Quadric Surfaces. Six basic types of quadric surfaces: ellipsoid. cone. elliptic paraboloid. hyperboloid of one sheet. hyperboloid of two sheets
|
|
- Myrtle Francis
- 6 years ago
- Views:
Transcription
1 Quadric Surfaces Six basic types of quadric surfaces: ellipsoid cone elliptic paraboloid hyperboloid of one sheet hyperboloid of two sheets hyperbolic paraboloid (A) (B) (C) (D) (E) (F) 1. For each surface, describe the traces of the surface in x = k, y = k, and z = k. Then pick the term from the list above which seems to most accurately describe the surface (we haven t learned any of these terms yet, but you should be able to make a good educated guess), and pick the correct picture of the surface. (a) x2 y2 = z. 1
2 Traces in x = k: parabolas Traces in y = k: parabolas Traces in z = k: hyperbolas (possibly a pair of lines) Solution. The trace in x = k of the surface is z = y2 parabola. The trace in y = k of the surface is z = x2 +, which is a downward-opening, which is an upward-opening parabola. The trace in z = k of the surface is x2 (a degenerate hyperbola) if k = 0. y2 = k, which is a hyperbola if k 0 and a pair of lines This surface is called a hyperbolic paraboloid, and it looks like picture (E). It is also sometimes called a saddle. (b) x2 + y z2 = 1. Traces in x = k: ellipses (possibly a point) or nothing Traces in y = k: ellipses (possibly a point) or nothing Traces in z = k: ellipses (possibly a point) or nothing Solution. The trace in x = k of the surface is y z2 = 1. Since 1 can be positive, zero, or negative (depending on what k is), this intersection can be an ellipse, a point (a degenerate ellipse), or nothing. The trace in y = k of the surface is x2 + z2 = 1 25, which is an ellipse, a point, or nothing. The trace in z = k of the surface is x2 + y2 25 = 1, which is an ellipse, a point, or nothing. The picture which matches this description is (A), and this surface is an ellipsoid. (c) x2 + y2 = z 2. Traces in x = k: parabolas Traces in y = k: parabolas Traces in z = k: ellipses (possibly a point) or nothing Solution. The trace in x = k of this surface is z = 2y2 The trace in y = k of this surface is z = x , which is a parabola., which is a parabola. The trace in z = k of this surface is x2 + y2 = k 2, which is an ellipse when k > 0. If k = 0, it is a point (a degenerate ellipse), and if k < 0, it is nothing. This matches picture (C), and this surface is called an elliptic paraboloid. (d) z2 x2 y2 = 1. 2
3 Traces in x = k: hyperbolas (never a pair of lines) Traces in y = k: hyperbolas (never a pair of lines) Traces in z = k: ellipses (possibly a point) or nothing Solution. The trace in x = k of this surface is z2 1 + k 2 is always positive, this is never a pair of lines. y2 = 1 +, which is a hyperbola. Since The trace in y = k of this surface is z2 x2 = 1 +, which is a hyperbola (and never a pair of lines). The trace in z = k of this surface is x 2 + y2 = 1. Since negative, this intersection can be an ellipse, a point, or nothing. 1 can be positive, zero, or The pictures (B), (D), and (F) all show surfaces whose traces in x = k and y = k are hyperbolas and whose traces in z = k are ellipses. Of these three graphs, (D) is the only one in which some traces in z = k are nothing, so the correct picture is (D). This surface is an example of a hyperboloid of two sheets. (e) x 2 + y2 = z2. Traces in x = k: hyperbolas (possibly a pair of lines) Traces in y = k: hyperbolas (possibly a pair of lines) Traces in z = k: ellipses (possibly a point) Solution. The trace in x = k of this surface is z2 y2 =. Since k 2 can be either non-zero or zero, this is either a hyperbola or a pair of lines. (It is only a pair of lines when k = 0; otherwise, it is a hyperbola.) The trace in y = k of this surface is z2 x2 =, which is again either a hyperbola or a pair of lines (a pair of lines only when k = 0). The trace in z = k of this surface is x 2 + y2 =. Since is either positive or 0, this intersection is either an ellipse or a point (but never nothing, and it is only a point when k = 0). This matches picture (F) and is an example of an cone. (f) x2 + y2 z2 = 1. Traces in x = k: hyperbolas (possibly a pair of lines) Traces in y = k: hyperbolas (possibly a pair of lines) Traces in z = k: ellipses (never a point) Solution. The trace in x = k of this surface is y 2 z2 = 1. Since 1 or zero, this is either a hyperbola or a pair of lines. can be non-zero The trace in y = k of this surface is x2 lines. z2 = 1, which is either a hyperbola or a pair of 3
