B. Incorrect! Presence of all three variables in the equation makes it not cylindrical.
|
|
|
- Shavonne McDonald
- 5 years ago
- Views:
Transcription
1 Calculus III - Problem Drill 08: Clindrical and Quadric Surfaces Question No. of 0 Instructions: () Read the problem and answer choices carefull () Work the problems on paper as needed () Pick the. Which one of the following equations in gives a clindrical surface as its graph? (A) + + = 0 Question #0 (B) 5 = 0 (C) + 4 = 0 (D) = e A. Incorrect! This equation has an empt set as its graph. B. Incorrect! Presence of all three variables in the equation makes it not clindrical. C. Incorrect! Presence of all three variables in the equation makes it not clindrical. D. Correct! This is a clinder. The ais of clinder is -ais. Its generator is a curve other plane = constant. = e, on - plane, or on an E. Incorrect! e of (A) ~ (D) is correct. The variable is missing from the equation, so the surface etends indefinitel in -direction. The rulings are straight lines parallel to -ais, and the all pass through the generator = e. The computer generated graph is displaed below.
2 Question No. of 0 Instructions: () Read the problem and answer choices carefull () Work the problems on paper as needed () Pick the. Which one of the following equations in gives a hperboloid of one sheet as its graph? Question #0 (A) = (B) + = 4 8 (C) = 0 5 (D) + = A. Correct! It is in the form + = with b c a 6 a =, b= and c = 5 5 B. Incorrect! This equation is in the form: + + = C. Incorrect! This equation is in the form: + = D. Incorrect! This equation is in the form: c + = b E. Incorrect! e of (A) ~ (D) is correct = + = So ( ) ( 5) ( 4) 6 a =, b= and c = 5 5 Trace-wise: = 0 + =, an ellipse. ( ) ( 5) = 0 =, a hperbola. ( 5) ( 4) = 0 =, a hperbola. The graph is shown below. ( ) ( 4)
3 Question No. of 0 Instructions: () Read the problem and answer choices carefull () Work the problems on paper as needed () Pick the. Which one of the following equations in gives an elliptic cone as its graph? (A) + = 4 (B) + = Question #0 (C) + = 9 5 (D) = A. Incorrect! This equation is in the form: B. Incorrect! This equation is in the form: a + = c + + = C. Incorrect! This equation is missing one of the variables. D. Incorrect! This equation gives an empt set as its graph. E. Correct! None of (A) ~ (D) is correct. A. B. C. + = + = is an equation of an elliptic paraboloid. 4 + = = is an equation of an ellipsoid. 9 4 ( ) ( ) ( ) + = 9 + = is an equation of an elliptic clinder. 5 5 ( ) D. = 7 is impossible to meet in 6 5, since 0 (,, ) 6 5
4 Question No. 4 of 0 Instructions: () Read the problem and answer choices carefull () Work the problems on paper as needed () Pick the 4. Which one of the following equations in (A) + 4 = 0 (B) + 5 = 0 gives a hperbolic paraboloid as its graph? Question #04 (C) = 4 (D) = 0 A. Incorrect! This equation is in the form: B. Incorrect! This equation is in the form: C. Correct! This equation is in the form: a + = b + = = c a D. Incorrect! This equation gives the origin as its graph. E. Incorrect! e of (A) ~ (D) is correct. = 4 = 0 =, a parabola. = 0 =, a parabola. 4 = =±, a pair of lines. = 0 > 0 4 =, a hperbola = 0 < 0 4 = ( ) ( ) 0 0, a hperbola. We get:
5 Question No. 5 of 0 Instructions: () Read the problem and answer choices carefull () Work the problems on paper as needed () Pick the Question #05 5. Which one of the following equations in (A) + = (B) + + = (C) + + = (D) + = 0 gives a hperboloid of two sheets as its graph? A. Correct! This equation is in the form: B. Incorrect! This equation is in the form: C. Incorrect! This equation is in the form: D. Incorrect! This equation is in the form: E. Incorrect! e of (A) ~ (D) is correct. + = b c a + = b c a + + = + = b c a + = = ( ) ( ), a hperbola. = 0 = ( ) ( ) = 0 =, a hperbola. ( ), an ellipse. = 0 > + = > 0 ( ) ( ) = 0, ±, 0 = 0 < no graph. on a graph.
