DOCSLIB.ORG

Section 14.2 Horizontal Cross Sections of Graphs and Level Curves

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Section 14.2 Horizontal Cross Sections of Graphs and Level Curves Section 14.2 (3/23/08) Horizontal cross sections of graphs and level curves Overview: In the last section we analyzed graphs of functions of two variables by studying their vertical cross sections. Here we study horizontal cross sections of graphs and the associated level curves of functions. Topics: Horizontal cross sections of graphs • Level curves • Estimating function values from level curves • Topographical maps and other contour curves • Horizontal cross sections of graphs In Section 14.1 we determined the shape of the surface z = x2 + y2 in Figure 1 by studying its vertical cross sections in planes y = x and x = c perpendicular to the y- and x-axes. In the next example we look at the cross sections of this surface in horizontal planes z = c perpendicular to the z-axis. FIGURE 1 Example 1 Determine the shape of the graph of z = x2 + y2 in Figure 1 by studying its horizontal cross sections. Solution Horizontal planes have the equations z = c with constant c. Consequently, the horizontal cross sections of the surface z = x2 + y2 are given by the equations, 2 2 z = x + y ( z = c. Setting z = c in the first of these equation yields the equivalent equations for the cross section, 2 2 c = x + y (1) ( z = c. 288 Section 14.2, Horizontal cross sections of graphs and level curves p. 289 (3/23/08) If c is positive, then x2 + y2 = c is the circle of radius √c in an xy-plane and the cross section (1) of the surface is the circle of radius √c in the plane z = c with its center at c on the z-axis. If c = 0 then the curve (1) is the origin of xyz-space. The cross section is empty (has no points in it) if c is negative. Since the radius of the circular cross section at z = c increases as c> 0 increases and the plane z = c rises, the surface has the bowl shape in Figure 2. y 4 9 6 3 4 x Level curves of f(x,y)= x2 + y2 FIGURE 2 FIGURE 3 Level curves The horizontal cross sections of the surface in Figure 2 are at z = 0, 1, 2, 3,..., 10. The lowest cross section is the point at the origin. The other cross sections are circles; if we drop them down to the xy-plane, we obtain the ten concentric circles in Figure 3. The point and circles are curves on which f(x,y)= x2 + y2 is constant. They are called level curves or contour curves of the function. The function has the value 0 at the origin and the values 1, 2, 3,..., 10 on the circles in Figure 3. The numbers 3, 6, and 9 on three of the circles indicate the values of the function on those circles. To visualize the surface in Figure 2 from the level curves, imagine that the xy-plane in Figure 3 is horizontal in xyz-space, that the innermost circle in Figure 3 is lifted one unit to z = 1, the next circle is lifted two units to z = 2, the next circle, labeled “3” is lifted to z = 3, and so forth. This gives the horizontal cross sections of the surface which determine its shape in Figure 3. It is because the vertical cross sections of the surface z = x2 + y2 in Figure 2 are parabolas and its horizontal cross sections are circles, that it is called a circular paraboloid. This is a special type of elliptical paraboloid, whose horizontal cross sections are ellipses. p. 290 (3/23/08) Section 14.2, Horizontal cross sections of graphs and level curves Here is a general definition of level curves. Definition 1 The level curves (contour curves) of z = f(x,y) are the curves in the xy-plane where the function is constant (Figure 4). They have the equations f(x,y)= c with constants c. FIGURE 4 Example 2 Determine the shape of the surface z = y2 x2 in Figure 5 by studying its horizontal − cross sections. FIGURE 5 Section 14.2, Horizontal cross sections of graphs and level curves p. 291 (3/23/08) Solution The horizontal cross sections of the surface z = y2 x2 are given by the equations, − 2 2 z = y x − ( z = c. Setting z = c in the first equation gives the equivalent equations, 2 2 c = y x − (2) ( z = c. 2 2 For c = 0, the first of equations (2) reads y x = 0 and gives y = x. The − ± level curve consists of the lines y = x and y = x in