Chapter 10: Conic Sections; Polar Coordinates; Parametric Equations

Size: px
Start display at page:

Download "Chapter 10: Conic Sections; Polar Coordinates; Parametric Equations"

Transcription

1 Chapter 10: Conic Sections; Polar Coordinates; Parametric Equations Section 10.1 Geometry of Parabola, Ellipse, Hyperbola a. Geometric Definition b. Parabola c. Ellipse d. Hyperbola e. Translations f. Distance Between a Point and Line g. Parabolic Mirrors h. Optical Consequences i. Elliptical Reflectors j. Hyperbolic Reflectors Section 10.2 Polar Coordinates a. Illustrative Figure b. Assigning Polar Coordinates c. Properties 1 and 2 d. Property 3 e. Relation to Rectangular Coordinates f. Properties Relating Polar and Rectangular Coordinates g. Simple Sets h. Symmetry Section 10.3 Sketching Curves in Polar Coordinates a. Spiral of Archimedes b. Example c. Lines d. Circles e. Limaçons f. Lemniscates g. Petal Curves h. Intersection of Polar Curves Section 10.4 Area in Polar Coordinates a. Computing Area b. Properties Section 10.5 Curves Given Parametrically a. Parameterized Curve b. Straight Lines c. Ellipses and Circles d. Hyperbolas Section 10.6 Tangents to Curves Given Parametrically a. Assumptions b. Properties Section 10.7 Arc Length and Speed a. Length of a Curve b. Formula c. Length of the Graph of f d. Geometric Significance of dx/ds and dy/ds e. Speed Along a Plane Curve Section 10.7 The Area of a Surface of Revolution; The Centroid of a Curve; Pappus s Theorem on Surface Area a. The Area of a Surface of Revolution b. Computing Area c. Centroid of a Curve d. Formulas e. Pappus s Theorem on Surface Area

2 Geometry Of Parabolas Geometric Definition

3 Geometry Of Parabolas Parabola Standard Position F on the positive y-axis, l horizontal. Then F has coordinates of the form (c, 0) with c > 0 and l has equation x = c. Derivation of the Equation A point P(x, y) lies on the parabola iff d 1 = d 2, which here means ( ) 2 2 x + y c = y+ c This equation simplifies to x 2 = 4cy. Terminology A parabola has a focus, a directrix, a vertex, and an axis.

4 Geometry Of Ellipses Ellipse Standard Position F 1 and F 2 on the x-axis at equal distances c from the origin. Then F 1 is at ( c, 0) and F 2 at (c, 0). With d 1 and d 2 as in the defining figure, set d 1 + d 2 = 2a. Equation x y a a c = Setting 2 2 b= a c, we have x a y b = Terminology An ellipse has two foci, F 1 and F 2, a major axis, a minor axis, and four vertices. The point at which the axes of the ellipse intersect is called the center of the ellipse.

5 Geometry Of Hyperbolas Hyperbola Standard Position F 1 and F 2 on the x-axis at equal distances c from the origin. Then F 1 is at ( c, 0) and F 2 at (c, 0). With d 1 and d 2 as in the defining figure, set d 1 d 2 = 2a Equation x y a c a 2 2 = Setting 2 2 b= c a, we have Terminology A hyperbola has two foci, F 1 and F 2, two vertices, a transverse axis that joins the two vertices, and two asymptotes. The midpoint of the transverse axis is called the center of the hyperbola. x a y = b

6 Geometry Of Parabola, Ellipse, Hyperbola Translations Suppose that x 0 and y 0 are real numbers and S is a set in the xy-plane. By replacing each point (x, y) of S by (x + x 0, y + y 0 ), we obtain a set S which is congruent to S and obtained from S without any rotation. Such a displacement is called a translation. The translation (x, y) (x + x 0, y + y 0 ) applied to a curve C with equation E(x, y) = 0 results in a curve C with equation E(x x 0, y y 0 ) = 0.

7 Geometry Of Parabola, Ellipse, Hyperbola The distance between the origin and any line l : Ax + By + C = 0 is given by the formula d C ( 0, l) = 2 2 A + B By means of a translation we can show that the distance between any point P(x 0, y 0 ) and the line l : Ax + By + C = 0 is given by the formula

8 Geometry Of Parabola, Ellipse, Hyperbola Parabolic Mirrors Take a parabola and revolve it about its axis. This gives you a parabolic surface. A curved mirror of this form is called a parabolic mirror. Such mirrors are used in searchlights (automotive headlights, flashlights, etc.) and in reflecting telescopes.