6 Question No. 6 of 0 Instructions: () Read the problem and answer choices carefull () Work the problems on paper as needed () Pick the 6. Which one of the following equations in gives an ellipsoid as its graph? Question #06 (A) + + = (B) + 5 = 4 (C) + + = (D) = A. Incorrect! This equation is obtained b translation along ais of one of the quadric surface. Which one is it? And how much shifting is taking place? B. Incorrect! Two pluses and one minus for coefficients of squared terms make it no ellipsoid. C. Correct! This gives an origin. D. Correct! This is trick. Since the problem does not ask to identif canonical form, this is indeed an ellipsoid, with translations. E. Incorrect! e of (A) ~ (D) is correct. B completing the squares for and, we get: ( ) ( + ) = + + = + 4+ = 7 We see that this equation represents an ellipsoid in canonical form, if we shift the and aes b to positive -direction and down to the negative -direction. i.e., with and +, we get: + + = =, which is a canonical ellipse in the translated coordinate 7 4 sstem. ( ) ( + ) So + + = or 7 4 ( 0,, ) in the original coordinate sstem = is an ellipsoid, centered at
7 Question No. 7 of 0 Instructions: () Read the problem and answer choices carefull () Work the problems on paper as needed () Pick the 7. Which one of the following equations in gives an elliptic paraboloid as its graph? Question #07 (A) + = 9 7 (B) = 9 7 (C) + = (D) + = A. Incorrect! You shall get a hperbolic paraboloid with shifting. B. Correct! This involves shifting along -ais. C. Incorrect! This shall give ou a two sheet hperboloid, shifted 4 units upward in positive -direction, of the canonical form. D. Incorrect! This shall give ou a one sheet hperboloid. E. Correct! None of (A) ~ (D) is correct. = = , then we obtain the canonical form of an elliptic paraboloid in translated So if we let coordinate sstem. = Sketch = + and shift the graph units toward positive -direction. 9 7
8 Question No. 8 of 0 Instructions: () Read the problem and answer choices carefull () Work the problems on paper as needed () Pick the 8. Which one of the following surfaces is the one for the equation (A) (B) + =? Y Question #08 (C) (D) Y Y Y A. Incorrect! You need to check the sign of term. B. Correct! Shifting of the canonical form in the negative -direction is taking the place for this equation. C. Incorrect! This one is the canonical version of the equation. D. Incorrect! You need to check the term involving. E. Incorrect! e of (A) ~ (D) is correct. + = + = + = So b letting +, we see that paraboloid in the shifted coordinate sstem. This means that the graph of ( 54) one unit to the negative -direction. Which gives: ( ) = is in canonical form of the equation of hperbolic + = is obtained from that of = b shifting it ( 54)
9 Question No. 9 of 0 Instructions: () Read the problem and answer choices carefull () Work the problems on paper as needed () Pick the 9. Which one of the following is the level curves at several heights of the surface described b =? 4 (A) (B) Question #09 (C) (D) A. Correct! The level curves are hperboloids, including the degenerate one. B. Incorrect! The equation does not generate ellipses for level curves. C. Incorrect! The equation does not generate parabolas when is set constant. In fact, these are for the positive branch of =, i.e., for 4 = 4+. D. Incorrect! The equation does not generate straight lines when is set constant, ecept for = 0, but there will be a pair of lines through origin. E. Incorrect! e of (A) ~ (D) is correct. Projection onto - plane of traces of the surface on the plane = 0 for various constants 0 is b definition the level curves for the surface at the height 0. This surface is for a hperbolic paraboloid. Y Level Curves at =-0, -5, 0, 5 and 0
10 Question No. 0 of 0 Instructions: () Read the problem and answer choices carefull () Work the problems on paper as needed () Pick the 0. Which one of the following equations describes an elliptic cone that is obtained from translation of the canonical form? Question #0 (A) (B) (C) = + + = + = + (D) = + + A. Incorrect! This, once again is a two sheets hperboloid, with a translation. B. Incorrect! This gives a shifted hperbolic paraboloid. C. Incorrect! This is a one sheet hperboloid. D. Incorrect! This gives a shifted elliptic paraboloid. E. Correct! None of the given answers is correct. The elliptic cone has the standard form: + = As for = + +, we have ( ) So ou get ( ) = + = = and this is a two sheets hperboloid with a translation. As for translation. = +, we have = and this is a hperbolic paraboloid with a As for form. As for = +, we have = + +, we have + =, and it is a one sheet hperboloid in canonical ( ) ( ) + = + = + = +, which is a an elliptic paraboloid with a translation. If, for instance, we consider = + +, then ( ) + = + = +, and hence this would have been an elliptic cone with a translation.
Math 323 Exam 2 - Practice Problem Solutions. 2. Given the vectors a = 1,2,0, b = 1,0,2, and c = 0,1,1, compute the following:
Math 323 Eam 2 - Practice Problem Solutions 1. Given the vectors a = 2,, 1, b = 3, 2,4, and c = 1, 4,, compute the following: (a) A unit vector in the direction of c. u = c c = 1, 4, 1 4 =,, 1+16+ 17 17
(6, 4, 0) = (3, 2, 0). Find the equation of the sphere that has the line segment from P to Q as a diameter.