Figure 6. − y2 x2 = c y2 x2 = c y2 x2 = c y − y − y − c = 0 c> 0 c< 0 x x x y = x y = √c + x2 x = c + y2 ± ± ± − p FIGURE6 FIGURE7 FIGURE8 2 2 For c > 0, the first of equations (2) gives y = c + x and then y = √c + x2. ± The level curve consists of the two halves of the hyperbola in Figure 7. For c < 0, we obtain x2 = c + y2 and then x = c + y2. The level − ± − curve consists of the two halves of the hyperbola in Figure 8. The level curves for c = 0, 1, 2, 4 and 6 are shown in Figure 9. p ± ± ± ± p. 292 (3/23/08) Section 14.2, Horizontal cross sections of graphs and level curves The horizontal cross sections on the surface in Figure 10 are obtained from the level curves in a horizontal xy-plane in Figure 9 by raising the level curve y2 x2 = c − for c = 0, 1, 2, 3,... 6 up c units and lowering the level curve for c = 1, 2, 3,..., 6 − − − − down c units. This gives the surface the shape in Figure 10. | | y 4 3 0 0 −3 −3 4 x 3 Level curves of z = y2 x2 − FIGURE 9 FIGURE 10 The surface in Figure 10 is called a hyperbolic parabola because its vertical cross sections are parabolas and its horizontal cross sections are hyperbolas. Example 3 Figure 11 shows the “boot-shaped” graph of the function h(x,y)= y 1 y3 1 x2 with − 12 − 4 the vertical cross sections that we found in Section 14.1 and with its horizontal cross sections at the integer values z = 5, 4, 3,..., 4, 5. The corresponding level curves − − − of h are in Figure 11. Explain how the level curves relate to the graph. y z = 1 6 z = −5 z = −1 2 2 6 x z = −1 z = 1 z = 5 (To be redrawn) FIGURE11 FIGURE12 Section 14.2, Horizontal cross sections of graphs and level curves p. 293 (3/23/08) Solution Imagine that the surface is sitting in a container that is being filled with water. The bottom edge of the surface is at z = 5 and has the shape of the top level curve in − Figure 12. As the water rises to z = 4, z = 3, and z = 2, it reaches the next three − − − cross sections in Figure 12, which have the shape of the next three level curves from the top in Figure 11. At z = 1 the water line is in two parts. One part goes around the “toe” of the − boot; the other is on the “instep” of the boot. The corresponding level curve is in the two parts labeled z = 1 in Figure 12. The next two curves, at z = 0 and z = 1, are − also in two parts. Finally, the highest cross sections, at z = 2, 3, 4, and 5, are only on the “shin” of the boot. They correspond to the bottom four level curves in Figure 11. Estimating function values from level curves Level curves of a function, as in Figure 13, show where the function has each of the z-values for the given curves, and we can estimate the function’s values at other points from values on nearby level curves, y 3 80 63 3 60 6 x Level curves of P − − FIGURE 13 3 40 − Example 4 The level curves in Figure 13 are of the function z = P (x,y). What are the approximate values of (a) P (0, 1) and (b) P ( 2, 1)? − − Solution (a) The level curves in Figure 14 are where P has the values 40, 50, 60, 70, and 80. The point (0, 1) on the negative y-axis is on the the level curve where P = 50, so − P (0, 1) = 50. − (b) The point ( 2, 1) on the upper left in Figure 15 is about halfway between the − level curves where P = 70 and P = 80. We conclude that P ( 2, 1) 75. (Different − ≈ estimates give different answers.) y 3 80 63 3 60 6 x − − FIGURE 14 3 40 − p. 294 (3/23/08) Section 14.2, Horizontal cross sections of graphs and level curves Topographical maps and other contour curves The next example involves a topographical map showing level curves of the elevation on the mountain. Example 5 Figure 15 shows a topographical map of Mt. Shasta in northern California and Figure 16 is a photograph of the mountain. The numbers on the contour curves are the elevation above sea level. What is the approximate elevation of the highest point on the mountain? Solution The contour curves drawn with heavy lines in Figure 14 are 2,500 feet apart, and the other curves represent elevations 500 feet apart. The summit, therefore, is approxi- mately 14,000 feet above sea level. FIGURE15 FIGURE16 Weather maps show isotherms, which are level curves of temperature, and isobars, which are level curves of barometric pressure.
