# Parametric Equations and the Parabola (Extension 1)

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Parametric Equations and the Parabola (Extension 1) Parametric Equations Parametric equations are a set of equations in terms of a parameter that represent a relation. Each value of the parameter, when evaluated in the parametric equations, corresponds to a point along the curve of the relation. To convert equations from parametric form into a single relation, the parameter needs to be eliminated by solving simultaneous equations. Locus of a Point If the position of a point is given in terms of the parameter, to find the locus of the point is to find the equation of the path that the point takes as the parameter s value changes. If the point s x- or y-coordinate is independent of the parameter, then a vertical or horizontal line is the locus, although it might be restricted to just part of the line. If not, eliminate the parameter by solving the equations simultaneously. Parametric Representation of a Parabola Parametric equations x = 2ap (1) y = ap 2 (2) A variable point on the parabola is given by (2ap, ap 2 ), for constant a and parameter p. Conversion into Cartesian equation Rearrange (1) to give: Then substitute (3) into (2): p = x 2a (3) ( x 2 y = a 2a) = x2 4a x = 4ay which is the equation of a parabola with vertex (0, 0) and focal length a. Gradient of Tangent The gradient of the tangent to x 2 = 4ay at P (2ap, ap 2 ) is p. y = x2 4a m = y = x 2a = 2ap 2a = p (4) Copyright c 2006 Enoch Lau. All Rights Reserved. For personal use only. 1 of 6

2 Equation of Tangent The equation of the tangent to x 2 = 4ay at P (2ap, ap 2 ) is y = px ap Find the gradient of the tangent at P ; using (4): m = p 2. Use y y 1 = m(x x 1 ): y ap 2 = p(x 2ap) y = px ap 2 (5) Equation of Normal The equation of the normal to x 2 = 4ay at P (2ap, ap 2 ) is x + py = ap 3 + 2ap. 1. Find the gradient of the normal at P : m = 1 p 2. Use y y 1 = m(x x 1 ): y ap 2 = 1 (x 2ap) p py ap 3 = x + 2ap x + py = ap 3 + 2ap (6) Intersection of Tangents The intersection of the tangents to x 2 = 4ay at P (2ap, ap 2 ) and Q(2aq, aq 2 ) is (a(p + q), apq). 1. Obtain the equations of the tangents using (5): y = px ap 2 (7) y = qx aq 2 (8) 2. Substitute (7) into (8): px ap 2 = qx aq 2 px qx = ap 2 aq 2 x(p q) = a(p q)(p + q) x = a(p + q) (9) 3. Substitute (9) back into (7): y = p(ap + aq) ap 2 = apq (10) Copyright c 2006 Enoch Lau. All Rights Reserved. For personal use only. 2 of 6

3 Intersection of Normals The intersection of the normals to x 2 = 4ay at P (2ap, ap 2 ) and Q(2aq, aq 2 ) is ( apq(p + q), a(p 2 + pq + q 2 + 2)). 1. Obtain the equations of the normals using (6): 2. Subtract (12) from (11): 3. Substitute (13) back into (11): Equation of Chord x + py = ap 3 + 2ap (11) x + qy = aq 3 + 2aq (12) py qy = ap 3 aq 3 + 2ap 2aq y(p q) = a(p q)(p 2 + pq + q 2 ) + 2a(p q) y = a(p 2 + pq + q 2 + 2) (13) x + pa(p 2 + pq + q 2 + 2) = ap 3 + 2ap x + ap 3 + ap 2 q + apq 2 + 2ap = ap 3 + 2ap x = (ap 2 q + apq 2 ) = apq(p + q) (14) The equation of the chord from P (2ap, ap 2 ) to Q(2aq, aq 2 ) is y 1 2x(p + q) + apq = 0. Using y y 1 x x 1 = y 2 y 1 x 2 x 1 : y ap 2 x 2ap = aq2 ap 2 2aq 2ap (y ap 2 )2a(q p) = (x 2ap)a(q p)(q + p) 2(y ap 2 ) = (x 2ap)(p + q) 2y 2ap 2 = px + qx 2ap 2 2apq y + apq = 1 x(p + q) 2 y 1 x(p + q) + apq = 0 (15) 2 Focal Chord Properties A focal chord is a chord that goes through the focus of the parabola at (0, a). pq = 1 1. Obtain the equation of the chord using (15): 2. Substitute in (0, a): y 1 x(p + q) + apq = 0 2 a + apq = 0 apq = a pq = 1 (16) Copyright c 2006 Enoch Lau. All Rights Reserved. For personal use only. 3 of 6

