SECTION 11-5 Parametric Equations. Parametric Equations and Plane Curves. Parametric Equations and Plane Curves Projectile Motion Cycloid
|
|
- Derek Cook
- 7 years ago
- Views:
Transcription
1 - Parametric Equations 9. A hperbola with foci (, ) and (6, ) and vertices (, ) and (, ).. An ellipse with foci (, ) and (, 6) and vertices (, ) and (, ).. A parabola with ais the ais and passing through the points (, ) and (, ).. A parabola with verte at (6, ), ais the line, and passing through the point (, 7).. An ellipse with vertices (, ), and (, ) that passes through the origin.. A hperbola with vertices at (, ), and (, ) that passes through the point (, ). C In Problems, find the coordinates of an foci relative to the original coordinate sstem:. Problem 6. Problem 6 7. Problem 7. Problem 9. Problem. Problem In Problems, use a graphing utilit to find the coordinates of all points of intersection to two decimal places.. 7, ,. 7,., 6 SECTION - Parametric Equations Parametric Equations and Plane Curves Projectile Motion Ccloid Parametric Equations and Plane Curves Consider the two equations t t t t () Each value of t determines a value of, a value of, and hence, an ordered pair (, ). To graph the set of ordered pairs (, ) determined b letting t assume all real values, we construct Table listing selected values of t and the corresponding values of and. Then we plot the ordered pairs (, ) and connect them with a continuous curve, as shown in Figure. The variable t is called a parameter and does not appear on the graph. Equations () are called parametric equations because both and are epressed in terms of the parameter t. The graph of the ordered pairs (, ) is called a plane curve. TABLE t FIGURE Graph of t, t t, t. In some cases it is possible to eliminate the parameter b solving one of the equations for t and substituting into the other. In the eample just considered, solving the first equation for t in terms of, we have t
2 Additional Topics in Analtic Geometr Then, substituting the result into the second equation, we obtain ( ) ( ) We recognize this as the equation of a parabola, as we would guess from Figure. In other cases, it ma not be eas or possible to eliminate the parameter to obtain an equation in just and. For eample, for t log t t e t t ou will not find it possible to solve either equation for t in terms of functions we have considered. Is there more than one parametric representation for a plane curve? The answer is es. In fact, there is an unlimited number of parametric representations for the same plane curve. The following are two additional representations of the parabola in Figure. t t t t t t t () t () The concepts introduced in the preceding discussion are summarized in Definition. DEFINITION Parametric Equations and Plane Curves A plane curve is the set of points (, ) determined b the parametric equations f(t) g(t) where the parameter t varies over an interval I and the functions f and g are both defined on the interval I. Wh are we interested in parametric representations of plane curves? It turns out that this approach is more general than using equations with two variables as we have been doing. In addition, the approach generalizes to curves in three- and higher-dimensional spaces. Other important reasons for using parametric representations of plane curves will be brought out in the discussion and eamples that follow.
3 - Parametric Equations EXAMPLE Graphing Parametric Equations and Eliminating the Parameter Graph the plane curve given parametricall b cos sin () Identif the curve b eliminating the parameter. Solution Construct a table and graph: /6 / / / /6 7/6 / / / /6 FIGURE Graph of cos, sin,. To eliminate the parameter, we solve the first equation in () for cos, the second for sin, and substitute into the Pthagorean identit cos sin : The graph is an ellipse (Fig. ). cos and sin cos sin 6 6 Matched Problem Graph the plane curve given parametricall b cos, sin,. Identif the curve b eliminating the parameter. EXPLORE-DISCUSS Graph one period ( ) of each of the three plane curves given parametricall b cos sin cos sin cos sin Identif the curves b eliminating the parameter.