Load more

Recommended publications
  • Quadratic Approximation at a Stationary Point Let F(X, Y) Be a Given
    Multivariable Calculus Grinshpan Quadratic approximation at a stationary point Let f(x; y) be a given function and let (x0; y0) be a point in its domain. Under proper differentiability conditions one has f(x; y) = f(x0; y0) + fx(x0; y0)(x − x0) + fy(x0; y0)(y − y0) 1 2 1 2 + 2 fxx(x0; y0)(x − x0) + fxy(x0; y0)(x − x0)(y − y0) + 2 fyy(x0; y0)(y − y0) + higher−order terms: Let (x0; y0) be a stationary (critical) point of f: fx(x0; y0) = fy(x0; y0) = 0. Then 2 2 f(x; y) = f(x0; y0) + A(x − x0) + 2B(x − x0)(y − y0) + C(y − y0) + higher−order terms, 1 1 1 1 where A = 2 fxx(x0; y0);B = 2 fxy(x0; y0) = 2 fyx(x0; y0), and C = 2 fyy(x0; y0). Assume for simplicity that (x0; y0) = (0; 0) and f(0; 0) = 0. [ This can always be achieved by translation: f~(x; y) = f(x0 + x; y0 + y) − f(x0; y0). ] Then f(x; y) = Ax2 + 2Bxy + Cy2 + higher−order terms. Thus, provided A, B, C are not all zero, the graph of f near (0; 0) resembles the quadric surface z = Ax2 + 2Bxy + Cy2: Generically, this quadric surface is either an elliptic or a hyperbolic paraboloid. We distinguish three scenarios: * Elliptic paraboloid opening up, (0; 0) is a point of local minimum. * Elliptic paraboloid opening down, (0; 0) is a point of local maximum. * Hyperbolic paraboloid, (0; 0) is a saddle point. It should certainly be possible to tell which case we are dealing with by looking at the coefficients A, B, and C, and this is the idea behind the Second Partials Test.
    [Show full text]
  • Chapter 11. Three Dimensional Analytic Geometry and Vectors
    Chapter 11. Three dimensional analytic geometry and vectors. Section 11.5 Quadric surfaces. Curves in R2 : x2 y2 ellipse + =1 a2 b2 x2 y2 hyperbola − =1 a2 b2 parabola y = ax2 or x = by2 A quadric surface is the graph of a second degree equation in three variables. The most general such equation is Ax2 + By2 + Cz2 + Dxy + Exz + F yz + Gx + Hy + Iz + J =0, where A, B, C, ..., J are constants. By translation and rotation the equation can be brought into one of two standard forms Ax2 + By2 + Cz2 + J =0 or Ax2 + By2 + Iz =0 In order to sketch the graph of a quadric surface, it is useful to determine the curves of intersection of the surface with planes parallel to the coordinate planes. These curves are called traces of the surface. Ellipsoids The quadric surface with equation x2 y2 z2 + + =1 a2 b2 c2 is called an ellipsoid because all of its traces are ellipses. 2 1 x y 3 2 1 z ±1 ±2 ±3 ±1 ±2 The six intercepts of the ellipsoid are (±a, 0, 0), (0, ±b, 0), and (0, 0, ±c) and the ellipsoid lies in the box |x| ≤ a, |y| ≤ b, |z| ≤ c Since the ellipsoid involves only even powers of x, y, and z, the ellipsoid is symmetric with respect to each coordinate plane. Example 1. Find the traces of the surface 4x2 +9y2 + 36z2 = 36 1 in the planes x = k, y = k, and z = k. Identify the surface and sketch it. Hyperboloids Hyperboloid of one sheet. The quadric surface with equations x2 y2 z2 1.