Solutions Review for Eam #1 Math 1260 1. Consider the points P = (2, 5, 1) and Q = (4, 1, 1). (a) Find the distance from P to Q. Solution. dist(p, Q) = (4 2) 2 + (1 + 5) 2 + (1 + 1) 2 = 4 + 36 + 4 = 44
Chapter 12 Review Vector. MATH 126 (Section 9.5) Vector and Scalar The University of Kansas 1 / 30
Chapter 12 Review Vector MATH 126 (Section 9.5) Vector and Scalar The University of Kansas 1 / 30 iclicker 1: Let v = PQ where P = ( 2, 5) and Q = (1, 2). Which of the following vectors with the given
Not for reproduction
ROTATION OF AES For a discussion of conic sections, see Review of Conic Sections In precalculus or calculus ou ma have studied conic sections with equations of the form A C D E F Here we show that the
ES.182A Problem Section 11, Fall 2018 Solutions
Problem 25.1. (a) z = 2 2 + 2 (b) z = 2 2 ES.182A Problem Section 11, Fall 2018 Solutions Sketch the following quadratic surfaces. answer: The figure for part (a) (on the left) shows the z trace with =
APPENDIX D Rotation and the General Second-Degree Equation
APPENDIX D Rotation and the General Second-Degree Equation Rotation of Aes Invariants Under Rotation After rotation of the - and -aes counterclockwise through an angle, the rotated aes are denoted as the
MATH 1020 WORKSHEET 12.1 & 12.2 Vectors in the Plane
MATH 100 WORKSHEET 1.1 & 1. Vectors in the Plane Find the vector v where u =, 1 and w = 1, given the equation v = u w. Solution. v = u w =, 1 1, =, 1 +, 4 =, 1 4 = 0, 5 Find the magnitude of v = 4, 3 Solution.
Math 21a: Multivariable calculus. List of Worksheets. Harvard University, Spring 2009
Math 2a: Multivariable calculus Harvard Universit, Spring 2009 List of Worksheets Vectors and the Dot Product Cross Product and Triple Product Lines and Planes Functions and Graphs Quadric Surfaces Vector-Valued
Equations for Some Hyperbolas
Lesson 1-6 Lesson 1-6 BIG IDEA From the geometric defi nition of a hperbola, an equation for an hperbola smmetric to the - and -aes can be found. The edges of the silhouettes of each of the towers pictured
Mathematics 309 Conic sections and their applicationsn. Chapter 2. Quadric figures. ai,j x i x j + b i x i + c =0. 1. Coordinate changes
Mathematics 309 Conic sections and their applicationsn Chapter 2. Quadric figures In this chapter want to outline quickl how to decide what figure associated in 2D and 3D to quadratic equations look like.
The details of the derivation of the equations of conics are com-
Part 6 Conic sections Introduction Consider the double cone shown in the diagram, joined at the verte. These cones are right circular cones in the sense that slicing the double cones with planes at right-angles
Lesson 9.1 Using the Distance Formula
Lesson. Using the Distance Formula. Find the eact distance between each pair of points. a. (0, 0) and (, ) b. (0, 0) and (7, ) c. (, 8) and (, ) d. (, ) and (, 7) e. (, 7) and (8, ) f. (8, ) and (, 0)
Ready To Go On? Skills Intervention 10-1 Introduction to Conic Sections
Find this vocabular word in Lesson 10-1 and the Multilingual Glossar. Graphing Parabolas and Hperbolas on a Calculator A is a single curve, whereas a has two congruent branches. Identif and describe each
MATH 16C: MULTIVARIATE CALCULUS
April 12, 2010 7.1: The 3-dimensional coordinate system Thus, d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 + (z 2 z 1 ) 2. EXAMPLE: Find the distance between ( 3,2,5) and (4, 1,2). d = (( 3) 4) 2 + (2 ( 1)) 2 + (5
Attributes and Transformations of Quadratic Functions VOCABULARY. Maximum value the greatest. Minimum value the least. Parabola the set of points in a
- Attributes and Transformations of Quadratic Functions TEKS FCUS VCABULARY TEKS ()(B) Write the equation of a parabola using given attributes, including verte, focus, directri, ais of smmetr, and direction
CK- 12 Algebra II with Trigonometry Concepts 1
10.1 Parabolas with Verte at the Origin Answers 1. up. left 3. down 4.focus: (0, -0.5), directri: = 0.5 5.focus: (0.065, 0), directri: = -0.065 6.focus: (-1.5, 0), directri: = 1.5 7.focus: (0, ), directri:
f x, y x 2 y 2 2x 6y 14. Then
SECTION 11.7 MAXIMUM AND MINIMUM VALUES 645 absolute minimum FIGURE 1 local maimum local minimum absolute maimum Look at the hills and valles in the graph of f shown in Figure 1. There are two points a,
Section 9.1 Video Guide Distance and Midpoint Formulas
Objectives: 1. Use the Distance Formula 2. Use the Midpoint Formula Section 9.1 Video Guide Distance and Midpoint Formulas Section 9.1 Objective 1: Use the Distance Formula Video Length 8:27 1. Eample:
CALCULUS 4 QUIZ #2 REVIEW / SPRING 09
CALCULUS QUIZ # REVIEW / SPRING 09 (.) Determine the following about the given quaric surfaces. (a.) Ientif & Sketch the quaric surface: +. 9 In planes parallel to the -plane an planes parallel to the
SOLUTIONS TO HOMEWORK ASSIGNMENT #2, Math 253
SOLUTIONS TO HOMEWORK ASSIGNMENT #, Math 5. Find the equation of a sphere if one of its diameters has end points (, 0, 5) and (5, 4, 7). The length of the diameter is (5 ) + ( 4 0) + (7 5) = =, so the
IAS 3.1 Conic Sections
Year 13 Mathematics IAS 3.1 Conic Sections Robert Lakeland & Carl Nugent Contents Achievement Standard.................................................. The Straight Line.......................................................