4 Additional Topics in Analtic Geometr EXAMPLE Parametric Equations for Conic Sections Find parametric equations for the conic section with the given equation: (A) 9 (B) 6 7 Solutions (A) B completing the square in and we obtain the standard form ( ) ( ). So the graph is an ellipse with center (, ) and 9 major ais on the line. Since cos sin, a parametric representation with parameter is obtained b letting cos, sin : cos sin (B) B completing the square in and we obtain the standard form ( ) ( ). So the graph is a hperbola with center (, ) and 6 transverse ais on the line. Since sec tan, a parametric representation with parameter is obtained b letting sec, tan : sec tan, k, k an integer FIGURE sec, tan. Note that when the parametric equations are graphed using a graphing utilit in connected mode, the graph appears to show the asmptotes of the hperbola (see Fig. ). Matched Problem Find parametric equations for the conic section with the given equation: (A) (B) Projectile Motion Newton s laws and advanced mathematics can be used to determine the path of a projectile. If v is the initial speed of the projectile at an angle with the horizontal (see Fig. ) and air resistance is neglected, then the path of the projectile is given b (v cos )t (v sin )t.9t t b ()
5 - Parametric Equations The parameter t represents time in seconds, and and are distances measured in meters. Solving the first equation in () for t in terms of, substituting into the second equation, and simplifing results in the following equation:.9 (tan ) (6) v cos You should verif this b suppling the omitted details. FIGURE Projectile motion. v v cos v sin We recognize equation (6) as a parabola. This equation in and describes the path the projectile follows but tells us little else about its flight. On the other hand, the parametric equations () not onl determine the path of the projectile but also tell us where it is at an time t. Furthermore, using concepts from phsics and calculus, the parametric equations can be used to determine the velocit and acceleration of the projectile at an time t. This illustrates another advantage of using parametric representations of plane curves. The range of a projectile is the distance from the point of firing to the point of impact. If we keep the initial speed v of the projectile constant and var the angle in Figure, we obtain different parabolic paths followed b the projectile and different ranges. The maimum range is obtained when. Furthermore, assuming that the projectile alwas stas in the same vertical plane, then there are points in the air and on the ground that the projectile cannot reach, irrespective of the angle used,. Using more advanced mathematics, it can be shown that the reachable region is separated from the nonreachable region b a parabola called an envelope of the other parabolas (see Fig. ). FIGURE Reachable region of a projectile. Envelope Ccloid We now consider an unusual curve called a ccloid, which has a fairl simple parametric representation and a ver complicated representation in terms of and onl. The path traced b a point on the rim of a circle that rolls along a line is called a ccloid. To derive parametric equations for a ccloid we roll a circle of radius a along the ais with the tracing point P on the rim starting at the origin (see Figure 6).
6 6 Additional Topics in Analtic Geometr FIGURE 6 Ccloid. P(, ) a C Q O R S Since the circle rolls along the ais without slipping (refer to Figure 6), we see that d(o, S) arc PS a in radians (7) where S is the point of contact between the circle and the ais. Referring to triangle CPQ, we see that Using these results, we have d(o, R) d(o, S) d(r, S) (arc PS) d(p, Q) d(p, Q) a sin / () d(q, C) a cos / (9) a a sin Use equations (7) and (). d(r, P) d(s, C) d(q, C) a a cos Use equation (9) and the fact that d(s, C) a. Even though in equations () and (9) was restricted so that /, it can be shown that the derived parametric equations generate the whole ccloid for. The graph specifies a periodic function with period a. Thus, in general, we have Theorem. Theorem Parametric Equations for a Ccloid For a circle of radius a rolled along the ais, the resulting ccloid generated b a point on the rim starting at the origin is given b a a sin a a cos
7 - Parametric Equations 7 P Q FIGURE 7 Ccloid path. The ccloid is a good eample of a curve that is ver difficult to represent without the use of a parameter. A ccloid has a ver interesting phsical propert. An object sliding without friction from a point P to a point Q lower than P, but not on the same vertical line as P, will arrive at Q in a shorter time traveling along a ccloid than on an other path (see Fig. 7). EXPLORE-DISCUSS (A) Let Q be a point b units from the center of a wheel of radius a, where b a. If the wheel rolls along the ais with the tracing point Q starting at (, a b), eplain wh parametric equations for the path of Q are given b a b sin a b cos (B) Use a graphing utilit to graph the paths of a point on the rim of a wheel of radius, and a point halfwa between the rim and center, as the wheel makes two complete revolutions rolling along the ais. Answers to Matched Problems. 6; circle. (A) 7 cos, 6 sin, (B) tan, sec,, k, k an integer EXERCISE - A In Problems, plot each plane curve b use of a table of values (see Eample ). Obtain an equation in and b eliminating the parameter, and identif the curve. (In this eercise set, the interval for the parameter is the whole real line, unless stated to the contrar.). t, t. t, t. t, t. t, t. t, t 6. t, t 7. t, t. t, t 9. t, t. t, t B In Problems, obtain an equation in and b eliminating the parameter. Use the simpler of the two forms to plot the curve. Name the curve if it is a curve we have identified.. sin, cos. sin, cos. sin, cos. sin, cos. t, ; t t