    [Show full text]
  • Calculus & Analytic Geometry
    TQS 126 Spring 2008 Quinn Calculus & Analytic Geometry III Quadratic Equations in 3-D Match each function to its graph 1. 9x2 + 36y2 +4z2 = 36 2. 4x2 +9y2 4z2 =0 − 3. 36x2 +9y2 4z2 = 36 − 4. 4x2 9y2 4z2 = 36 − − 5. 9x2 +4y2 6z =0 − 6. 9x2 4y2 6z =0 − − 7. 4x2 + y2 +4z2 4y 4z +36=0 − − 8. 4x2 + y2 +4z2 4y 4z 36=0 − − − cone • ellipsoid • elliptic paraboloid • hyperbolic paraboloid • hyperboloid of one sheet • hyperboloid of two sheets 24 TQS 126 Spring 2008 Quinn Calculus & Analytic Geometry III Parametric Equations (§10.1) and Vector Functions (§13.1) Definition. If x and y are given as continuous function x = f(t) y = g(t) over an interval of t-values, then the set of points (x, y)=(f(t),g(t)) defined by these equation is a parametric curve (sometimes called aplane curve). The equations are parametric equations for the curve. Often we think of parametric curves as describing the movement of a particle in a plane over time. Examples. x = 2cos t x = et 0 t π 1 t e y = 3sin t ≤ ≤ y = ln t ≤ ≤ Can we find parameterizations of known curves? the line segment circle x2 + y2 =1 from (1, 3) to (5, 1) Why restrict ourselves to only moving through planes? Why not space? And why not use our nifty vector notation? 25 Definition. If x, y, and z are given as continuous functions x = f(t) y = g(t) z = h(t) over an interval of t-values, then the set of points (x,y,z)= (f(t),g(t), h(t)) defined by these equation is a parametric curve (sometimes called a space curve).
    [Show full text]
  • Chapter 3 Quadratic Curves, Quadric Surfaces
    Chapter 3 Quadratic curves, quadric surfaces In this chapter we begin our study of curved surfaces. We focus on the quadric surfaces. To do this, we also need to look at quadratic curves, such as ellipses. We discuss: Equations and parametric descriptions of the plane quadratic curves: circles, ellipses, ² hyperbolas and parabolas. Equations and parametric descriptions of quadric surfaces, the 2{dimensional ana- ² logues of quadratic curves. We also discuss aspects of matrices, since they are relevant for our discussion. 3.1 Plane quadratic curves 3.1.1 From linear to quadratic equations Lines in the plane R2 are represented by linear equations and linear parametric descriptions. Degree 2 equations also correspond to curves you undoubtedly have come across before: circles, ellipses, hyperbolas and parabolas. This section is devoted to these curves. They will reoccur when we consider quadric surfaces, a class of fascinating shapes, since the intersection of a quadric surface with a plane consists of a quadratic curve. Lines di®er from quadratic curves in various respects, one of which is that all lines look the same (only their position in the plane may di®er), but that quadratic curves may truely di®er in shape. 3.1.2 The general equation of a quadratic curve The general equation of a line in R2 is ax + by = c. When we also allow terms of degree 2 in the variables x and y, i.e., x2, xy and y2, we obtain quadratic equations like x2 + y2 = 1, a circle. ² x2 + 2x + y = 3, a parabola; probably you recognize it as such if it is rewritten in the ² form y = 3 x2 2x or y = (x + 1)2 + 4.
    [Show full text]
  • Lesson 9: Volume and Cavalieri's Principle
    NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 9 M3 PRECALCULUS AND ADVANCED TOPICS Lesson 9: Volume and Cavalieri’s Principle Student Outcomes . Students will be able to give an informal argument using Cavalieri’s principle for the formula for the volume of a sphere and other solid figures (G-GMD.A.2). Lesson Notes The opening uses the idea of cross sections to establish a connection between the current lesson and the previous lessons. In particular, ellipses and hyperbolas are seen as cross sections of a cone, and Cavalieri’s volume principle is based on cross-sectional areas. This principle is used to explore the volume of pyramids, cones, and spheres. Classwork Opening (2 minutes) Scaffolding: . A cutout of a cone is available in Geometry, Module 3, Lesson 7 to make picturing this exercise easier. Use the cutout to model determining that a circle is a possible cross section of a cone. “Conic Sections” by Magister Mathematicae is licensed under CC BY-SA 3.0 http://creativecommons.org/licenses/by-sa/3.0/deed.en In the previous lesson, we saw how the ellipse, parabola, and hyperbola can come together in the context of a satellite orbiting a body such as a planet; we learned that the velocity of the satellite determines the shape of its orbit. Another context in which these curves and a circle arise is in slicing a cone. The intersection of a plane with a solid is called a cross section of the solid. Lesson 9: Volume and Cavalieri’s Principle Date: 2/9/15 144 This work is licensed under a © 2015 Common Core, Inc.