Triple Integrals. y x
Triple Integrals. (a) If is an solid (in space), what does the triple integral dv represent? Wh? (b) Suppose the shape of a solid object is described b the solid, and f(,, ) gives the densit of the object
REVIEW OF CONIC SECTIONS
REVIEW OF CONIC SECTIONS In this section we give geometric definitions of parabolas, ellipses, and hperbolas and derive their standard equations. The are called conic sections, or conics, because the result
Ma 227 Final Exam Solutions 5/8/03
Ma 7 Final Eam Solutions 5/8/3 Name: Lecture Section: I pledge m honor that I have abided b the Stevens Honor Sstem. ID: Directions: Answer all questions. The point value of each problem is indicated.
13.1. For further details concerning the physics involved and animations of the trajectories of the particles, see the following websites:
8 CHAPTER VECTOR FUNCTIONS N Some computer algebra sstems provide us with a clearer picture of a space curve b enclosing it in a tube. Such a plot enables us to see whether one part of a curve passes in
2-6. _ k x and y = _ k. The Graph of. Vocabulary. Lesson
Chapter 2 Lesson 2-6 BIG IDEA The Graph of = _ k and = _ k 2 The graph of the set of points (, ) satisfing = k_, with k constant, is a hperbola with the - and -aes as asmptotes; the graph of the set of
Not for reproduction
REVIEW OF CONIC SECTIONS In this section we give geometric definitions of parabolas, ellipses, and hperbolas and derive their standard equations. The are called conic sections, or conics, because the result
Coordinate geometry. + bx + c. Vertical asymptote. Sketch graphs of hyperbolas (including asymptotic behaviour) from the general
A Sketch graphs of = a m b n c where m = or and n = or B Reciprocal graphs C Graphs of circles and ellipses D Graphs of hperbolas E Partial fractions F Sketch graphs using partial fractions Coordinate
F dr y 2. F r t r t dt. a sin t a sin t a cos t a cos t a 2 cos 2 t a 2 sin 2 t. P dx Q dy yy. x C. follows that F is a conservative vector field.
6 CHAPTER 6 VECTOR CALCULU We now easil compute this last integral using the parametriation given b rt a cos t i a sin t j, t. Thus C F dr C F dr Frt rt dt a sin ta sin t a cos ta cos t a cos t a sin t
8.8 Conics With Equations in the Form
8.8 Conics With Equations in the Form ax + b + gx + f + c = 0 The CF-18 Hornet is a supersonic jet flown in Canada. It has a maximum speed of Mach 1.8. The speed of sound is Mach 1. When a plane like the
Problem set 6 Math 207A, Fall 2011 Solutions. 1. A two-dimensional gradient system has the form
Problem set 6 Math 207A, Fall 2011 s 1 A two-dimensional gradient sstem has the form x t = W (x,, x t = W (x, where W (x, is a given function (a If W is a quadratic function W (x, = 1 2 ax2 + bx + 1 2
10.4 Nonlinear Inequalities and Systems of Inequalities. OBJECTIVES 1 Graph a Nonlinear Inequality. 2 Graph a System of Nonlinear Inequalities.