8 Additional Topics in Analtic Geometr 6. t, ; t t 7. t, t; t. t, t 9. If A, C, and E, find parametric equations for A C D E F. Identif the curve.. If A, C, and D, find parametric equations for A C D E F. Identif the curve. C In Problems 6, obtain an equation in and b eliminating the parameter. Use the simpler of the two forms to plot the curve. Name the curve if it is a curve we have identified.. t, t ; t. e t, e t. cos, sin. sec, tan. 6. t t, t, t t t t Graph, using a calculator, one period ( ) of each ccloid in Problems 7 and. 7. sin, cos. sin, cos In Problems 9, use a graphing utilit to graph the parametric equations. Then eliminate the parameter and find the standard equation for the curve. Name the curve and find its center cos t, sin t, t. sec t, tan t, t, t. tan t, sec t, t, t. cos t, sin t, t. Find an equation of the form A C D E F that has the same graph as the parametric equations tan t, tan t,. t. Repeat Problem for cot t, (t cot t)/t, t, t. In Problems, find the standard form for each equation. Name the curve and find its center. Use parametric equations to graph the curve on a graphing utilit APPLICATIONS 9. Plane Motion. An object follows a path as given b sin 6t cos 6t where t is time in seconds and and are distances in feet. (A) What are the coordinates of the object when t. second? Compute answers to one decimal place. (B) Eliminate the parameter and graph the resulting equation in and. Identif the path.. Plane Motion. Repeat Problem 9 for sin t cos t t t. Projectile Motion. A projectile is fired with an initial speed of meters per second at an angle of to the horizontal. Neglecting air resistance, find: (A) The time of impact (B) The horizontal distance covered (range) in meters and kilometers at time of impact (C) The maimum height in meters of the projectile Compute all answers to three decimal places using a calculator.. Projectile Motion. Repeat Problem if the same projectile is fired at to the horizontal instead of. CHAPTER GROUP ACTIVITY Focal Chords Man of the applications of the conic sections are based on their reflective or focal properties. One of the interesting algebraic properties of the conic sections concerns their focal chords. If a line through a focus F contains two points G and H of a conic section, then the line segment GH is called a focal chord. Let G(, ) and H(, ) be points on the graph of a such that GH is a focal chord. Let u denote the length of GF and v the length of FH (see Fig. ).
9 Chapter Review 9 FIGURE Focal chord GH of the parabola a. G F u v H (a, a) (A) Use the distance formula to show that u a. (B) Show that G and H lie on the line a m, where m ( )/( ). (C) Solve a m for and substitute in a, obtaining a quadratic equation in. Eplain wh a. (D) Show that. u v a (u a) (E) Show that u v a. Eplain wh this implies that u v a, with equalit if and onl u a if u v a. (F) Which focal chord is the shortest? Is there a longest focal chord? (G) Is a constant for focal chords of the ellipse? For focal chords of the hperbola? Obtain evidence u v for our answers b considering specific eamples. (H) The conic section with focus at the origin, directri the line D, and eccentricit E has the DE polar equation r. Eplain how this polar equation makes it eas to show that u v E cos a for a parabola. Use the polar equation to determine the sum for a focal chord of an ellipse or u v hperbola. Chapter Review - CONIC SECTIONS; PARABOLA The plane curves obtained b intersecting a right circular cone with a plane are called conic sections. If the plane cuts clear through one nappe, then the intersection curve is called a circle if the plane is perpendicular to the ais and an ellipse if the plane is not perpendicular to the ais. If a plane cuts onl one nappe, but does not cut clear through, then the intersection curve is called a parabola. If a plane cuts through both nappes, but not through the verte, the resulting intersection curve is called a hperbola. A plane passing through the verte of the cone produces a degenerate conic a point, a line, or a pair of lines. The figure illustrates the four nondegenerate conics. Circle Ellipse Parabola Hperbola
Ax 2 Cy 2 Dx Ey F 0. Here we show that the general second-degree equation. Ax 2 Bxy Cy 2 Dx Ey F 0. y X sin Y cos P(X, Y) X
Rotation of Aes ROTATION OF AES Rotation of Aes For a discussion of conic sections, see Calculus, Fourth Edition, Section 11.6 Calculus, Earl Transcendentals, Fourth Edition, Section 1.6 In precalculus
More informationD.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review
D0 APPENDIX D Precalculus Review SECTION D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane An ordered pair, of real numbers has as its
More information1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered
Conic Sections. Distance Formula and Circles. More on the Parabola. The Ellipse and Hperbola. Nonlinear Sstems of Equations in Two Variables. Nonlinear Inequalities and Sstems of Inequalities In Chapter,
More informationTHE PARABOLA 13.2. section
698 (3 0) Chapter 3 Nonlinear Sstems and the Conic Sections 49. Fencing a rectangle. If 34 ft of fencing are used to enclose a rectangular area of 72 ft 2, then what are the dimensions of the area? 50.