4 If P Q is a focal chord, the tangents at P and Q will meet at the directrix at right angles. Using (9) and (10), the tangents will intersect at (a(p + q), apq) = (a(p + q), a), since pq = 1, by (16). The directrix of the parabola x 2 = 4ay is y = a, and so the point lies on the directrix. Also, m P X m QX = pq = 1, and hence P X QX, where X is the point of intersection. Chord of Contact For a given point P (x 0, y 0 ), the chord of contact (the chord that joins the two points whose tangents pass through P ) is xx 0 = 2a(y + y 0 ). 1. Suppose Q(x 1, y 1 ) and R(x 2, y 2 ) are the two points with tangents passing through P. The equation of the tangent at Q can be found by applying y y 1 = m(x x 1 ): y y 1 = x 1 2a (x x 1) 2a(y y 1 ) = xx 1 x 2 1 2a(y y 1 ) = xx 1 4ay 1 since the point lies on x 2 = 4ay xx 1 = 2a(y + y 1 ) (17) Similarly, the equation of the tangent at R is: xx 2 = 2a(y + y 2 ) (18) 2. However, since P (x 0, y 0 ) lies on both tangents, we substitute P into (17) and (18): x 0 x 1 = 2a(y 0 + y 1 ) (19) x 0 x 2 = 2a(y 0 + y 2 ) (20) 3. From (19) and (20), it is clear that Q and R lie on the line: xx 0 = 2a(y + y 0 ) (21) Copyright c 2006 Enoch Lau. All Rights Reserved. For personal use only. 4 of 6

5 Angle of Incidence Equals Angle of Reflection 1. Find the position of A by substituting x = 0 into the equation of the tangent at P from (5): y = ap 2 A is at (0, ap 2 ). 2. Find distances: AS = a + ap 2, SP = P M = a + ap Therefore, ASP is isosceles, and so SAP = β. 4. However, SAP = α (corresponding angles on parallel lines), and hence α = β. Parametric Representation of a Circle Parametric equations x = r cos θ (22) y = r sin θ (23) A variable point on the circle is given by (r cos θ, r sin θ), for constant r and parameter θ. Conversion into Cartesian equation Squaring (22) and (23) gives x 2 = r 2 cos 2 θ and y 2 = r 2 sin 2 θ, and thus: x 2 + y 2 = r 2 ( cos 2 θ + sin 2 θ ) = r 2 which is the equation of a circle with centre (0, 0) and radius r. Copyright c 2006 Enoch Lau. All Rights Reserved. For personal use only. 5 of 6

6 Parametric Representation of an Ellipse Parametric equations x = a cos θ (24) y = b sin θ (25) A variable point on the ellipse is given by (a cos θ, b sin θ), for constants a and b, and parameter θ. Conversion into Cartesian equation Squaring (24) and (25): Substitute (27) into (26): x 2 = a 2 cos 2 θ = a 2 ( 1 sin 2 θ ) (26) y 2 = b 2 sin 2 θ sin 2 θ = y2 b 2 (27) ( ) x 2 = a 2 1 y2 b 2 x 2 a 2 = 1 y2 b 2 x 2 a 2 + y2 b 2 = 1 which is the equation of an ellipse with centre (0, 0), length of major axis (which is along the x-axis) 2a and length of minor axis 2b. Ellipse as Generalisation of Circle A circle is an ellipse with equal lengths for the semi-major and semi-minor axes, and thus (24) and (25) reduce to (22) and (23) respectively, when a = b = r. Use of parameterisation It is convenient to express curves in parameterised form, because it allows you to differentiate and integrate termwise. For example, if we have the x- and y-coordinates of a particle expressed as a function of the parameter time, t: r(t) = (x(t), y(t)) (28) we can obtain the x- and y-components of the velocity and acceleration. These are, respectively: v(t) = (x (t), y (t)) (29) a(t) = (x (t), y (t)) (30) Copyright c 2006 Enoch Lau. All Rights Reserved. For personal use only. 6 of 6