Section 0. Nonlinear Inequalities and Sstems of Inequalities 6 CONCEPT EXTENSIONS For the eercises below, see the Concept Check in this section.. Without graphing, how can ou tell that the graph of + =
Mat 267 Engineering Calculus III Updated on 9/19/2010
Chapter 11 Partial Derivatives Section 11.1 Functions o Several Variables Deinition: A unction o two variables is a rule that assigns to each ordered pair o real numbers (, ) in a set D a unique real number
SECOND-DEGREE INEQUALITIES
60 (-40) Chapter Nonlinear Sstems and the Conic Sections 0 0 4 FIGURE FOR EXERCISE GETTING MORE INVOLVED. Cooperative learning. Let (, ) be an arbitrar point on an ellipse with foci (c, 0) and ( c, 0)
Dr. Back. Nov. 3, 2009
Dr. Back Nov. 3, 2009 Please Don t Rely on this File! In you re expected to work on these topics mostly without computer aid. But seeing a few better pictures can help understanding the concepts. A copy
ARE YOU READY FOR CALCULUS?? Name: Date: Period:
ARE YOU READY FOR CALCULUS?? Name: Date: Period: Directions: Complete the following problems. **You MUST show all work to receive credit.**(use separate sheets of paper.) Problems with an asterisk (*)
Lines, Conics, Tangents, Limits and the Derivative
Lines, Conics, Tangents, Limits and te Derivative Te Straigt Line An two points on te (,) plane wen joined form a line segment. If te line segment is etended beond te two points ten it is called a straigt
In order to take a closer look at what I m talking about, grab a sheet of graph paper and graph: y = x 2 We ll come back to that graph in a minute.
Module 7: Conics Lesson Notes Part : Parabolas Parabola- The parabola is the net conic section we ll eamine. We talked about parabolas a little bit in our section on quadratics. Here, we eamine them more
CALC 3 CONCEPT PACKET Complete
CALC 3 CONCEPT PACKET Complete Written by Jeremy Robinson, Head Instructor Find Out More +Private Instruction +Review Sessions WWW.GRADEPEAK.COM Need Help? Online Private Instruction Anytime, Anywhere
The telescopes at the W.M. Keck Observatory in Hawaii use hyperbolic mirrors.
UNIT 15 Conic Sections The telescopes at the W.M. Keck Observator in Hawaii use hperbolic mirrors. Copright 009, K1 Inc. All rights reserved. This material ma not be reproduced in whole or in part, including
Mathematics 10 Page 1 of 7 The Quadratic Function (Vertex Form): Translations. and axis of symmetry is at x a.
Mathematics 10 Page 1 of 7 Verte form of Quadratic Relations The epression a p q defines a quadratic relation called the verte form with a horizontal translation of p units and vertical translation of
Math 2412 Pre Calculus TEST 2 Prep Fall 2011
Math 41 Pre Calculus TEST Prep Fall 011 Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Find the eact value under the given conditions. 1) sin α
3) Find the distance for each set of ordered pairs (remember to provide EXACT answers): 5) x 2 + y 2 + 6x 6y + 9 = 0 A) Ellipse (x 1) 2
Algebra Chapter Review 1) State the Midpoint Formula: State the Distance Formula: ID: 1 Name Date ) Find the midpoint for each set of ordered pairs: a) (1, ), (, ) b) (-, 0), (-, 3) Period c) (, ), (-,
Summary and Vocabulary
Chapter 2 Chapter 2 Summar and Vocabular The functions studied in this chapter are all based on direct and inverse variation. When k and n >, formulas of the form = k n define direct-variation functions,
Solutions to Problem Sheet for Week 11
THE UNIVERSITY OF SYDNEY SCHOOL OF MATHEMATICS AND STATISTICS Solutions to Problem Sheet for Week MATH9: Differential Calculus (Advanced) Semester, 7 Web Page: sydney.edu.au/science/maths/u/ug/jm/math9/
C H A P T E R 9 Topics in Analytic Geometry
C H A P T E R Topics in Analtic Geometr Section. Circles and Parabolas.................... 77 Section. Ellipses........................... 7 Section. Hperbolas......................... 7 Section. Rotation
UNCORRECTED. To recognise the rules of a number of common algebraic relations: y = x 1 y 2 = x
5A galler of graphs Objectives To recognise the rules of a number of common algebraic relations: = = = (rectangular hperbola) + = (circle). To be able to sketch the graphs of these relations. To be able
Edexcel GCE A Level Maths. Further Maths 3 Coordinate Systems
Edecel GCE A Level Maths Further Maths 3 Coordinate Sstems Edited b: K V Kumaran kumarmaths.weebl.com 1 kumarmaths.weebl.com kumarmaths.weebl.com 3 kumarmaths.weebl.com 4 kumarmaths.weebl.com 5 1. An ellipse
Topic 1: Fractional and Negative Exponents x x Topic 2: Domain. 2x 9 2x y x 2 5x y logg 2x 12
AP Calculus BC Summer Packet Name INSTRUCTIONS: Where applicable, put your solutions in interval notation. Do not use any calculator (ecept on Topics 5:#5 & :#6). Please do all work on separate paper and