4 The trace in z = k of this surface is x2 + y2 = 1 +. Since 1 + always an ellipse (and never a point or nothing). is always positive, this is This description matches picture (B), and this surface is a hyperboloid of one sheet. 2. Sketch the surface y 2 + z 2 = 36. What type of quadric surface is it? Solution. Notice that the equation describing this surface does not involve x at all. Each trace of the surface in x = k is the ellipse y 2 + z 2 = 36, which can also be written as y2 + z2 = 1. Therefore, the surface looks like this: This is not any of the quadrics we have seen so far; it is a (quadric) cylinder. 3. Sketch the surface y 2 + 2y + z 2 = x 2. What type of quadric surface is it? Solution. We have a y 2 term as well as a y term, and we can make the equation simpler by completing the square. In this case, we accomplish that by adding 1 to both sides of the equation: To sketch, let s look at the traces: y 2 + 2y z 2 = x (y + 1) 2 + z 2 = x x 2 + (y + 1) 2 + z 2 = 1 The trace in x = k of the surface is (y + 1) 2 + z 2 = 1 + k 2, which describes a circle (centered at y = 1, z = 0 and having radius 1 + k 2 ). The trace in y = k of the surface is x 2 + z 2 = 1 (k + 1) 2, which is a hyperbola (or a pair of lines). The trace in z = k of the surface x 2 + (y + 1) 2 = 1 k 2, which is a hyperbola (or a pair of lines). Since we know the traces in x = k are circles centered at y = 1, z = 0, we can think of the surface as being obtained by rotating some sort of curve about the line y = 1, z = 0 (this line is parallel to the x-axis). To figure out what that curve is, we can look at the trace of the surface in either y = 1 or z = 0. Let s do the trace in z = 0: this is (y + 1) 2 x 2 = 1, which is the following hyperbola:
5 y x If we draw this in space (in the plane z = 0, since we re talking about the trace in z = 0), it looks like the left figure below. Therefore, the surface looks like the right figure. z x y This is a hyperboloid of one sheet.. What type of quadric surface is x 2 y 2 + z 2 + = 0? Solution. Let s look at the traces: The trace in x = k of this surface is y 2 z 2 = k 2 +, which is always a hyperbola. The trace in y = k of this surface is x 2 + z 2 = k 2, which is an ellipse, a point, or nothing, depending on k. The trace in z = k of this surface is y 2 x 2 = k 2 +, which is always a hyperbola. Since the traces in x = k and z = k are hyperbolas, we know the surface is either a hyperboloid or a cone. The fact that the traces in y = k can be nothing tells us that the surface must be a hyperboloid of two sheets. The surface looks like this: 5
6 6
4 = 1 which is an ellipse of major axis 2 and minor axis 2. Try the plane z = y2
12.6 Quadrics and Cylinder Surfaces: Example: What is y = x? More correctly what is {(x,y,z) R 3 : y = x}? It s a plane. What about y =? Its a cylinder surface. What about y z = Again a cylinder surface
More informationCylinders and Quadric Surfaces A cylinder is a three dimensional shape that is determined by
Cylinders and Quadric Surfaces A cylinder is a three dimensional shape that is determined by a two dimensional (plane) curve C in three dimensional space a line L in a plane not parallel to the one in
More informationQuadric Surfaces. Philippe B. Laval. Today KSU. Philippe B. Laval (KSU) Quadric Surfaces Today 1 / 24
Quadric Surfaces Philippe B. Laval KSU Today Philippe B. Laval (KSU) Quadric Surfaces Today 1 / 24 Introduction A quadric surface is the graph of a second degree equation in three variables. The general
More informationQuadric Surfaces. Philippe B. Laval. Spring 2012 KSU. Philippe B. Laval (KSU) Quadric Surfaces Spring /
.... Quadric Surfaces Philippe B. Laval KSU Spring 2012 Philippe B. Laval (KSU) Quadric Surfaces Spring 2012 1 / 15 Introduction A quadric surface is the graph of a second degree equation in three variables.