### The Parabola and the Circle

The Parabola and the Circle The following are several terms and definitions to aid in the understanding of parabolas. 1.) Parabola - A parabola is the set of all points (h, k) that are equidistant from

### Chapter 12. The Straight Line

302 Chapter 12 (Plane Analytic Geometry) 12.1 Introduction: Analytic- geometry was introduced by Rene Descartes (1596 1650) in his La Geometric published in 1637. Accordingly, after the name of its founder,

### Preliminary Mathematics (2U)

Preliminary Mathematics (2U) Week 2 Student Name:. Class code:.. Teacher name:.. DUX Week 2 Theory A Parabola: 2 A parabola may be defined as the locus of a point P(x, y) whose distance from a given fixed

### cos Newington College HSC Mathematics Ext 1 Trial Examination 2011 QUESTION ONE (12 Marks) (b) Find the exact value of if. 2 . 3

1 QUESTION ONE (12 Marks) Marks (a) Find tan x e 1 2 cos dx x (b) Find the exact value of if. 2 (c) Solve 5 3 2x 1. 3 (d) If are the roots of the equation 2 find the value of. (e) Use the substitution

### Test Bank Exercises in. 7. Find the intercepts, the vertical asymptote, and the slant asymptote of the graph of

Test Bank Exercises in CHAPTER 5 Exercise Set 5.1 1. Find the intercepts, the vertical asymptote, and the horizontal asymptote of the graph of 2x 1 x 1. 2. Find the intercepts, the vertical asymptote,

### Conic Sections in Cartesian and Polar Coordinates

Conic Sections in Cartesian and Polar Coordinates The conic sections are a family of curves in the plane which have the property in common that they represent all of the possible intersections of a plane

### decide, when given the eccentricity of a conic, whether the conic is an ellipse, a parabola or a hyperbola;

Conic sections In this unit we study the conic sections. These are the curves obtained when a cone is cut by a plane. We find the equations of one of these curves, the parabola, by using an alternative

### Assignment 3. Solutions. Problems. February 22.

Assignment. Solutions. Problems. February.. Find a vector of magnitude in the direction opposite to the direction of v = i j k. The vector we are looking for is v v. We have Therefore, v = 4 + 4 + 4 =.

### STRAIGHT LINES. , y 1. tan. and m 2. 1 mm. If we take the acute angle between two lines, then tan θ = = 1. x h x x. x 1. ) (x 2

STRAIGHT LINES Chapter 10 10.1 Overview 10.1.1 Slope of a line If θ is the angle made by a line with positive direction of x-axis in anticlockwise direction, then the value of tan θ is called the slope

### September 14, Conics. Parabolas (2).notebook

9/9/16 Aim: What is parabola? Do now: 1. How do we define the distance from a point to the line? Conic Sections are created by intersecting a set of double cones with a plane. A 2. The distance from the

5.1 The Unit Circle Copyright Cengage Learning. All rights reserved. Objectives The Unit Circle Terminal Points on the Unit Circle The Reference Number 2 The Unit Circle In this section we explore some

### Midterm Exam I, Calculus III, Sample A

Midterm Exam I, Calculus III, Sample A 1. (1 points) Show that the 4 points P 1 = (,, ), P = (, 3, ), P 3 = (1, 1, 1), P 4 = (1, 4, 1) are coplanar (they lie on the same plane), and find the equation of

### 10-5 Parabolas. Warm Up Lesson Presentation Lesson Quiz. Holt Algebra 2 2

10-5 Parabolas Warm Up Lesson Presentation Lesson Quiz 2 Warm Up 1. Given, solve for p when c = Find each distance. 2. from (0, 2) to (12, 7) 13 3. from the line y = 6 to (12, 7) 13 Objectives Write the

### Chapter 10: Topics in Analytic Geometry

Chapter 10: Topics in Analytic Geometry 10.1 Parabolas V In blue we see the parabola. It may be defined as the locus of points in the plane that a equidistant from a fixed point (F, the focus) and a fixed