More informationD.3. Angles and Degree Measure. Review of Trigonometric Functions
APPENDIX D Precalculus Review D7 SECTION D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions Solving Trigonometric
More informationSection 5-9 Inverse Trigonometric Functions
46 5 TRIGONOMETRIC FUNCTIONS Section 5-9 Inverse Trigonometric Functions Inverse Sine Function Inverse Cosine Function Inverse Tangent Function Summar Inverse Cotangent, Secant, and Cosecant Functions
More information9.5 CALCULUS AND POLAR COORDINATES
smi9885_ch09b.qd 5/7/0 :5 PM Page 760 760 Chapter 9 Parametric Equations and Polar Coordinates 9.5 CALCULUS AND POLAR COORDINATES Now that we have introduced ou to polar coordinates and looked at a variet
More information7.3 Parabolas. 7.3 Parabolas 505
7. Parabolas 0 7. Parabolas We have alread learned that the graph of a quadratic function f() = a + b + c (a 0) is called a parabola. To our surprise and delight, we ma also define parabolas in terms of
More informationSECTION 2.2. Distance and Midpoint Formulas; Circles
SECTION. Objectives. Find the distance between two points.. Find the midpoint of a line segment.. Write the standard form of a circle s equation.. Give the center and radius of a circle whose equation
More informationDISTANCE, CIRCLES, AND QUADRATIC EQUATIONS
a p p e n d i g DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS DISTANCE BETWEEN TWO POINTS IN THE PLANE Suppose that we are interested in finding the distance d between two points P (, ) and P (, ) in the
More informationINVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1
Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.
More informationWarm-Up y. What type of triangle is formed by the points A(4,2), B(6, 1), and C( 1, 3)? A. right B. equilateral C. isosceles D.
CST/CAHSEE: Warm-Up Review: Grade What tpe of triangle is formed b the points A(4,), B(6, 1), and C( 1, 3)? A. right B. equilateral C. isosceles D. scalene Find the distance between the points (, 5) and
More information2.1 Three Dimensional Curves and Surfaces
. Three Dimensional Curves and Surfaces.. Parametric Equation of a Line An line in two- or three-dimensional space can be uniquel specified b a point on the line and a vector parallel to the line. The
More informationREVIEW 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
More informationGraphing Quadratic Equations
.4 Graphing Quadratic Equations.4 OBJECTIVE. Graph a quadratic equation b plotting points In Section 6.3 ou learned to graph first-degree equations. Similar methods will allow ou to graph quadratic equations
More informationClick here for answers.
CHALLENGE PROBLEMS: CHALLENGE PROBLEMS 1 CHAPTER A Click here for answers S Click here for solutions A 1 Find points P and Q on the parabola 1 so that the triangle ABC formed b the -ais and the tangent
More informationACT Math Vocabulary. Altitude The height of a triangle that makes a 90-degree angle with the base of the triangle. Altitude
ACT Math Vocabular Acute When referring to an angle acute means less than 90 degrees. When referring to a triangle, acute means that all angles are less than 90 degrees. For eample: Altitude The height
More informationNorth Carolina Community College System Diagnostic and Placement Test Sample Questions
North Carolina Communit College Sstem Diagnostic and Placement Test Sample Questions 0 The College Board. College Board, ACCUPLACER, WritePlacer and the acorn logo are registered trademarks of the College
More informationSolving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form
SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving
More informationPhysics 53. Kinematics 2. Our nature consists in movement; absolute rest is death. Pascal
Phsics 53 Kinematics 2 Our nature consists in movement; absolute rest is death. Pascal Velocit and Acceleration in 3-D We have defined the velocit and acceleration of a particle as the first and second
More informationShake, Rattle and Roll
00 College Board. All rights reserved. 00 College Board. All rights reserved. SUGGESTED LEARNING STRATEGIES: Shared Reading, Marking the Tet, Visualization, Interactive Word Wall Roller coasters are scar
More informationStart Accuplacer. Elementary Algebra. Score 76 or higher in elementary algebra? YES
COLLEGE LEVEL MATHEMATICS PRETEST This pretest is designed to give ou the opportunit to practice the tpes of problems that appear on the college-level mathematics placement test An answer ke is provided
More informationChapter 6 Quadratic Functions
Chapter 6 Quadratic Functions Determine the characteristics of quadratic functions Sketch Quadratics Solve problems modelled b Quadratics 6.1Quadratic Functions A quadratic function is of the form where
More informationMath, Trigonometry and Vectors. Geometry. Trig Definitions. sin(θ) = opp hyp. cos(θ) = adj hyp. tan(θ) = opp adj. Here's a familiar image.