More informationSection 12.2: Quadric Surfaces
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
More information12.6 Cylinders and Quadric Surfaces
12 Vectors and the Geometry of Space 12.6 and Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. and We have already looked at two special types of surfaces:
More informationWhat you will learn today
What you will learn today Conic Sections (in 2D coordinates) Cylinders (3D) Quadric Surfaces (3D) Vectors and the Geometry of Space 1/24 Parabolas ellipses Hyperbolas Shifted Conics Conic sections result
More informationKey Idea. It is not helpful to plot points to sketch a surface. Mainly we use traces and intercepts to sketch
Section 12.7 Quadric surfaces 12.7 1 Learning outcomes After completing this section, you will inshaallah be able to 1. know what are quadric surfaces 2. how to sketch quadric surfaces 3. how to identify
More information1.6 Quadric Surfaces Brief review of Conic Sections 74 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE. Figure 1.18: Parabola y = 2x 2
7 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE Figure 1.18: Parabola y = x 1.6 Quadric Surfaces Figure 1.19: Parabola x = y 1.6.1 Brief review of Conic Sections You may need to review conic sections for
More informationDemo of some simple cylinders and quadratic surfaces
Demo of some simple cylinders and quadratic surfaces Yunkai Zhou Department of Mathematics Southern Methodist University (Prepared for Calculus-III, Math 2339) Acknowledgement: The very nice free software
More informationChapter 10. Exploring Conic Sections
Chapter 10 Exploring Conic Sections Conics A conic section is a curve formed by the intersection of a plane and a hollow cone. Each of these shapes are made by slicing the cone and observing the shape
More informationSection 2.5. Functions and Surfaces
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
More informationQuadric surface. Ellipsoid
Quadric surface Quadric surfaces are the graphs of any equation that can be put into the general form 11 = a x + a y + a 33z + a1xy + a13xz + a 3yz + a10x + a 0y + a 30z + a 00 where a ij R,i, j = 0,1,,
More informationChapter 15: Functions of Several Variables
Chapter 15: Functions of Several Variables Section 15.1 Elementary Examples a. Notation: Two Variables b. Example c. Notation: Three Variables d. Functions of Several Variables e. Examples from the Sciences
More informationMath 126C: Week 3 Review
Math 126C: Week 3 Review Note: These are in no way meant to be comprehensive reviews; they re meant to highlight the main topics and formulas for the week. Doing homework and extra problems is always the
More informationConic Sections: Parabolas
Conic Sections: Parabolas Why are the graphs of parabolas, ellipses, and hyperbolas called 'conic sections'? Because if you pass a plane through a double cone, the intersection of the plane and the cone
More informationALGEBRA II UNIT X: Conic Sections Unit Notes Packet
Name: Period: ALGEBRA II UNIT X: Conic Sections Unit Notes Packet Algebra II Unit 10 Plan: This plan is subject to change at the teacher s discretion. Section Topic Formative Work Due Date 10.3 Circles
More informationFunctions of Several Variables
. Functions of Two Variables Functions of Several Variables Rectangular Coordinate System in -Space The rectangular coordinate system in R is formed by mutually perpendicular axes. It is a right handed
More informationMAT175 Overview and Sample Problems
MAT175 Overview and Sample Problems The course begins with a quick review/overview of one-variable integration including the Fundamental Theorem of Calculus, u-substitutions, integration by parts, and
More informationRewrite the equation in the left column into the format in the middle column. The answers are in the third column. 1. y 4y 4x 4 0 y k 4p x h y 2 4 x 0
Pre-Calculus Section 1.1 Completing the Square Rewrite the equation in the left column into the format in the middle column. The answers are in the third column. 1. y 4y 4x 4 0 y k 4p x h y 4 x 0. 3x 3y
More informationParametrization. Surface. Parametrization. Surface Integrals. Dr. Allen Back. Nov. 17, 2014
Dr. Allen Back Nov. 17, 2014 Paraboloid z = x 2 + 4y 2 The graph z = F (x, y) can always be parameterized by Φ(u, v) =< u, v, F (u, v) >. Paraboloid z = x 2 + 4y 2 The graph z = F (x, y) can always be
More informationWe will be sketching 3-dimensional functions. You will be responsible for doing this both by hand and with Mathematica.