### Multivariable Calculus MA22S1. Dr Stephen Britton

Multivariable Calculus MAS Dr Stephen Britton September 3 Introduction The aim of this course to introduce you to the basics of multivariable calculus. The early part is dedicated to discussing curves

### Analyzing Piecewise Functions

Connecting Geometry to Advanced Placement* Mathematics A Resource and Strategy Guide Updated: 04/9/09 Analyzing Piecewise Functions Objective: Students will analyze attributes of a piecewise function including

### 17.1. Conic Sections. Introduction. Prerequisites. Learning Outcomes. Learning Style

Conic Sections 17.1 Introduction The conic sections (or conics) - the ellipse, the parabola and the hperbola - pla an important role both in mathematics and in the application of mathematics to engineering.

### Name: Class: Date: Conics Multiple Choice Post-Test. Multiple Choice Identify the choice that best completes the statement or answers the question.

Name: Class: Date: Conics Multiple Choice Post-Test Multiple Choice Identify the choice that best completes the statement or answers the question. 1 Graph the equation x = 4(y 2) 2 + 1. Then describe the

### Essential Question: What is the relationship among the focus, directrix, and vertex of a parabola?

Name Period Date: Topic: 9-3 Parabolas Essential Question: What is the relationship among the focus, directrix, and vertex of a parabola? Standard: G-GPE.2 Objective: Derive the equation of a parabola

### CIRCLE COORDINATE GEOMETRY

CIRCLE COORDINATE GEOMETRY (EXAM QUESTIONS) Question 1 (**) A circle has equation x + y = 2x + 8 Determine the radius and the coordinates of the centre of the circle. r = 3, ( 1,0 ) Question 2 (**) A circle

### Solutions to old Exam 1 problems

Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections

### Section 1.8 Coordinate Geometry

Section 1.8 Coordinate Geometry The Coordinate Plane Just as points on a line can be identified with real numbers to form the coordinate line, points in a plane can be identified with ordered pairs of

### Example 1. Example 1 Plot the points whose polar coordinates are given by

Polar Co-ordinates A polar coordinate system, gives the co-ordinates of a point with reference to a point O and a half line or ray starting at the point O. We will look at polar coordinates for points

### Senior Math Circles February 4, 2009 Conics I

1 University of Waterloo Faculty of Mathematics Conics Senior Math Circles February 4, 2009 Conics I Centre for Education in Mathematics and Computing The most mathematically interesting objects usually

### Chapter 10: Analytic Geometry

10.1 Parabolas Chapter 10: Analytic Geometry We ve looked at parabolas before when talking about the graphs of quadratic functions. In this section, parabolas are discussed from a geometrical viewpoint.

### Tangent and normal lines to conics

4.B. Tangent and normal lines to conics Apollonius work on conics includes a study of tangent and normal lines to these curves. The purpose of this document is to relate his approaches to the modern viewpoints

### Contents. 2 Lines and Circles 3 2.1 Cartesian Coordinates... 3 2.2 Distance and Midpoint Formulas... 3 2.3 Lines... 3 2.4 Circles...

Contents Lines and Circles 3.1 Cartesian Coordinates.......................... 3. Distance and Midpoint Formulas.................... 3.3 Lines.................................. 3.4 Circles..................................

### Engineering Drawing. Anup Ghosh

Engineering Drawing Anup Ghosh Divide a line in n equal segments. Divide a line in n equal segments. Divide a line in n equal segments. Divide a line in n equal segments. Divide a line in n equal segments.

### Section 13.5 Equations of Lines and Planes

Section 13.5 Equations of Lines and Planes Generalizing Linear Equations One of the main aspects of single variable calculus was approximating graphs of functions by lines - specifically, tangent lines.