    [Show full text]
  • Math 16C: Multivariate Calculus
    MATH 16C: MULTIVARIATE CALCULUS Jes´us De Loera, UC Davis April 12, 2010 Jes´us De Loera, UC Davis MATH 16C: MULTIVARIATE CALCULUS 7.1: The 3-dimensional coordinate system Jes´us De Loera, UC Davis MATH 16C: MULTIVARIATE CALCULUS 2 2 2 Thus, d = (x2 x1) + (y2 y1) + (z2 z1) . EXAMPLE: − − − Find the distancep between ( 3, 2, 5) and (4, 1, 2). − − d = (( 3) 4)2 + (2 ( 1))2 + (5 2)2 = √49+9+9= √67. q − − − − − Jes´us De Loera, UC Davis MATH 16C: MULTIVARIATE CALCULUS Mid-point between two points What is the mid-point between two points (x1, y1, z1) and (x2, y2, z2)? Let (xm, ym, zm) be the mid-point. Then xm must be half-way x1+x1 bewteen x1 and x2, so xm = 2 . y1+y2 Also, ym must be half-way between y1 and y2, so ym = 2 Similarly, zm must be half-way between z1 and z2, so z1+z2 zm = 2 . Thus, x1 + x1 y1 + y2 z1 + z2 (xm, ym, zm)= , , . 2 2 2 E.g. What is the midpoint bewteen ( 3, 2, 5) and (4, 1, 2)? − ( 3) + 4 2 + 1 5 + 2 1 3 1 (xm, ym, zm)= − , , = , , . 2 2 2 2 2 2 Jes´us De Loera, UC Davis MATH 16C: MULTIVARIATE CALCULUS Question: If we fix a point (a, b, c), what is the set of all points at a distance r from (a, b, c) called? Answer: A SPHERE centered at (a, b, c). (x a)2 + (y b)2 + (z c)2 = r (x a)2+(y b)2+(z c)2 = r 2.
    [Show full text]
  • Calculus III -- Fall 2008
    Section 12.2: Quadric Surfaces Goals : 1. To recognize and write equations of quadric surfaces 2. To graph quadric surfaces by hand Definitions: 1. A quadric surface is the three-dimensional graph of an equation that can (through appropriate transformations, if necessary), be written in either of the following forms: Ax 2 + By 2 + Cz 2 + J = 0 or Ax 2 + By 2 + Iz = 0 . 2. The intersection of a surface with a plane is called a trace of the surface in the plane. Notes : 1. There are 6 kinds of quadric surfaces. Scroll down to get an idea of what they look like. Keep in mind that each graph shown illustrates just one of many possible orientations of the surface. 2. The traces of quadric surfaces are conic sections (i.e. a parabola, ellipse, or hyperbola). 3. The key to graphing quadric surfaces is making use of traces in planes parallel to the xy , xz , and yz planes. 4. The following pages are from the lecture notes of Professor Eitan Angel, University of Colorado. Keep scrolling down (or press the Page Down key) to advance the slide show. Calculus III { Fall 2008 Lecture{QuadricSurfaces Eitan Angel University of Colorado Monday, September 8, 2008 E. Angel (CU) Calculus III 8 Sep 1 / 11 Now we will discuss second-degree equations (called quadric surfaces). These are the three dimensional analogues of conic sections. To sketch the graph of a quadric surface (or any surface), it is useful to determine curves of intersection of the surface with planes parallel to the coordinate planes.