Math, Trigonometr and Vectors Geometr Trig Definitions Here's a familiar image. To make predictive models of the phsical world, we'll need to make visualizations, which we can then turn into analtical
More informationSolutions to Exercises, Section 5.1
Instructor s Solutions Manual, Section 5.1 Exercise 1 Solutions to Exercises, Section 5.1 1. Find all numbers t such that ( 1 3,t) is a point on the unit circle. For ( 1 3,t)to be a point on the unit circle
More informationRotated Ellipses. And Their Intersections With Lines. Mark C. Hendricks, Ph.D. Copyright March 8, 2012
Rotated Ellipses And Their Intersections With Lines b Mark C. Hendricks, Ph.D. Copright March 8, 0 Abstract: This paper addresses the mathematical equations for ellipses rotated at an angle and how to
More informationsin(θ) = opp hyp cos(θ) = adj hyp tan(θ) = opp adj
Math, Trigonometr and Vectors Geometr 33º What is the angle equal to? a) α = 7 b) α = 57 c) α = 33 d) α = 90 e) α cannot be determined α Trig Definitions Here's a familiar image. To make predictive models
More informationREVIEW OF ANALYTIC GEOMETRY
REVIEW OF ANALYTIC GEOMETRY The points in a plane can be identified with ordered pairs of real numbers. We start b drawing two perpendicular coordinate lines that intersect at the origin O on each line.
More informationSection 6-3 Double-Angle and Half-Angle Identities
6-3 Double-Angle and Half-Angle Identities 47 Section 6-3 Double-Angle and Half-Angle Identities Double-Angle Identities Half-Angle Identities This section develops another important set of identities
More informationAnswers (Anticipation Guide and Lesson 10-1)
Answers (Anticipation Guide and Lesson 0-) Lesson 0- Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 0- NAME DATE PERID Lesson Reading Guide Midpoint and Distance Formulas Get
More informationCOMPONENTS OF VECTORS
COMPONENTS OF VECTORS To describe motion in two dimensions we need a coordinate sstem with two perpendicular aes, and. In such a coordinate sstem, an vector A can be uniquel decomposed into a sum of two
More information5.2 Inverse Functions
78 Further Topics in Functions. Inverse Functions Thinking of a function as a process like we did in Section., in this section we seek another function which might reverse that process. As in real life,
More informationPolynomial Degree and Finite Differences
CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial
More informationDownloaded from www.heinemann.co.uk/ib. equations. 2.4 The reciprocal function x 1 x
Functions and equations Assessment statements. Concept of function f : f (); domain, range, image (value). Composite functions (f g); identit function. Inverse function f.. The graph of a function; its
More informationCore Maths C2. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...
More informationAlgebra II. Administered May 2013 RELEASED
STAAR State of Teas Assessments of Academic Readiness Algebra II Administered Ma 0 RELEASED Copright 0, Teas Education Agenc. All rights reserved. Reproduction of all or portions of this work is prohibited
More informationAlgebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.
Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.
More informationSECTION 5-1 Exponential Functions
354 5 Eponential and Logarithmic Functions Most of the functions we have considered so far have been polnomial and rational functions, with a few others involving roots or powers of polnomial or rational
More information6. The given function is only drawn for x > 0. Complete the function for x < 0 with the following conditions:
Precalculus Worksheet 1. Da 1 1. The relation described b the set of points {(-, 5 ),( 0, 5 ),(,8 ),(, 9) } is NOT a function. Eplain wh. For questions - 4, use the graph at the right.. Eplain wh the graph
More informationC3: Functions. Learning objectives
CHAPTER C3: Functions Learning objectives After studing this chapter ou should: be familiar with the terms one-one and man-one mappings understand the terms domain and range for a mapping understand the
More informationColegio del mundo IB. Programa Diploma REPASO 2. 1. The mass m kg of a radio-active substance at time t hours is given by. m = 4e 0.2t.