### Free Response Questions Compiled by Kaye Autrey for face-to-face student instruction in the AP Calculus classroom

Free Response Questions 1969-005 Compiled by Kaye Autrey for face-to-face student instruction in the AP Calculus classroom 1 AP Calculus Free-Response Questions 1969 AB 1 Consider the following functions

### Class 11 Maths Chapter 10. Straight Lines

1 P a g e Class 11 Maths Chapter 10. Straight Lines Coordinate Geometry The branch of Mathematics in which geometrical problem are solved through algebra by using the coordinate system, is known as coordinate

### Locus and the Parabola

Locus and the Parabola TERMINOLOGY Ais: A line around which a curve is reflected eg the ais of symmetry of a parabola Cartesian equation: An equation involving two variables and y Chord: An interval joining

### Assignment 5 Math 101 Spring 2009

Assignment 5 Math 11 Spring 9 1. Find an equation of the tangent line(s) to the given curve at the given point. (a) x 6 sin t, y t + t, (, ). (b) x cos t + cos t, y sin t + sin t, ( 1, 1). Solution. (a)

### The Locus of the Focus of a Rolling Parabola

The Locus of the Focus of a Rolling Parabola Anurag Agarwal & James Marengo October 6, 2009 1 Introduction The catenary can easily be mistaken for a parabola. Even Galileo made this error. In his Discorsi

### Parametric Curves, Vectors and Calculus. Jeff Morgan Department of Mathematics University of Houston

Parametric Curves, Vectors and Calculus Jeff Morgan Department of Mathematics University of Houston jmorgan@math.uh.edu Online Masters of Arts in Mathematics at the University of Houston http://www.math.uh.edu/matweb/grad_mam.htm

### EER#21- Graph parabolas and circles whose equations are given in general form by completing the square.

EER#1- Graph parabolas and circles whose equations are given in general form by completing the square. Circles A circle is a set of points that are equidistant from a fixed point. The distance is called

### www.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates

Further Pure Summary Notes. Roots of Quadratic Equations For a quadratic equation ax + bx + c = 0 with roots α and β Sum of the roots Product of roots a + b = b a ab = c a If the coefficients a,b and c

### Lesson 19: Equations for Tangent Lines to Circles

Student Outcomes Given a circle, students find the equations of two lines tangent to the circle with specified slopes. Given a circle and a point outside the circle, students find the equation of the line

### 7-1 Parabolas 62/87,21 62/87,21 62/87,21 62/87,21

For each equation, identify the vertex, focus, axis of symmetry, and directrix. Then graph the parabola. (x 3) = 1(y 7) 6/87,1 The equation is in standard form and the squared term is x, which means that

### Mathematics Extension 1

Girraween High School 05 Year Trial Higher School Certificate Mathematics Extension General Instructions Reading tjmc - 5 mjnutcs Working time- hours Write using black or blue pen Black pen is preferred

### Differentiation of vectors

Chapter 4 Differentiation of vectors 4.1 Vector-valued functions In the previous chapters we have considered real functions of several (usually two) variables f : D R, where D is a subset of R n, where

### Roots and Coefficients of a Quadratic Equation Summary

Roots and Coefficients of a Quadratic Equation Summary For a quadratic equation with roots α and β: Sum of roots = α + β = and Product of roots = αβ = Symmetrical functions of α and β include: x = and

### Linear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices

MATH 30 Differential Equations Spring 006 Linear algebra and the geometry of quadratic equations Similarity transformations and orthogonal matrices First, some things to recall from linear algebra Two

### Conics. Find the equation of the parabola which has its vertex at the origin and its focus at point F in the following cases.

Conics 1 Find the equation of the parabola which has its vertex at the origin and its focus at point F in the following cases. a) F(, 0) b) F(0,-4) c) F(-3,0) d) F(0, 5) In the Cartesian plane, represent

### Parametric Curves. EXAMPLE: Sketch and identify the curve defined by the parametric equations

Section 9. Parametric Curves 00 Kiryl Tsishchanka Parametric Curves Suppose that x and y are both given as functions of a third variable t (called a parameter) by the equations x = f(t), y = g(t) (called

### Homework 3 Model Solution Section

Homework 3 Model Solution Section 12.6 13.1. 12.6.3 Describe and sketch the surface + z 2 = 1. If we cut the surface by a plane y = k which is parallel to xz-plane, the intersection is + z 2 = 1 on a plane,

### 1.(6pts) Find symmetric equations of the line L passing through the point (2, 5, 1) and perpendicular to the plane x + 3y z = 9.