Review polar coordinates before 9.7. Section 9.6 Functions and Surfaces We will be sketching 3-dimensional functions. You will be responsible for doing this both b hand and with Mathematica. Remember:
More information13.1 2/20/2018. Conic Sections. Conic Sections: Parabolas and Circles
13 Conic Sections 13.1 Conic Sections: Parabolas and Circles 13.2 Conic Sections: Ellipses 13.3 Conic Sections: Hyperbolas 13.4 Nonlinear Systems of Equations 13.1 Conic Sections: Parabolas and Circles
More informationLecture 34: Curves defined by Parametric equations
Curves defined by Parametric equations When the path of a particle moving in the plane is not the graph of a function, we cannot describe it using a formula that express y directly in terms of x, or x
More informationChapter 3. Quadric hypersurfaces. 3.1 Quadric hypersurfaces Denition.
Chapter 3 Quadric hypersurfaces 3.1 Quadric hypersurfaces. 3.1.1 Denition. Denition 1. In an n-dimensional ane space A; given an ane frame fo;! e i g: A quadric hypersurface in A is a set S consisting
More informationChapter 9. Linear algebra applications in geometry
Chapter 9. Linear algebra applications in geometry C.O.S. Sorzano Biomedical Engineering August 25, 2013 9. Linear algebra applications in geometry August 25, 2013 1 / 73 Outline 9 Linear algebra applications
More informationChapter 10 Test Review
Name: Class: Date: Chapter 10 Test Review Short Answer 1. Write an equation of a parabola with a vertex at the origin and a focus at ( 2, 0). 2. Write an equation of a parabola with a vertex at the origin
More informationUnit 3 Functions of Several Variables
Unit 3 Functions of Several Variables In this unit, we consider several simple examples of multi-variable functions, quadratic surfaces and projections, level curves and surfaces, partial derivatives of
More informationü 12.1 Vectors Students should read Sections of Rogawski's Calculus [1] for a detailed discussion of the material presented in this section.
Chapter 12 Vector Geometry Useful Tip: If you are reading the electronic version of this publication formatted as a Mathematica Notebook, then it is possible to view 3-D plots generated by Mathematica
More informationName. Center axis. Introduction to Conic Sections
Name Introduction to Conic Sections Center axis This introduction to conic sections is going to focus on what they some of the skills needed to work with their equations and graphs. year, we will only
More informationPlanes Intersecting Cones: Static Hypertext Version
Page 1 of 12 Planes Intersecting Cones: Static Hypertext Version On this page, we develop some of the details of the plane-slicing-cone picture discussed in the introduction. The relationship between the
More informationAssignment Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Assignment.1-.3 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) The arch beneath a bridge is semi-elliptical, a one-way
More informationThe point (x, y) lies on the circle of radius r and center (h, k) iff. x h y k r
NOTES +: ANALYTIC GEOMETRY NAME LESSON. GRAPHS OF EQUATIONS IN TWO VARIABLES (CIRCLES). Standard form of a Circle The point (x, y) lies on the circle of radius r and center (h, k) iff x h y k r Center:
More information7. r = r = r = r = r = 2 5
Exercise a: I. Write the equation in standard form of each circle with its center at the origin and the given radius.. r = 4. r = 6 3. r = 7 r = 5 5. r = 6. r = 6 7. r = 0.3 8. r =.5 9. r = 4 0. r = 3.
More informationStandard Equation of a Circle
Math 335 Trigonometry Conics We will study all 4 types of conic sections, which are curves that result from the intersection of a right circular cone and a plane that does not contain the vertex. (If the
More information1.) Write the equation of a circle in standard form with radius 3 and center (-3,4). Then graph the circle.
Welcome to the world of conic sections! http://www.youtube.com/watch?v=bfonicn4bbg Some examples of conics in the real world: Parabolas Ellipse Hyperbola Your Assignment: Circle -Find at least four pictures
More informationDescriptive Geometry 2
By Pál Ledneczki Ph.D. 2007 Spring Semester 1 Conic Sections Ellipse Parabola Hyperbola 2007 Spring Semester 2 Intersection of Cone and Plane: Ellipse The intersection of a cone of revolution and a plane
More information9/30/2014 FIRST HOURLY PRACTICE VIII Math 21a, Fall Name:
9/30/2014 FIRST HOURLY PRACTICE VIII Math 21a, Fall 2014 Name: MWF 9 Oliver Knill MWF 9 Chao Li MWF 10 Gijs Heuts MWF 10 Yu-Wen Hsu MWF 10 Yong-Suk Moon MWF 11 Rosalie Belanger-Rioux MWF 11 Gijs Heuts
More informationWe start by looking at a double cone. Think of this as two pointy ice cream cones that are connected at the small tips:
Math 1330 Chapter 8 Conic Sections In this chapter, we will study conic sections (or conics). It is helpful to know exactly what a conic section is. This topic is covered in Chapter 8 of the online text.