Map Projections & Coordinate Systems 9/10/2013. Why? M. Helper GEO327G/386G, UT Austin 2. M. Helper GEO327G/386G, UT Austin 4
Map Projections & Coordinates Laing the earth flat Wh? Need convenient means of measuring and comparing distances, directions, areas, shapes. Traditional surveing instruments measure in meters or feet,
Analytic Geometry in Two and Three Dimensions
CHAPTER 8 Analtic Geometr in Two and Three Dimensions 8.1 Conic Sections and Parabolas 8.2 Ellipses 8.3 Hperbolas 8.4 Translation and Rotation of Aes 8.5 Polar Equations of Conics 8.6 Three-Dimensional
Instructor: Imelda Valencia Course: A3 Honors Pre Calculus
Student: Date: Instructor: Imelda Valencia Course: A3 Honors Pre Calculus 01 017 Assignment: Summer Homework for those who will be taking FOCA 017 01 onl available until Sept. 15 1. Write the epression
17.3. Parametric Curves. Introduction. Prerequisites. Learning Outcomes
Parametric Curves 7.3 Introduction In this Section we eamine et another wa of defining curves - the parametric description. We shall see that this is, in some was, far more useful than either the Cartesian
MATHEMATICS 200 December 2011 Final Exam Solutions
MATHEMATICS December 11 Final Eam Solutions 1. Consider the function f(, ) e +4. (a) Draw a contour map of f, showing all tpes of level curves that occur. (b) Find the equation of the tangent plane to
Name of the Student:
Engineering Mathematics 016 SUBJECT NAME : Engineering Mathematics - I SUBJECT CODE : MA111 MATERIAL NAME : Universit Questions REGULATION : R008 WEBSITE : wwwhariganeshcom UPDATED ON : Januar 016 TEXTBOOK
Identifying second degree equations
Chapter 7 Identifing second degree equations 71 The eigenvalue method In this section we appl eigenvalue methods to determine the geometrical nature of the second degree equation a 2 + 2h + b 2 + 2g +
Tetragonal or hexagonal symmetry in modeling of failure criteria for transversely isotropic materials
Tetragonal or heagonal smmetr in modeling of failure criteria for transversel isotropic materials MSc Michal Adamsi Supervisor: Prof. Artur Ganarsi Abstract Present wor deals with modeling of failure criteria
and ( x, y) in a domain D R a unique real number denoted x y and b) = x y = {(, ) + 36} that is all points inside and on
Mat 7 Calculus III Updated on 10/4/07 Dr. Firoz Chapter 14 Partial Derivatives Section 14.1 Functions o Several Variables Deinition: A unction o two variables is a rule that assigns to each ordered pair
17. Find the moments of inertia I x, I y, I 0 for the lamina of. 4. D x, y 0 x a, 0 y b ; CAS. 20. D is enclosed by the cardioid r 1 cos ; x, y 3
SCTION 2.5 TRIPL INTGRALS 69 2.4 XRCISS. lectric charge is distributed over the rectangle, 2 so that the charge densit at, is, 2 2 (measured in coulombs per square meter). Find the total charge on the
Map Projections & Coordinate Systems 9/7/2017
Map Projections & Coordinate Sstems Laing the Earth Flat Wh? Need convenient means of measuring and comparing distances, directions, areas, shapes. Traditional surveing instruments measure in meters or
Conic Sections CHAPTER OUTLINE. The Circle Ellipses and Hyperbolas Second-Degree Inequalities and Nonlinear Systems FIGURE 1
088_0_p676-7 /7/0 :5 PM Page 676 (FPG International / Telegraph Colour Librar) Conic Sections CHAPTER OUTLINE. The Circle. Ellipses and Hperbolas.3 Second-Degree Inequalities and Nonlinear Sstems O ne
Vectors and the Geometry of Space
Chapter 12 Vectors and the Geometr of Space Comments. What does multivariable mean in the name Multivariable Calculus? It means we stud functions that involve more than one variable in either the input
Question Booklet (Parts A and B)
Mathematics 565-506 Secondar V June 01 Competenc Uses Mathematical Reasoning Science Option Question Booklet (Parts A and B) Instructions 1. Write all the required information in the Student Booklet. Answers
Find the distance between the pair of points. 2) (7, -7) and (3, -5) A) 12 3 units B) 2 5 units C) 6 units D) 12 units B) 8 C) 63 2
Sample Departmental Final - Math 9 Write the first five terms of the sequence whose general term is given. 1) a n = n 2 - n 0, 2,, 12, 20 B) 2,, 12, 20, 30 C) 0, 3, 8, 1, 2 D) 1,, 9, 1, 2 Find the distance
Vertex Form of a Parabola
Verte Form of a Parabola In this investigation ou will graph different parabolas and compare them to what is known as the Basic Parabola. THE BASIC PARABOLA Equation = 2-3 -2-1 0 1 2 3 verte? What s the
Exercises. . Evaluate f (, ) xydawhere. 2, 1 x< 3, 0 y< (Answer: 14) 3, 3 x 4, 0 y 2. (Answer: 12) ) f x. ln 2 1 ( ) = (Answer: 1)
Eercises Eercises. Let = {(, : 4, }. Evaluate f (,,
x = y = A = (1) (2) = 2.