REPASO. The mass m kg of a radio-active substance at time t hours is given b m = 4e 0.t. Write down the initial mass. The mass is reduced to.5 kg. How long does this take?. The function f is given b f()
More informationSolutions 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
More informationSECTION 7-4 Algebraic Vectors
7-4 lgebraic Vectors 531 SECTIN 7-4 lgebraic Vectors From Geometric Vectors to lgebraic Vectors Vector ddition and Scalar Multiplication Unit Vectors lgebraic Properties Static Equilibrium Geometric vectors
More information10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED
CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations
More informationLINEAR FUNCTIONS OF 2 VARIABLES
CHAPTER 4: LINEAR FUNCTIONS OF 2 VARIABLES 4.1 RATES OF CHANGES IN DIFFERENT DIRECTIONS From Precalculus, we know that is a linear function if the rate of change of the function is constant. I.e., for
More informationIdentifying second degree equations
Chapter 7 Identifing second degree equations 7.1 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 +
More informationDouble Integrals in Polar Coordinates
Double Integrals in Polar Coordinates. A flat plate is in the shape of the region in the first quadrant ling between the circles + and +. The densit of the plate at point, is + kilograms per square meter
More informationSECTION 2-2 Straight Lines
- Straight Lines 11 94. Engineering. The cross section of a rivet has a top that is an arc of a circle (see the figure). If the ends of the arc are 1 millimeters apart and the top is 4 millimeters above
More informationMATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60
MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60 A Summar of Concepts Needed to be Successful in Mathematics The following sheets list the ke concepts which are taught in the specified math course. The sheets
More information1.6. Piecewise Functions. LEARN ABOUT the Math. Representing the problem using a graphical model
. Piecewise Functions YOU WILL NEED graph paper graphing calculator GOAL Understand, interpret, and graph situations that are described b piecewise functions. LEARN ABOUT the Math A cit parking lot uses
More informationRotation of Axes 1. Rotation of Axes. At the beginning of Chapter 5 we stated that all equations of the form
Rotation of Axes 1 Rotation of Axes At the beginning of Chapter we stated that all equations of the form Ax + Bx + C + Dx + E + F =0 represented a conic section, which might possibl be degenerate. We saw
More informationHigher. Polynomials and Quadratics 64
hsn.uk.net Higher Mathematics UNIT OUTCOME 1 Polnomials and Quadratics Contents Polnomials and Quadratics 64 1 Quadratics 64 The Discriminant 66 3 Completing the Square 67 4 Sketching Parabolas 70 5 Determining
More informationNew York State Student Learning Objective: Regents Geometry
New York State Student Learning Objective: Regents Geometry All SLOs MUST include the following basic components: Population These are the students assigned to the course section(s) in this SLO all students
More informationSection 2-3 Quadratic Functions
118 2 LINEAR AND QUADRATIC FUNCTIONS 71. Celsius/Fahrenheit. A formula for converting Celsius degrees to Fahrenheit degrees is given by the linear function 9 F 32 C Determine to the nearest degree the
More information15.1. Exact Differential Equations. Exact First-Order Equations. Exact Differential Equations Integrating Factors
SECTION 5. Eact First-Order Equations 09 SECTION 5. Eact First-Order Equations Eact Differential Equations Integrating Factors Eact Differential Equations In Section 5.6, ou studied applications of differential
More informationSTRAND: ALGEBRA Unit 3 Solving Equations
CMM Subject Support Strand: ALGEBRA Unit Solving Equations: Tet STRAND: ALGEBRA Unit Solving Equations TEXT Contents Section. Algebraic Fractions. Algebraic Fractions and Quadratic Equations. Algebraic
More informationLines and Planes 1. x(t) = at + b y(t) = ct + d
1 Lines in the Plane Lines and Planes 1 Ever line of points L in R 2 can be epressed as the solution set for an equation of the form A + B = C. The equation is not unique for if we multipl both sides b
More informationSection V.2: Magnitudes, Directions, and Components of Vectors
Section V.: Magnitudes, Directions, and Components of Vectors Vectors in the plane If we graph a vector in the coordinate plane instead of just a grid, there are a few things to note. Firstl, directions
More informationFunctions and Graphs CHAPTER INTRODUCTION. The function concept is one of the most important ideas in mathematics. The study
Functions and Graphs CHAPTER 2 INTRODUCTION The function concept is one of the most important ideas in mathematics. The stud 2-1 Functions 2-2 Elementar Functions: Graphs and Transformations 2-3 Quadratic
More informationDIFFERENTIAL EQUATIONS
DIFFERENTIAL EQUATIONS 379 Chapter 9 DIFFERENTIAL EQUATIONS He who seeks f methods without having a definite problem in mind seeks f the most part in vain. D. HILBERT 9. Introduction In Class XI and in
More informationExponential and Logarithmic Functions
Chapter 6 Eponential and Logarithmic Functions Section summaries Section 6.1 Composite Functions Some functions are constructed in several steps, where each of the individual steps is a function. For eample,
More informationCore Maths C3. Revision Notes
Core Maths C Revision Notes October 0 Core Maths C Algebraic fractions... Cancelling common factors... Multipling and dividing fractions... Adding and subtracting fractions... Equations... 4 Functions...