More informationWe start by looking at a double cone. Think of this as two pointy ice cream cones that are connected at the small tips:
Math 1330 Conic Sections In this chapter, we will study conic sections (or conics). It is helpful to know exactly what a conic section is. This topic is covered in Chapter 8 of the online text. We start
More informationCurvilinear Coordinates
Curvilinear Coordinates Cylindrical Coordinates A 3-dimensional coordinate transformation is a mapping of the form T (u; v; w) = hx (u; v; w) ; y (u; v; w) ; z (u; v; w)i Correspondingly, a 3-dimensional
More informationQuadratic Functions. *These are all examples of polynomial functions.
Look at: f(x) = 4x-7 f(x) = 3 f(x) = x 2 + 4 Quadratic Functions *These are all examples of polynomial functions. Definition: Let n be a nonnegative integer and let a n, a n 1,..., a 2, a 1, a 0 be real
More informationHYPERBOLA. Going off on a TANGENT!
HYPERBOLA Going off on a TANGENT! RECALL THAT THE HYPERBOLA IS A CONIC SECTION A LAMP CASTS A HYPERBOLIC BEAM OF LIGHT NUCLEAR COOLING TOWERS TORNADO TOWER, QATAR KOBE PORT TOWER, JAPAN RULED HYPERBOLOID
More information9.3 Hyperbolas and Rotation of Conics
9.3 Hyperbolas and Rotation of Conics Copyright Cengage Learning. All rights reserved. What You Should Learn Write equations of hyperbolas in standard form. Find asymptotes of and graph hyperbolas. Use
More informationDr. Allen Back. Nov. 19, 2014
Why of Dr. Allen Back Nov. 19, 2014 Graph Picture of T u, T v for a Lat/Long Param. of the Sphere. Why of Graph Basic Picture Why of Graph Why Φ(u, v) = (x(u, v), y(u, v), z(u, v)) Tangents T u = (x u,
More informationDr. Allen Back. Nov. 21, 2014
Dr. Allen Back of Nov. 21, 2014 The most important thing you should know (e.g. for exams and homework) is how to setup (and perhaps compute if not too hard) surface integrals, triple integrals, etc. But
More informationYou may know these...
You may know these... Chapter 1: Multivariables Functions 1.1 Functions of Two Variables 1.1.1 Function representations 1.1. 3-D Coordinate System 1.1.3 Graph of two variable functions 1.1.4 Sketching
More informationName: Class: Date: Conics Multiple Choice Pre-Test. Multiple Choice Identify the choice that best completes the statement or answers the question.
Name: Class: Date: Conics Multiple Choice Pre-Test Multiple Choice Identify the choice that best completes the statement or answers the question. 1 Graph the equation x 2 + y 2 = 36. Then describe the
More informationConic Sections. College Algebra
Conic Sections College Algebra Conic Sections A conic section, or conic, is a shape resulting from intersecting a right circular cone with a plane. The angle at which the plane intersects the cone determines
More information8.2 Graph and Write Equations of Parabolas
8.2 Graph and Write Equations of Parabolas Where is the focus and directrix compared to the vertex? How do you know what direction a parabola opens? How do you write the equation of a parabola given the
More information3.7. Vertex and tangent
3.7. Vertex and tangent Example 1. At the right we have drawn the graph of the cubic polynomial f(x) = x 2 (3 x). Notice how the structure of the graph matches the form of the algebraic expression. The
More informationMultivariable Calculus
Multivariable Calculus Chapter 10 Topics in Analytic Geometry (Optional) 1. Inclination of a line p. 5. Circles p. 4 9. Determining Conic Type p. 13. Angle between lines p. 6. Parabolas p. 5 10. Rotation
More informationA function: A mathematical relationship between two variables (x and y), where every input value (usually x) has one output value (usually y)
SESSION 9: FUNCTIONS KEY CONCEPTS: Definitions & Terminology Graphs of Functions - Straight line - Parabola - Hyperbola - Exponential Sketching graphs Finding Equations Combinations of graphs TERMINOLOGY
More information2. Give an example of a non-constant function f(x, y) such that the average value of f over is 0.