    [Show full text]
  • Tilted Planes and Curvature in Three-Dimensional Space: Explorations of Partial Derivatives
    Tilted Planes and Curvature in Three-Dimensional Space: Explorations of Partial Derivatives By Andrew Grossfield, Ph. D., P. E. Abstract Many engineering students encounter and algebraically manipulate partial derivatives in their fluids, thermodynamics or electromagnetic wave theory courses. However it is possible that unless these students were properly introduced to these symbols, they may lack the insight that could be obtained from a geometric or visual approach to the equations that contain these symbols. We accept the approach that just as the direction of a curve at a point in two-dimensional space is described by the slope of the straight line tangent to the curve at that point, the orientation of a surface at a point in 3-dimensional space is determined by the orientation of the plane tangent to the surface at that point. A straight line has only one direction described by the same slope everywhere along its length; a tilted plane has the same orientation everywhere but has many slopes at each point. In fact the slope in almost every direction leading away from a point is different. How do we conceive of these differing slopes and how can they be evaluated? We are led to conclude that the derivative of a multivariable function is the gradient vector, and that it is wrong and misleading to define the gradient in terms of some kind of limiting process. This approach circumvents the need for all the unnecessary delta-epsilon arguments about obvious results, while providing visual insight into properties of multi-variable derivatives and differentials. This paper provides the visual connection displaying the remarkably simple and beautiful relationships between the gradient, the directional derivatives and the partial derivatives.
    [Show full text]
  • Multivariable Calculus Math 21A
    Multivariable Calculus Math 21a Harvard University Spring 2004 Oliver Knill These are some class notes distributed in a multivariable calculus course tought in Spring 2004. This was a physics flavored section. Some of the pages were developed as complements to the text and lectures in the years 2000-2004. While some of the pages are proofread pretty well over the years, others were written just the night before class. The last lecture was "calculus beyond calculus". Glued with it after that are some notes from "last hours" from previous semesters. Oliver Knill. 5/7/2004 Lecture 1: VECTORS DOT PRODUCT O. Knill, Math21a VECTOR OPERATIONS: The ad- ~u + ~v = ~v + ~u commutativity dition and scalar multiplication of ~u + (~v + w~) = (~u + ~v) + w~ additive associativity HOMEWORK: Section 10.1: 42,60: Section 10.2: 4,16 vectors satisfy "obvious" properties. ~u + ~0 = ~0 + ~u = ~0 null vector There is no need to memorize them. r (s ~v) = (r s) ~v scalar associativity ∗ ∗ ∗ ∗ VECTORS. Two points P1 = (x1; y1), Q = P2 = (x2; y2) in the plane determine a vector ~v = x2 x1; y2 y1 . We write here for multiplication (r + s)~v = ~v(r + s) distributivity in scalar h − − i ∗ It points from P1 to P2 and we can write P1 + ~v = P2. with a scalar but usually, the multi- r(~v + w~) = r~v + rw~ distributivity in vector COORDINATES. Points P in space are in one to one correspondence to vectors pointing from 0 to P . The plication sign is left out. 1 ~v = ~v the one element numbers ~vi in a vector ~v = (v1; v2) are also called components or of the vector.
    [Show full text]
  • Quadric Surfaces the University of Kansas 1 / 23 2 2 2 X  Y  Ellipse of Semi-Major, Semi-Minor Axes A, B: a + B = 1
    1 circle of radius r: x 2 + y 2 = r 2 2 2 2 x y ellipse of semi-major, semi-minor axes a; b: a + b = 1 2 2 3 x y hyperbola of semi-major, semi-minor axes a; b: a − b = 1 4 parabola: y = ax 2 Considering these conic sections as three-dimensional objects by letting z vary, we obtain the family of cylinders. Conic Sections Recall the usual conic sections from the Cartesian plane. MATH 127 (Section 12.6) Quadric Surfaces The University of Kansas 1 / 23 2 2 2 x y ellipse of semi-major, semi-minor axes a; b: a + b = 1 2 2 3 x y hyperbola of semi-major, semi-minor axes a; b: a − b = 1 4 parabola: y = ax 2 Considering these conic sections as three-dimensional objects by letting z vary, we obtain the family of cylinders. Conic Sections Recall the usual conic sections from the Cartesian plane. 1 circle of radius r: x 2 + y 2 = r 2 MATH 127 (Section 12.6) Quadric Surfaces The University of Kansas 1 / 23 2 2 3 x y hyperbola of semi-major, semi-minor axes a; b: a − b = 1 4 parabola: y = ax 2 Considering these conic sections as three-dimensional objects by letting z vary, we obtain the family of cylinders. Conic Sections Recall the usual conic sections from the Cartesian plane. 1 circle of radius r: x 2 + y 2 = r 2 2 2 2 x y ellipse of semi-major, semi-minor axes a; b: a + b = 1 MATH 127 (Section 12.6) Quadric Surfaces The University of Kansas 1 / 23 4 parabola: y = ax 2 Considering these conic sections as three-dimensional objects by letting z vary, we obtain the family of cylinders.