More informationPlane Stress Transformations
6 Plane Stress Transformations ASEN 311 - Structures ASEN 311 Lecture 6 Slide 1 Plane Stress State ASEN 311 - Structures Recall that in a bod in plane stress, the general 3D stress state with 9 components
More informationImplicit Differentiation
Revision Notes 2 Calculus 1270 Fall 2007 INSTRUCTOR: Peter Roper OFFICE: LCB 313 [EMAIL: roper@math.utah.edu] Standard Disclaimer These notes are not a complete review of the course thus far, and some
More informationax 2 by 2 cxy dx ey f 0 The Distance Formula The distance d between two points (x 1, y 1 ) and (x 2, y 2 ) is given by d (x 2 x 1 )
SECTION 1. The Circle 1. OBJECTIVES The second conic section we look at is the circle. The circle can be described b using the standard form for a conic section, 1. Identif the graph of an equation as
More informationA CLASSROOM NOTE ON PARABOLAS USING THE MIRAGE ILLUSION
A CLASSROOM NOTE ON PARABOLAS USING THE MIRAGE ILLUSION Abstract. The present work is intended as a classroom note on the topic of parabolas. We present several real world applications of parabolas, outline
More informationMEMORANDUM. All students taking the CLC Math Placement Exam PLACEMENT INTO CALCULUS AND ANALYTIC GEOMETRY I, MTH 145:
MEMORANDUM To: All students taking the CLC Math Placement Eam From: CLC Mathematics Department Subject: What to epect on the Placement Eam Date: April 0 Placement into MTH 45 Solutions This memo is an
More informationMath 259 Winter 2009. Recitation Handout 1: Finding Formulas for Parametric Curves
Math 259 Winter 2009 Recitation Handout 1: Finding Formulas for Parametric Curves 1. The diagram given below shows an ellipse in the -plane. -5-1 -1-3 (a) Find equations for (t) and (t) that will describe
More informationCHAPTER 10 SYSTEMS, MATRICES, AND DETERMINANTS
CHAPTER 0 SYSTEMS, MATRICES, AND DETERMINANTS PRE-CALCULUS: A TEACHING TEXTBOOK Lesson 64 Solving Sstems In this chapter, we re going to focus on sstems of equations. As ou ma remember from algebra, sstems
More informationThe Distance Formula and the Circle
10.2 The Distance Formula and the Circle 10.2 OBJECTIVES 1. Given a center and radius, find the equation of a circle 2. Given an equation for a circle, find the center and radius 3. Given an equation,
More informationTrigonometry Review Workshop 1
Trigonometr Review Workshop Definitions: Let P(,) be an point (not the origin) on the terminal side of an angle with measure θ and let r be the distance from the origin to P. Then the si trig functions
More informationCHALLENGE PROBLEMS: CHAPTER 10. Click here for solutions. Click here for answers.
CHALLENGE PROBLEMS CHALLENGE PROBLEMS: CHAPTER 0 A Click here for answers. S Click here for solutions. m m m FIGURE FOR PROBLEM N W F FIGURE FOR PROBLEM 5. Each edge of a cubical bo has length m. The bo
More informationSupporting Australian Mathematics Project. A guide for teachers Years 11 and 12. Algebra and coordinate geometry: Module 2. Coordinate geometry
1 Supporting Australian Mathematics Project 3 4 5 6 7 8 9 1 11 1 A guide for teachers Years 11 and 1 Algebra and coordinate geometr: Module Coordinate geometr Coordinate geometr A guide for teachers (Years
More informationSection 7.2 Linear Programming: The Graphical Method
Section 7.2 Linear Programming: The Graphical Method Man problems in business, science, and economics involve finding the optimal value of a function (for instance, the maimum value of the profit function
More informationIn this this review we turn our attention to the square root function, the function defined by the equation. f(x) = x. (5.1)
Section 5.2 The Square Root 1 5.2 The Square Root In this this review we turn our attention to the square root function, the function defined b the equation f() =. (5.1) We can determine the domain and
More informationZeros of Polynomial Functions. The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra. zero in the complex number system.