Midterm 3 Review Short Answer 2. Give an example of a non-constant function f(x, y) such that the average value of f over is 0. 3. Compute the Riemann sum for the double integral where for the given grid
More informationCK 12 Algebra II with Trigonometry Concepts 1
10.1 Parabolas with Vertex at the Origin Answers 1. up 2. left 3. down 4.focus: (0, 0.5), directrix: y = 0.5 5.focus: (0.0625, 0), directrix: x = 0.0625 6.focus: ( 1.25, 0), directrix: x = 1.25 7.focus:
More information1. no trace exists correct. 2. hyperbola : z 2 y 2 = ellipse : y z2 = ellipse : 5. circle : y 2 +z 2 = 2
grandi (rg38778) Homework 5 grandi () This print-out should have 3 questions. Multiple-choice questions may continue on the next column or page find all choices before answering.. points Classify the quadric
More informationDesign and Communication Graphics
An approach to teaching and learning Design and Communication Graphics Solids in Contact Syllabus Learning Outcomes: Construct views of up to three solids having curved surfaces and/or plane surfaces in
More informationSubstituting a 2 b 2 for c 2 and using a little algebra, we can then derive the standard equation for an ellipse centred at the origin,
Conics onic sections are the curves which result from the intersection of a plane with a cone. These curves were studied and revered by the ancient Greeks, and were written about extensively by both Euclid
More informationSurfaces. U (x; y; z) = k. Indeed, many of the most familiar surfaces are level surfaces of functions of 3 variables.
Surfaces Level Surfaces One of the goals of this chapter is to use di erential calculus to explore surfaces, in much the same way that we used di erential calculus to study curves in the rst chapter. In
More informationVectors and the Geometry of Space
Vectors and the Geometry of Space In Figure 11.43, consider the line L through the point P(x 1, y 1, z 1 ) and parallel to the vector. The vector v is a direction vector for the line L, and a, b, and c
More informationUnit 12 Topics in Analytic Geometry - Classwork
Unit 1 Topics in Analytic Geometry - Classwork Back in Unit 7, we delved into the algebra and geometry of lines. We showed that lines can be written in several forms: a) the general form: Ax + By + C =
More informationform. We will see that the parametric form is the most common representation of the curve which is used in most of these cases.
Computer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture - 36 Curve Representation Welcome everybody to the lectures on computer graphics.
More informationRay Tracer I: Ray Casting Due date: 12:00pm December 3, 2001
Computer graphics Assignment 5 1 Overview Ray Tracer I: Ray Casting Due date: 12:00pm December 3, 2001 In this assignment you will implement the camera and several primitive objects for a ray tracer. We
More informationAssignment 3/17/15. Section 10.2(p 568) 2 12 (E) (E)
Section 10.2 Warm Up Assignment 3/17/15 Section 10.2(p 568) 2 12 (E) 24 40 (E) Objective We are going to find equations for parabolas identify the vertex, focus, and directrix of a parabola The parabola
More informationPut your initials on the top of every page, in case the pages become separated.
Math 1201, Fall 2016 Name (print): Dr. Jo Nelson s Calculus III Practice for 1/2 of Final, Midterm 1 Material Time Limit: 90 minutes DO NOT OPEN THIS BOOKLET UNTIL INSTRUCTED TO DO SO. This exam contains
More information10.2: Parabolas. Chapter 10: Conic Sections. Conic sections are plane figures formed by the intersection of a double-napped cone and a plane.
Conic sections are plane figures formed b the intersection of a double-napped cone and a plane. Chapter 10: Conic Sections Ellipse Hperbola The conic sections ma be defined as the sets of points in the
More information5. y 2 + z 2 + 4z = 0 correct. 6. z 2 + x 2 + 2x = a b = 4 π
M408D (54690/95/00), Midterm #2 Solutions Multiple choice questions (20 points) See last two pages. Question #1 (25 points) Dene the vector-valued function r(t) = he t ; 2; 3e t i: a) At what point P (x
More informationSketching graphs of polynomials
Sketching graphs of polynomials We want to draw the graphs of polynomial functions y = f(x). The degree of a polynomial in one variable x is the highest power of x that remains after terms have been collected.
More informationAlgebra II Lesson 10-5: Hyperbolas Mrs. Snow, Instructor
Algebra II Lesson 10-5: Hyperbolas Mrs. Snow, Instructor In this section, we will look at the hyperbola. A hyperbola is a set of points P in a plane such that the absolute value of the difference between
More informationSection 4.4: Parabolas
Objective: Graph parabolas using the vertex, x-intercepts, and y-intercept. Just as the graph of a linear equation y mx b can be drawn, the graph of a quadratic equation y ax bx c can be drawn. The graph
More informationPre-Calculus. 2) Find the equation of the circle having (2, 5) and (-2, -1) as endpoints of the diameter.