.(6pts Find symmetric equations of the line L passing through the point (, 5, and perpendicular to the plane x + 3y z = 9. (a x = y + 5 3 = z (b x (c (x = ( 5(y 3 = z + (d x (e (x + 3(y 3 (z = 9 = y 3

### For each equation, identify the vertex, focus, axis of symmetry, and directrix. Then graph the parabola.

For each equation, identify the vertex, focus, axis of symmetry, and directrix. Then graph the parabola. 1. (x 3) 2 = 12(y 7) The equation is in standard form and the squared term is x, which means that

10.9 Systems Of Inequalities Copyright Cengage Learning. All rights reserved. Objectives Graphing an Inequality Systems of Inequalities Systems of Linear Inequalities Application: Feasible Regions 2 Graphing

### Astromechanics Two-Body Problem (Cont)

5. Orbit Characteristics Astromechanics Two-Body Problem (Cont) We have shown that the in the two-body problem, the orbit of the satellite about the primary (or vice-versa) is a conic section, with the

### L 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has

The line L through the points A and B is parallel to the vector AB = 3, 2, and has parametric equations x = 3t + 2, y = 2t +, z = t Therefore, the intersection point of the line with the plane should satisfy:

### 4.4 Transforming Circles

Specific Curriculum Outcomes. Transforming Circles E13 E1 E11 E3 E1 E E15 analyze and translate between symbolic, graphic, and written representation of circles and ellipses translate between different

### Section 10.2 The Parabola

243 Section 10.2 The Parabola Recall that the graph of f(x) = x 2 is a parabola with vertex of (0, 0) and axis of symmetry of x = 0: x = 0 f(x) = x 2 vertex: (0, 0) In this section, we want to expand our

### Determine whether the following lines intersect, are parallel, or skew. L 1 : x = 6t y = 1 + 9t z = 3t. x = 1 + 2s y = 4 3s z = s

Homework Solutions 5/20 10.5.17 Determine whether the following lines intersect, are parallel, or skew. L 1 : L 2 : x = 6t y = 1 + 9t z = 3t x = 1 + 2s y = 4 3s z = s A vector parallel to L 1 is 6, 9,

### 1.7 Cylindrical and Spherical Coordinates

56 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.7 Cylindrical and Spherical Coordinates 1.7.1 Review: Polar Coordinates The polar coordinate system is a two-dimensional coordinate system in which the

1.2 GRAPHS OF EQUATIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Sketch graphs of equations. Find x- and y-intercepts of graphs of equations. Use symmetry to sketch graphs

Date Period Unit 0: Quadratic Relations DAY TOPIC Distance and Midpoint Formulas; Completing the Square Parabolas Writing the Equation 3 Parabolas Graphs 4 Circles 5 Exploring Conic Sections video This

### Techniques of Differentiation Selected Problems. Matthew Staley

Techniques of Differentiation Selected Problems Matthew Staley September 10, 011 Techniques of Differentiation: Selected Problems 1. Find /dx: (a) y =4x 7 dx = d dx (4x7 ) = (7)4x 6 = 8x 6 (b) y = 1 (x4

### Name: x 2 + 4x + 4 = -6y p = -6 (opens down) p = -3/2 = (x + 2) 2 = -6y + 6. (x + 2) 2 = -6(y - 1) standard form: (-2, 1) Vertex:

Write the equation of the parabola in standard form and sketch the graph of the parabola, labeling all points, and using the focal width as a guide for the width of the parabola. Find the vertex, focus,

### Second Year Algebra. Gobel, Gordon, et. al. Walla Walla High School

Second Year Algebra Gobel, Gordon, et al Walla Walla High School c 010 Contents 1 Quadratic Equations and Functions 1 11 Simple quadratic equations 1 Solving quadratic equations by factoring 3 13 Completing

### Mid-Chapter Quiz: Lessons 9-1 through 9-4. Find the midpoint of the line segment with endpoints at the given coordinates (7, 4), ( 1, 5)

Find the midpoint of the line segment with endpoints at the given coordinates (7, 4), (1, 5) Let (7, 4) be (x 1, y 1 ) and (1, 5) be (x 2, y 2 ). (2, 9), (6, 0) Let (2, 9) be (x 1, y 1 ) and (6, 0) be