MATH 3 Final Eam (Version 1) Solutions May, 8 S. F. Ellermeyer Name Instructions. Your work on this eam will be graded according to two criteria: mathematical correctness and clarity of presentation. In
Quadratic Functions Objective: To be able to graph a quadratic function and identify the vertex and the roots.
Name: Quadratic Functions Objective: To be able to graph a quadratic function and identif the verte and the roots. Period: Quadratic Function Function of degree. Usuall in the form: We are now going to
Map Projections & Coordinate Systems 1/25/2018
Map Projections & Coordinate Sstems Laing the Earth Flat How? Projections transformation of curved earth to a flat map; sstematic rendering of the lat. & lon. graticule to rectangular coordinate sstem.
Chapter 13. Overview. The Quadratic Formula. Overview. The Quadratic Formula. The Quadratic Formula. Lewinter & Widulski 1. The Quadratic Formula
Chapter 13 Overview Some More Math Before You Go The Quadratic Formula The iscriminant Multiplication of Binomials F.O.I.L. Factoring Zero factor propert Graphing Parabolas The Ais of Smmetr, Verte and
POPULAR QUESTIONS IN ADVANCED CALCULUS
GRIET(AUTONOMOU) POPULAR QUETION IN ADVANED ALULU UNIT-. If u = f(e z, e z, e u u u ) then prove that. z. If z u, Prove that u u u. zz. If r r e cos, e sin then show that r u u e [ urr u ]. 4. Find J,
MATH UN1201, Section 3 (11:40am 12:55pm) - Midterm 1 February 14, 2018 (75 minutes)
Name: Instructor: Shrenik Shah MATH UN1201, Section 3 (11:40am 12:55pm) - Midterm 1 February 14, 2018 (75 minutes) This examination booklet contains 6 problems plus an additional extra credit problem.
3. TRANSLATED PARABOLAS
3. TRANSLATED PARABOLAS The Parabola with Verte V(h, k) and Aes Parallel to the ais Consider the concave up parabola with verte V(h, k) shown below. This parabola is obtained b translating the parabola
STUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE. Functions & Graphs
STUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE Functions & Graphs Contents Functions and Relations... 1 Interval Notation... 3 Graphs: Linear Functions... 5 Lines and Gradients... 7 Graphs: Quadratic
104Math. Find the equation of the parabola and sketch it in the exercises 10-18:
KING SAUD UNIVERSITY COLEGE OF SCIENCE DEPARTMENT OF MATHEMATICS Math Prof Messaoud Bounkhel List of Eercises: Chapter [Parabola] Find the elements of the parabola and sketch it in the eercises -9: ( )
Mathematics A A* Type Questions 1H
Write our name here Surname Other names In the stle of: Edecel GCSE Centre Number Mathematics A A* Tpe Questions 1H Etra topics that occur less frequentl, for students working towards an A* Candidate Number
Solutions to Practice Exam 2
Solutions to Practice Eam Problem : For each of the following, set up (but do not evaluate) iterated integrals or quotients of iterated integral to give the indicated quantities: Problem a: The average
INVESTIGATE the Math
. Graphs of Reciprocal Functions YOU WILL NEED graph paper coloured pencils or pens graphing calculator or graphing software f() = GOAL Sketch the graphs of reciprocals of linear and quadratic functions.
Chapter Summary. How does Chapter 10 fit into the BIGGER PICTURE of algebra?
Page of 5 0 Chapter Summar WHAT did ou learn? Find the distance between two points. (0.) Find the midpoint of the line segment connecting two points. (0.) Use distance and midpoint formulas in real-life
SVKM s NMIMS. Mukesh Patel School of Technology Management & Engineering, Vile Parle, Mumbai
Mukesh Patel School of Technolog Management & Engineering Page SVKM s NMIMS Mukesh Patel School of Technolog Management & Engineering, Vile Parle, Mumbai- 456 Tutorial Manual Academic Year : 4-5 Program:
Mth Quadratic functions and quadratic equations
Mth 0 - Quadratic functions and quadratic equations Name Find the product. 1) 8a3(2a3 + 2 + 12a) 2) ( + 4)( + 6) 3) (3p - 1)(9p2 + 3p + 1) 4) (32 + 4-4)(2-3 + 3) ) (4a - 7)2 Factor completel. 6) 92-4 7)
AQA Level 2 Further mathematics Number & algebra. Section 3: Functions and their graphs
AQA Level Further mathematics Number & algebra Section : Functions and their graphs Notes and Eamples These notes contain subsections on: The language of functions Gradients The equation of a straight