Pre-Calculus Conic Review Name Block Date Circles: 1) Determine the center and radius of each circle. a) ( x 5) + ( y + 6) = 11 b) x y x y + 6 + 16 + 56 = 0 ) Find the equation of the circle having (,
More informationWHAT YOU SHOULD LEARN
GRAPHS OF EQUATIONS WHAT YOU SHOULD LEARN Sketch graphs of equations. Find x- and y-intercepts of graphs of equations. Use symmetry to sketch graphs of equations. Find equations of and sketch graphs of
More informationCHAPTER 8 QUADRATIC RELATIONS AND CONIC SECTIONS
CHAPTER 8 QUADRATIC RELATIONS AND CONIC SECTIONS Big IDEAS: 1) Writing equations of conic sections ) Graphing equations of conic sections 3) Solving quadratic systems Section: Essential Question 8-1 Apply
More informationUnit 5: Quadratic Functions
Unit 5: Quadratic Functions LESSON #5: THE PARABOLA GEOMETRIC DEFINITION DIRECTRIX FOCUS LATUS RECTUM Geometric Definition of a Parabola Quadratic Functions Geometrically, a parabola is the set of all
More informationThis is called the vertex form of the quadratic equation. To graph the equation
Name Period Date: Topic: 7-5 Graphing ( ) Essential Question: What is the vertex of a parabola, and what is its axis of symmetry? Standard: F-IF.7a Objective: Graph linear and quadratic functions and show
More informationAssignment Assignment for Lesson 11.1
Assignment Assignment for Lesson.1 Name Date Conics? Conics as Cross Sections Determine the conic section that results from the intersection of the double-napped cone shown and each plane described. 1.
More informationCurves, Tangent Planes, and Differentials ( ) Feb. 26, 2012 (Sun) Lecture 9. Partial Derivatives: Signs on Level Curves, Tangent
Lecture 9. Partial Derivatives: Signs on Level Curves, Tangent Planes, and Differentials ( 11.3-11.4) Feb. 26, 2012 (Sun) Signs of Partial Derivatives on Level Curves Level curves are shown for a function
More informationMath 155, Lecture Notes- Bonds
Math 155, Lecture Notes- Bonds Name Section 10.1 Conics and Calculus In this section, we will study conic sections from a few different perspectives. We will consider the geometry-based idea that conics
More informationWorksheet 2.1: Introduction to Multivariate Functions (Functions of Two or More Independent Variables)
Boise State Math 275 (Ultman) Worksheet 2.1: Introduction to Multivariate Functions (Functions of Two or More Independent Variables) From the Toolbox (what you need from previous classes) Know the meaning
More informationMath 20C. Lecture Examples.
Math 0C. Lecture Eamples. (8/30/08) Section 14.1, Part 1. Functions of two variables Definition 1 A function f of the two variables and is a rule = f(,) that assigns a number denoted f(,), to each point
More informationGeometry: Conic Sections
Conic Sections Introduction When a right circular cone is intersected by a plane, as in figure 1 below, a family of four types of curves results. Because of their relationship to the cone, they are called
More informationCSC418 / CSCD18 / CSC2504
5 5.1 Surface Representations As with 2D objects, we can represent 3D objects in parametric and implicit forms. (There are also explicit forms for 3D surfaces sometimes called height fields but we will
More informationPre-Calculus Guided Notes: Chapter 10 Conics. A circle is
Name: Pre-Calculus Guided Notes: Chapter 10 Conics Section Circles A circle is _ Example 1 Write an equation for the circle with center (3, ) and radius 5. To do this, we ll need the x1 y y1 distance formula:
More informationAccelerated Pre-Calculus Unit 1 Task 1: Our Only Focus: Circles & Parabolas Review
Accelerated Pre-Calculus Unit 1 Task 1: Our Only Focus: Circles & Parabolas Review Name: Date: Period: For most students, you last learned about conic sections in Analytic Geometry, which was a while ago.