### Review Sheet for Test 1

Review Sheet for Test 1 Math 261-00 2 6 2004 These problems are provided to help you study. The presence of a problem on this handout does not imply that there will be a similar problem on the test. And

### Copyrighted Material. Chapter 1 DEGREE OF A CURVE

Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two

### Figure 1: Volume between z = f(x, y) and the region R.

3. Double Integrals 3.. Volume of an enclosed region Consider the diagram in Figure. It shows a curve in two variables z f(x, y) that lies above some region on the xy-plane. How can we calculate the volume

### Section 10.4: Motion in Space: Velocity and Acceleration

1 Section 10.4: Motion in Space: Velocity and Acceleration Velocity and Acceleration Practice HW from Stewart Textbook (not to hand in) p. 75 # 3-17 odd, 1, 3 Given a vector function r(t ) = f (t) i +

### Section summaries. d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2. 1 + y 2. x1 + x 2

Chapter 2 Graphs Section summaries Section 2.1 The Distance and Midpoint Formulas You need to know the distance formula d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 and the midpoint formula ( x1 + x 2, y ) 1 + y 2

### www.sakshieducation.com

LENGTH OF THE PERPENDICULAR FROM A POINT TO A STRAIGHT LINE AND DISTANCE BETWEEN TWO PAPALLEL LINES THEOREM The perpendicular distance from a point P(x 1, y 1 ) to the line ax + by + c 0 is ax1+ by1+ c

### Area Between Curves. The idea: the area between curves y = f(x) and y = g(x) (if the graph of f(x) is above that of g(x)) for a x b is given by

MATH 42, Fall 29 Examples from Section, Tue, 27 Oct 29 1 The First Hour Area Between Curves. The idea: the area between curves y = f(x) and y = g(x) (if the graph of f(x) is above that of g(x)) for a x

### Applications of Integration to Geometry

Applications of Integration to Geometry Volumes of Revolution We can create a solid having circular cross-sections by revolving regions in the plane along a line, giving a solid of revolution. Note that

### AB2.2: Curves. Gradient of a Scalar Field

AB2.2: Curves. Gradient of a Scalar Field Parametric representation of a curve A a curve C in space can be represented by a vector function r(t) = [x(t), y(t), z(t)] = x(t)i + y(t)j + z(t)k This is called

### Section 11.1: Vectors in the Plane. Suggested Problems: 1, 5, 9, 17, 23, 25-37, 40, 42, 44, 45, 47, 50

Section 11.1: Vectors in the Plane Page 779 Suggested Problems: 1, 5, 9, 17, 3, 5-37, 40, 4, 44, 45, 47, 50 Determine whether the following vectors a and b are perpendicular. 5) a = 6, 0, b = 0, 7 Recall

### GEOMETRY & INEQUALITIES. (1) State the Triangle Inequality. In addition, give an argument explaining why it should be true.

GEOMETRY & INEQUALITIES LAMC INTERMEDIATE GROUP - 2/09/14 The Triangle Inequality! (1) State the Triangle Inequality. In addition, give an argument explaining why it should be true. Given the three side

### AP Calculus AB 2010 Free-Response Questions Form B

AP Calculus AB 2010 Free-Response Questions Form B The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity.

### PROBLEM SET. Practice Problems for Exam #1. Math 1352, Fall 2004. Oct. 1, 2004 ANSWERS

PROBLEM SET Practice Problems for Exam # Math 352, Fall 24 Oct., 24 ANSWERS i Problem. vlet R be the region bounded by the curves x = y 2 and y = x. A. Find the volume of the solid generated by revolving

### Drawing the Parabolic Trajectory of an Object under Gravity

Drawing the Parabolic Trajectory of an Object under Gravity by Chao-Kuei Hung Email: ckhung@drgeo.eu February 13, 2016 Abstract When you throw an object into the air, its trajectory is a parabola. How

### x 2 + y 2 = 1 y 1 = x 2 + 2x y = x 2 + 2x + 1

Implicit Functions Defining Implicit Functions Up until now in this course, we have only talked about functions, which assign to every real number x in their domain exactly one real number f(x). The graphs

