b = 1 (dash), b = 2 (dot) 13

14 In summary, cross-sections are either ellipse or hyperbola. However, those hyperbola changes from horizontal to vertical as the cross-section parallel to xy plane moving upward. Example 5.6. Ellipsoid x 2 + y2 9 + z2 4 = 1 It is easy to see cross-sections from all three directions are ellipses. Example 5.7. Hyperboloid of One Sheet x 2 + y2 4 z2 4 = 1 14

15 Let us look at traces in all three directions. Along xy plane z = c the trace x 2 + y2 4 z2 4 = 1 z = c, x 2 + y2 4 = 1 + c2 4 is a ellipse on xy plane with the standard form x 2 Along yz plane, the trace is y 2 Ãr! 2 + Ã r! 2 = c c2 4 4 x 2 + y2 4 z2 4 = 1 x = a, or hyperbola y 2 4 z2 4 = 1 a2 15

16 on yz plane. As a moves across a = 1, i.e., as (1 a 2 ) changes signs, the direction of opening of the hyperbola changes from horizontal (or y axis, when 1 a 2 > 0) to vertical (or z axis if 1 a 2 < 0). Similarly, the traces on zx plane, is hyperbola x 2 + y2 4 z2 4 = 1 y = b, x 2 z2 4 = 1 b2 4 µ on xz plane whose direction changes when 1 b2 changes signs. 4 Example 5.8. Hyperboloid of Two Sheets x 2 + y2 4 z2 4 = 1. 16

17 The traces along three directions are, respectively, x 2 + y2 4 = c2 c2 1 (z = c) ellipse if > 0 x 2 z2 4 = 1 b2 4 (y = b) hyperbola (opening along z axis) y 2 4 z2 4 = 1 a2 (x = a) hyperbola (opening along z axis) Note that here there is not directional change. ² Classi cation of Quadratic Surfaces Consider in general quadratic equations of three variables Ax 2 + By 2 + Cz 2 + Dx + Ey + F z + G + Hxy + Iyz + Jzx = 0. By a rotation, it can be reduced to Ax 2 + By 2 + Cz 2 + Dx + Ey + F z + G = 0. We then complete squares, if possible. There are several cases analogous to 2Dsituations. (1) If ABC 6= 0,it reduces to A (x h) 2 + B (y k) 2 + C (z l) 2 = R. The signs of A, B, C, R determine shapes of surfaces. We suppose that R 6= 0. (a) A, B, C have the same sign (either all positive or all three are negative). In this case, we have ellipsoid with the standard form (x h) 2 + a 2 (y k)2 (z l)2 + = 1 b 2 c 2 C (h, k, l) = Center of ellipsoid a = half axis in x axis direction b = half axis in y axis direction c = half axis in z axis direction. 17

18 For simpli cation, we take h = k = l = 0 : x 2 a 2 + y2 b 2 + z2 c 2 = 1. We use traces to see the graph. Set z = l be a constant. The cross section in the direction parallel to xy plane is x 2 a 2 + y2 b 2 + z2 c 2 = 1 z = l or x 2 a 2 + y2 b 2 = 1 l2 c 2 z = l. If jlj c,this is an ellipse. If jlj > c, then 1 l2 c 2 < 0 so there is no solution for the system and the curve is empty. Similarly, in other directions, all cross-sections are ellipses or the empty set. (b) A, B, C don t have the same signs. Assuming that AB > 0. The equation A (x h) 2 + B (y k) 2 + C (z l) 2 = R. reduces to either (x h) 2 (y k)2 (z l)2 + a 2 b 2 c 2 or (x h) 2 + a 2 If R = 0,then we have (y k)2 (z l)2 b 2 c 2 = 1 (Hyperboloid of One Sheet, z axis is axis of symmetry) = 1 (Hyperboloid of Two Sheets) A (x h) 2 + B (y k) 2 + C (z l) 2 = 0, and depending on the signs of A, B, C,its graph is a cone. For instance, 2x 2 + 3y 2 4z 2 = 0 18

19 is a cone centered at (0, 0, 0).Its axis is parallel to z axis. (3) Assume that only one of three numbers A, B, C is zero. For simplicity, assuming C = 0, but AB 6= 0.The equation reduce to Ax 2 + By 2 + Cz 2 + Dx + Ey + F z + G = 0 A (x h) 2 + B (y k) 2 = F (z l). This is an elliptic paraboloid if AB > 0 and a hyperbolic paraboloid if AB < 0. We summarize by the Table 2 in page 682: (1) Ellipsoid: x 2 a + y2 2 b + z2 2 c =

20 An ellipsoid becomes a sphere if a = b = c. (2) Elliptic Paraboloid z c = x2 a 2 + y2 b 2 (opening up if c > 0, down if c < 0) y b = x2 a 2 +z2 c 2 (opening towards positiv if y direction if b > 0, opposite if b < 0) 20

21 x a = y2 b 2 +z2 c 2 (opening towards positiv if x direction if a > 0, opposite if a < 0) (3) Hyperbolic Paraboloid (Saddle) z c = x2 a y2 2 b 2 21

22 (4) Cone y b = z2 c x2 2 a 2 x a = y2 b z2 2 c 2 z 2 c = x2 2 a + y2 2 b 2 y 2 b = x2 2 a + z2 2 c 2 x 2 a = y2 2 b + z2 2 c 2 (5) Hyperboloid of One Sheet x 2 a 2 + y2 b 2 z2 c 2 = 1 22

23 x 2 a 2 y2 b 2 + z2 c 2 = 1 x2 a + y2 2 b + z2 2 c = 1 2 (6) Hyperboloid of Two Sheets x 2 a 2 + y2 b 2 z2 c 2 = 1 23

24 x 2 a 2 y2 b 2 + z2 c 2 = 1 x2 a 2 + y2 b 2 + z2 c 2 = 1 Homework: 24

25 1. Find and sketch the domain of the function. p y 4x 2 (a) f (x, y) = x 2 1 (b) g (x, y) = p 4 x 2 y 2 + ln (x 2 + y 2 1) 2. Identify and sketch the trace x = k, y = k, and z = k, and then use these traces to sketch the graph of y = x 2 + 4z 2 3. Identify (i.e., spell the name, openning and axis of symmetry, if any) and sketch the graph. (a) 4x 2 + y 2 4z 2 = 4 (b) 2y 2 + z 2 + 4x = 0 (c) 4x 2 y 2 4z 2 = 4 (d) y = p 16 x 2 z 2 (hint: square both sides) (e) x = p y 2 + 2z 2 (f) x 2 + 4y 2 + 2z 2 = 4 4. Give a concrete example. (hint: square both sides) (a) An elliptical paraboloid openning to the negative x axis with x axis as its axis of symmetry. (b) One branch of a hyperboloid with two sheets whose axis of symmetry is y axis. The branch is open to the negative y axis. (c) The upper-half of a hyperboloid with one sheet whose axis of symmetry is x axis. 5. Find an equation for the surface consisting of all points P (x, y, z) for which the distance from P to the x axis is twice the distance from P to the yz plane. identify the surface. 25