D. Correct! This is the correct answer. It is found by dy/dx = (dy/dt)/(dx/dt).
Calculus II - Problem Solving Drill 4: Calculus for Parametric Equations Question No. of 0 Instructions: () Read the problem and answer choices carefully () Work the problems on paper as. Find dy/dx where
Math 180 Chapter 10 Lecture Notes. Professor Miguel Ornelas
Math 180 Chapter 10 Lecture Notes Professor Miguel Ornelas 1 M. Ornelas Math 180 Lecture Notes Section 10.1 Section 10.1 Parabolas Definition of a Parabola A parabola is the set of all points in a plane
Saturday X-tra X-Sheet: 8. Inverses and Functions
Saturda X-tra X-Sheet: 8 Inverses and Functions Ke Concepts In this session we will ocus on summarising what ou need to know about: How to ind an inverse. How to sketch the inverse o a graph. How to restrict
Three-Dimensional Space; Vectors
Chapter 3 Three-Dimensional Space; Vectors 3.1 Rectangular Coordinates in 3-Space; Spheres Rectangular Coordinate Sstems To begin, consider three mutuall perpendicular coordinate lines, called the -ais,
17.3. Parametric Curves. Introduction. Prerequisites. Learning Outcomes
Parametric Curves 17.3 Introduction In this section we eamine et another wa of defining curves - the parametric description. We shall see that this is, in some was, far more useful than either the Cartesian
Algebra 2 End of Term Final REVIEW
Algebra End of Term Final REVIEW DO NOT WRITE IN TEST BOOKLET. 1. Graph. a. c. x x x x. Express as a single logarithm. Simplif, if possible. a. 5 c. 3 3. The volume V of a clinder varies jointl with the
S4 National 5 Maths Chapter 10
S4 National 5 Maths Chapter 10 Two men are sitting in the basket of a balloon. For hours, they have been drifting through a thick layer of clouds, and they have lost orientation completely. Suddenly, the
C. Incorrect! This symbol means greater than or equal to or at least. D. Correct! This symbol means at most or less than or equal to.
SAT Math - Problem Drill 10: Inequalities No. 1 of 10 1. Choose the inequality symbol that means at most. (A) > (B) < (C) (D) (E) This symbol means greater than. This symbol means less than. This symbol
x = 1 n (x 1 + x 2 + +x n )
3 PARTIAL DERIVATIVES Science Photo Librar Three-dimensional surfaces have high points and low points that are analogous to the peaks and valles of a mountain range. In this chapter we will use derivatives
( 7, 3) means x = 7 and y = 3. ( 7, 3) works in both equations so. Section 5 1: Solving a System of Linear Equations by Graphing
Section 5 : Solving a Sstem of Linear Equations b Graphing What is a sstem of Linear Equations? A sstem of linear equations is a list of two or more linear equations that each represents the graph of a
MATH 2083 FINAL EXAM REVIEW The final exam will be on Wednesday, May 4 from 10:00am-12:00pm.
MATH 2083 FINAL EXAM REVIEW The final exam will be on Wednesday, May 4 from 10:00am-12:00pm. Bring a calculator and something to write with. Also, you will be allowed to bring in one 8.5 11 sheet of paper
Math 101 chapter six practice exam MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Math 1 chapter si practice eam MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Which equation matches the given calculator-generated graph and description?
Functions of Several Variables
Chapter 1 Functions of Several Variables 1.1 Introduction A real valued function of n variables is a function f : R, where the domain is a subset of R n. So: for each ( 1,,..., n ) in, the value of f is
y = f(x + 4) a) Example: A repeating X by using two linear equations y = ±x. b) Example: y = f(x - 3). The translation is
Answers Chapter Function Transformations. Horizontal and Vertical Translations, pages to. a h, k h, k - c h -, k d h 7, k - e h -, k. a A (-,, B (-,, C (-,, D (,, E (, A (-, -, B (-,, C (,, D (, -, E (,
Department of Mathematical and Statistical Sciences University of Alberta
MATH 4 (R) Winter 8 Intermediate Calculus I Solutions to Problem Set #5 Completion Date: Frida Februar 5, 8 Department of Mathematical and Statistical Sciences Universit of Alberta Question. [Sec.., #
Precalculus Honors - AP Calculus A Information and Summer Assignment
Precalculus Honors - AP Calculus A Information and Summer Assignment General Information: Competenc in Algebra and Trigonometr is absolutel essential. The calculator will not alwas be available for ou
D u f f x h f y k. Applying this theorem a second time, we have. f xx h f yx k h f xy h f yy k k. f xx h 2 2 f xy hk f yy k 2
93 CHAPTER 4 PARTIAL DERIVATIVES We close this section b giving a proof of the first part of the Second Derivatives Test. Part (b) has a similar proof. PROOF OF THEOREM 3, PART (A) We compute the second-order
The coordinates of the vertex of the corresponding parabola are p, q. If a > 0, the parabola opens upward. If a < 0, the parabola opens downward.
Mathematics 10 Page 1 of 8 Quadratic Relations in Vertex Form The expression y ax p q defines a quadratic relation in form. The coordinates of the of the corresponding parabola are p, q. If a > 0, the