### Applications of Integration Day 1

Applications of Integration Day 1 Area Under Curves and Between Curves Example 1 Find the area under the curve y = x2 from x = 1 to x = 5. (What does it mean to take a slice?) Example 2 Find the area under

### Tangent line of a circle can be determined once the tangent point or the slope of the line is known.

Worksheet 7: Tangent Line of a Circle Name: Date: Tangent line of a circle can be determined once the tangent point or the slope of the line is known. Straight line: an overview General form : Ax + By

### Biggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress

Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation

### 7-2 Ellipses and Circles

Graph the ellipse given by each equation. 1. + = 1 The ellipse is in standard form, where h = 2, k = 0, a = or 7, b = or 3, and c = or about 6.3. The orientation is vertical because the y term contains

### 1916 New South Wales Leaving Certificate

1916 New South Wales Leaving Certificate Typeset by AMS-TEX New South Wales Department of Education PAPER I Time Allowed: Hours 1. Prove that the radical axes of three circles taken in pairs are concurrent.

### What is a parabola? It is geometrically defined by a set of points or locus of points that are

Section 6-1 A Parable about Parabolas Name: What is a parabola? It is geometrically defined by a set of points or locus of points that are equidistant from a point (the focus) and a line (the directrix).

### 2015 Mathematics. New Higher Paper 1. Finalised Marking Instructions

National Qualifications 0 0 Mathematics New Higher Paper Finalised Marking Instructions Scottish Qualifications Authority 0 The information in this publication may be reproduced to support SQA qualifications

### Mathematics 1. Lecture 5. Pattarawit Polpinit

Mathematics 1 Lecture 5 Pattarawit Polpinit Lecture Objective At the end of the lesson, the student is expected to be able to: familiarize with the use of Cartesian Coordinate System. determine the distance

### 2. THE x-y PLANE 7 C7

2. THE x-y PLANE 2.1. The Real Line When we plot quantities on a graph we can plot not only integer values like 1, 2 and 3 but also fractions, like 3½ or 4¾. In fact we can, in principle, plot any real

### Extra Problems for Midterm 2

Extra Problems for Midterm Sudesh Kalyanswamy Exercise (Surfaces). Find the equation of, and classify, the surface S consisting of all points equidistant from (0,, 0) and (,, ). Solution. Let P (x, y,

PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient

### *X100/12/02* X100/12/02. MATHEMATICS HIGHER Paper 1 (Non-calculator) NATIONAL QUALIFICATIONS 2014 TUESDAY, 6 MAY 1.00 PM 2.30 PM

X00//0 NTIONL QULIFITIONS 0 TUESY, 6 MY.00 PM.0 PM MTHEMTIS HIGHER Paper (Non-calculator) Read carefully alculators may NOT be used in this paper. Section Questions 0 (0 marks) Instructions for completion

### Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis

Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis 2. Polar coordinates A point P in a polar coordinate system is represented by an ordered pair of numbers (r, θ). If r >

### Section 10.7 Parametric Equations

299 Section 10.7 Parametric Equations Objective 1: Defining and Graphing Parametric Equations. Recall when we defined the x- (rcos(θ), rsin(θ)) and y-coordinates on a circle of radius r as a function of

### Engineering Geometry

Engineering Geometry Objectives Describe the importance of engineering geometry in design process. Describe coordinate geometry and coordinate systems and apply them to CAD. Review the right-hand rule.

### Functions and Equations

Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c

### Mathematics Chapter 8 and 10 Test Summary 10M2

Quadratic expressions and equations Expressions such as x 2 + 3x, a 2 7 and 4t 2 9t + 5 are called quadratic expressions because the highest power of the variable is 2. The word comes from the Latin quadratus

### Projectile Motion. directions simultaneously. deal with is called projectile motion. ! An object may move in both the x and y

Projectile Motion! An object may move in both the x and y directions simultaneously! The form of two-dimensional motion we will deal with is called projectile motion Assumptions of Projectile Motion! The

### Finding generalized parabolas. Generalized Parabolas

Generalized Parabolas Dan Joseph, Gregory Hartman, and Caleb Gibson Dan Joseph (josephds@vmi.edu) is an Associate Professor of Mathematics at the Virginia Military Institute, where he received his BS in