More informationChapter 7 curve. 3. x=y-y 2, x=0, about the y axis. 6. y=x, y= x,about y=1
Chapter 7 curve Find the volume of the solid obtained by rotating the region bounded by the given cures about the specified line. Sketch the region, the solid, and a typical disk or washer.. y-/, =, =;
More informationMath 21a Final Exam Solutions Spring, 2009
Math a Final Eam olutions pring, 9 (5 points) Indicate whether the following statements are True or False b circling the appropriate letter No justifications are required T F The (vector) projection of
More informationTo sketch the graph we need to evaluate the parameter t within the given interval to create our x and y values.
Module 10 lesson 6 Parametric Equations. When modeling the path of an object, it is useful to use equations called Parametric equations. Instead of using one equation with two variables, we will use two
More informationASSIGNMENT BOOKLET. Bachelor s Degree Programme. Analytical Geometry
ASSIGNMENT BOOKLET MTE-05 Bachelor s Degree Programme Analytical Geometry (Valid from st January, 0 to st December, 0) School of Sciences Indira Gandhi National Open University Maidan Garhi New Delhi-0068
More informationCURVE SKETCHING EXAM QUESTIONS
CURVE SKETCHING EXAM QUESTIONS Question 1 (**) a) Express f ( x ) in the form ( ) 2 f x = x + 6x + 10, x R. f ( x) = ( x + a) 2 + b, where a and b are integers. b) Describe geometrically the transformations
More informationUnit 8, Ongoing Activity, Little Black Book of Algebra II Properties
Unit 8, Ongoing Activity, Little Black Book of Algebra II Properties Little Black Book of Algebra II Properties Unit 8 Conic Sections 8.1 Circle write the definition, provide examples of both the standard
More informationMATH 1020 WORKSHEET 10.1 Parametric Equations
MATH WORKSHEET. Parametric Equations If f and g are continuous functions on an interval I, then the equations x ft) and y gt) are called parametric equations. The parametric equations along with the graph
More information2.3. Graphing Calculators; Solving Equations and Inequalities Graphically
2.3 Graphing Calculators; Solving Equations and Inequalities Graphically Solving Equations and Inequalities Graphically To do this, we must first draw a graph using a graphing device, this is your TI-83/84
More informationMATH 2023 Multivariable Calculus
MATH 2023 Multivariable Calculus Problem Sets Note: Problems with asterisks represent supplementary informations. You may want to read their solutions if you like, but you don t need to work on them. Set
More information9/30/2014 FIRST HOURLY PRACTICE VII Math 21a, Fall Name:
9/30/2014 FIRST HOURLY PRACTICE VII Math 21a, Fall 2014 Name: MWF 9 Oliver Knill MWF 9 Chao Li MWF 10 Gijs Heuts MWF 10 Yu-Wen Hsu MWF 10 Yong-Suk Moon MWF 11 Rosalie Belanger-Rioux MWF 11 Gijs Heuts MWF
More informationUNIT 3B CREATING AND GRAPHING EQUATIONS Lesson 4: Solving Systems of Equations Instruction
Prerequisite Skills This lesson requires the use of the following skills: graphing multiple equations on a graphing calculator graphing quadratic equations graphing linear equations Introduction A system
More informationRay casting. Ray casting/ray tracing
Ray casting Ray casting/ray tracing Iterate over pixels, not objects Effects that are difficult with Z-buffer, are easy with ray tracing: shadows, reflections, transparency, procedural textures and objects
More informationPure Math 30: Explained!
www.puremath30.com 5 Conics Lesson Part I - Circles Circles: The standard form of a circle is given by the equation (x - h) +(y - k) = r, where (h, k) is the centre of the circle and r is the radius. Example
More informationDescriptive Geometry 2
By Pál Ledneczki Ph.D. Table of contents 1. Intersection of cone and plane 2. Perspective image of circle 3. Tangent planes and surface normals 4. Intersection of surfaces 5. Ellipsoid of revolution 6.
More informationHOW CAN I USE MAPLE TO HELP MY STUDENTS LEARN MULTIVARIATE CALCULUS?
HOW CAN I USE MAPLE TO HELP MY STUDENTS LEARN MULTIVARIATE CALCULUS? Thomas G. Wangler Benedictine University 5700 College Road, Lisle, IL 6053-0900 twangler@ben.edu Introduction Maple is a powerful software
More informationPART I. Answer each of the following. è1è Let! u = h2;,1;,2i and! v = h3; 1;,1i. Calculate: èaè! u, 2! v èbè the dot product of! u and! v ècè! u æ! v
MATH 241: FINAL EXAM Name Instructions and Point Values: Put your name in the space provided above. Check that your test contains 14 diæerent pages including one blank page. Work each problem below and
More information