    [Show full text]
  • Sect. 13.6 (Cylinders & Quadric Surfaces)
    Sect. 13.6 (Cylinders & Quadric Surfaces) —MATH 2421 Kawai Sect. 13.6 (Cylinders & Quadric Surfaces) 1. Cylindrical surfaces. If we have the circle in the xy-plane: x2 + y2 = 4, then what happens if we use this equation in 3D? Answer: We say that z can do anything. Thus, we obtain a surface, and the only constraint is that the projection down onto the xy-plane must be that circle. We get the infinte tennis ball can. We say that the generating curve is in the xy-plane. Thus, the equation will only have x and y in it, and z is allowed to take on any real value. We say that this is a “right”circular cylinder because the generating curve is copied at right angles to the plane that contains the generating curve. This copying process is called extrusion, and we say that the right cylinder is extruded from the xy-plane. We create rigatoni pasta this way. In this course, we only care about cylindrical surfaces that are orthogonal to simple planes. Thus, we have two more possible orientations: x2 + z2 = 4 AND y2 + z2 = 4. [You try it!] 2. Cylindrical surfaces can come in a variety of shapes. (a) Parabolic cylinders: y = x2, x = y2, etc. There are 6 possible orientations. (b) Sinusoidal cylinders: y = sin (x) , etc. (c) Planes can be regarded as cylindrical surfaces (walls). The line y = x is in the xy-plane, but the plane y = x is a vertical plane, orthogonal to the xy-plane. In theory, this could be classified as a right cylindrical wall.
    [Show full text]
  • Second Partials Test Let F(X, Y) Be a Smooth Function and Let (X0,Y0)
    Multivariate Calculus Grinshpan Second Partials Test Let f(x; y) be a smooth function and let (x0; y0) be a point in its domain. It is possible to write f(x; y) = f(x0; y0) + fx(x0; y0)(x − x0) + fy(x0; y0)(y − y0) 1 2 1 2 + 2 fxx(x0; y0)(x−x0) +fxy(x0; y0)(x−x0)(y−y0)+ 2 fyy(x0; y0)(y−y0) + higher order terms: If (x0; y0) is a stationary point of f then df = 0 there and we have 1 2 1 2 f(x; y) = f(x0; y0) + 2 fxx(x0; y0)(x−x0) +fxy(x0; y0)(x−x0)(y−y0)+ 2 fyy(x0; y0)(y−y0) + higher order terms: Assume for simplicity that (x0; y0) = (0; 0) and f(0; 0) = 0. 0 0 0 0 (This can always be done by translating variables: f~(x ; y ) = f(x0 + x ; y0 + y ) − f(x0; y0).) Then letting 1 1 1 1 A = 2 fxx(0; 0);B = 2 fxy(0; 0) = 2 fyx(0; 0);C = 2 fyy(0; 0); we see that f(x; y) = Ax2 + 2Bxy + Cy2 + higher order terms: Thus the graph of f near (0; 0) resembles the quadric surface z = Ax2 + 2Bxy + Cy2: This quadric surface can either be an elliptic or a hyperbolic paraboloid. If it is an elliptic paraboloid opening up, (0; 0) is a point of local minimum. If it is an elliptic paraboloid opening down, (0; 0) is a point of local maximum. If it is a hyperbolic paraboloid, (0; 0) is a saddle point.
    [Show full text]