_.qd /7/ 9:6 AM Page 69 Section. Zeros of Polnomial Functions 69. Zeros of Polnomial Functions What ou should learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polnomial
More information5.3 Graphing Cubic Functions
Name Class Date 5.3 Graphing Cubic Functions Essential Question: How are the graphs of f () = a ( - h) 3 + k and f () = ( 1_ related to the graph of f () = 3? b ( - h) 3 ) + k Resource Locker Eplore 1
More informationProject: OUTFIELD FENCES
1 Project: OUTFIELD FENCES DESCRIPTION: In this project you will work with the equations of projectile motion and use mathematical models to analyze a design problem. Two softball fields in Rolla, Missouri
More informationCOMPLEX STRESS TUTORIAL 3 COMPLEX STRESS AND STRAIN
COMPLX STRSS TUTORIAL COMPLX STRSS AND STRAIN This tutorial is not part of the decel unit mechanical Principles but covers elements of the following sllabi. o Parts of the ngineering Council eam subject
More informationG. GRAPHING FUNCTIONS
G. GRAPHING FUNCTIONS To get a quick insight int o how the graph of a function looks, it is very helpful to know how certain simple operations on the graph are related to the way the function epression
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More information2.6. The Circle. Introduction. Prerequisites. Learning Outcomes
The Circle 2.6 Introduction A circle is one of the most familiar geometrical figures and has been around a long time! In this brief Section we discuss the basic coordinate geometr of a circle - in particular
More informationThnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks
Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson
More informationQuadratic Equations and Functions
Quadratic Equations and Functions. Square Root Propert and Completing the Square. Quadratic Formula. Equations in Quadratic Form. Graphs of Quadratic Functions. Verte of a Parabola and Applications In
More informationName Class. Date Section. Test Form A Chapter 11. Chapter 11 Test Bank 155
Chapter Test Bank 55 Test Form A Chapter Name Class Date Section. Find a unit vector in the direction of v if v is the vector from P,, 3 to Q,, 0. (a) 3i 3j 3k (b) i j k 3 i 3 j 3 k 3 i 3 j 3 k. Calculate
More informationDefinition: A vector is a directed line segment that has and. Each vector has an initial point and a terminal point.
6.1 Vectors in the Plane PreCalculus 6.1 VECTORS IN THE PLANE Learning Targets: 1. Find the component form and the magnitude of a vector.. Perform addition and scalar multiplication of two vectors. 3.
More informationBiggar 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
More informationSample Problems. Practice Problems
Lecture Notes Circles - Part page Sample Problems. Find an equation for the circle centered at (; ) with radius r = units.. Graph the equation + + = ( ).. Consider the circle ( ) + ( + ) =. Find all points
More information135 Final Review. Determine whether the graph is symmetric with respect to the x-axis, the y-axis, and/or the origin.
13 Final Review Find the distance d(p1, P2) between the points P1 and P2. 1) P1 = (, -6); P2 = (7, -2) 2 12 2 12 3 Determine whether the graph is smmetric with respect to the -ais, the -ais, and/or the
More information13 CALCULUS OF VECTOR-VALUED FUNCTIONS
CALCULUS OF VECTOR-VALUED FUNCTIONS. Vector-Valued Functions LT Section 4.) Preliminar Questions. Which one of the following does not parametrize a line? a) r t) 8 t,t,t b) r t) t i 7t j + t k c) r t)
More informationMore Equations and Inequalities
Section. Sets of Numbers and Interval Notation 9 More Equations and Inequalities 9 9. Compound Inequalities 9. Polnomial and Rational Inequalities 9. Absolute Value Equations 9. Absolute Value Inequalities
More informationSAMPLE. Polynomial functions
Objectives C H A P T E R 4 Polnomial functions To be able to use the technique of equating coefficients. To introduce the functions of the form f () = a( + h) n + k and to sketch graphs of this form through
More informationGraphing Linear Equations
6.3 Graphing Linear Equations 6.3 OBJECTIVES 1. Graph a linear equation b plotting points 2. Graph a linear equation b the intercept method 3. Graph a linear equation b solving the equation for We are
More informationSolutions to Homework 10
Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x
More information2014 2015 Geometry B Exam Review
Semester Eam Review 014 015 Geometr B Eam Review Notes to the student: This review prepares ou for the semester B Geometr Eam. The eam will cover units 3, 4, and 5 of the Geometr curriculum. The eam consists
More informationAlgebra. Exponents. Absolute Value. Simplify each of the following as much as possible. 2x y x + y y. xxx 3. x x x xx x. 1. Evaluate 5 and 123
Algebra Eponents Simplify each of the following as much as possible. 1 4 9 4 y + y y. 1 5. 1 5 4. y + y 4 5 6 5. + 1 4 9 10 1 7 9 0 Absolute Value Evaluate 5 and 1. Eliminate the absolute value bars from
More information5.1. A Formula for Slope. Investigation: Points and Slope CONDENSED
CONDENSED L E S S O N 5.1 A Formula for Slope In this lesson ou will learn how to calculate the slope of a line given two points on the line determine whether a point lies on the same line as two given
More informationcos 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
More informationEstimated Pre Calculus Pacing Timeline
Estimated Pre Calculus Pacing Timeline 2010-2011 School Year The timeframes listed on this calendar are estimates based on a fifty-minute class period. You may need to adjust some of them from time to
More information2.5 Library of Functions; Piecewise-defined Functions
SECTION.5 Librar of Functions; Piecewise-defined Functions 07.5 Librar of Functions; Piecewise-defined Functions PREPARING FOR THIS SECTION Before getting started, review the following: Intercepts (Section.,
More information