9 Geometry Of Parabola, Ellipse, Hyperbola

10 Geometry Of Parabola, Ellipse, Hyperbola Elliptical Reflectors

11 Geometry Of Parabola, Ellipse, Hyperbola Hyperbolic Reflectors

12 Polar Coordinates

13 Polar Coordinates

14 Polar Coordinates Polar coordinates are not unique. Many pairs [r, θ] can represent the same point. (1) If r = 0, it does not matter how we choose θ. The resulting point is still the pole: (2) Geometrically there is no distinction between angles that differ by an integer multiple of 2π. Consequently:

15 Polar Coordinates (3) Adding π to the second coordinate is equivalent to changing the sign of the first coordinate:

16 Polar Coordinates Relation to Rectangular Coordinates The relation between polar coordinates [r, θ] and rectangular coordinates (x, y) is given by the following equations:

17 Polar Coordinates Unless x = 0, and, under all circumstances,

18 Polar Coordinates Here are some simple sets specified in polar coordinates. (1) The circle of radius a centered at the origin is given by the equation r = a. The interior of the circle is given by r < a and the exterior by r > a. (2) The line that passes through the origin with an inclination of α radians has polar equation θ = α. (3) For a 0, the vertical line x = a has polar equation r cos θ = a or, equivalently, r = a sec θ (4) For b 0, the horizontal line y = b has polar equation r sin θ = b or, equivalently, r = b csc θ.

19 Polar Coordinates Symmetry

20 Sketching Curves in Polar Coordinates Example Sketch the curve r = θ, θ 0 in polar coordinates. Solution At θ = 0, r = 0; at θ = ¼π, r = ¼ π; at θ = ½π, r = ½ π; and so on. The curve is shown in detail from θ = 0 to θ = 2π in Figure It is an unending spiral, the spiral of Archimedes. More of the spiral is shown on a smaller scale in the right part of the figure.

21 Sketching Curves in Polar Coordinates Example Sketch the curve r = cos 2θ in polar coordinates. Solution Since the cosine function has period 2π, the function r = cos 2θ has period π. Thus it may seem that we can restrict ourselves to sketching the curve from θ = 0 to θ = π. But this is not the case. To obtain the complete curve, we must account for r in every direction; that is, from θ = 0 to θ = 2π. Translating Figure into polar coordinates [r, θ], we obtain a sketch of the curve r = cos 2θ in polar coordinates (Figure ). The sketch is developed in eight stages.

22 Sketching Curves in Polar Coordinates Lines : θ = a, r = a sec θ, r = a csc θ.

23 Sketching Curves in Polar Coordinates Circles : r = a, r = a sin θ, r = a cos θ.

24 Sketching Curves in Polar Coordinates Limaçons : r = a + b sin θ, r = a + b cos θ.

25 Sketching Curves in Polar Coordinates Lemniscates: r² = a sin 2θ, r² = a cos 2θ

26 Sketching Curves in Polar Coordinates Petal Curves: r = a sin nθ, r = a cos nθ, integer n. If n is odd, there are n petals. If n is even, there are 2n petals.

27 Sketching Curves in Polar Coordinates The Intersection of Polar Curves The fact that a single point has many pairs of polar coordinates can cause complications. In particular, it means that a point [r 1, θ 1 ] can lie on a curve given by a polar equation although the coordinates r 1 and θ 1 do not satisfy the equation. For example, the coordinates of [2, π] do not satisfy the equation r 2 = 4 cos θ: r 2 = 2 2 = 4 but 4 cos θ = 4 cos π = 4. Nevertheless the point [2, π] does lie on the curve r 2 = 4 cos θ. We know this because [2, π] = [ 2, 0] and the coordinates of [ 2, 0] do satisfy the equation: r 2 = ( 2) 2 = 4, 4 cos θ = 4 cos 0 = 4 In general, a point P[r 1, θ 1 ] lies on a curve given by a polar equation if it has at least one polar coordinate representation [r, θ] with coordinates that satisfy the equation. The difficulties are compounded when we deal with two or more curves.

28 Area in Polar Coordinates

29 Area in Polar Coordinates

30 Curves given Parametrically Assume a pair of functions x = x(t), y = y(t) is differentiable on the interior of an interval I. At the endpoints of I (if any) we require only one-sided continuity. For each number t in I we can interpret (x(t), y(t)) as the point with x-coordinate x(t) and y-coordinate y(t). Then, as t ranges over I, the point (x(t), y(t)) traces out a path in the xy-plane. We call such a path a parametrized curve and refer to t as the parameter.

31 Curves given Parametrically Straight Lines Given that (x 0, y 0 ) = (x 1, y 1 ), the functions parametrize the line that passes through the points (x 0, y 0 ) and (x 1, y 1 ).

32 Curves given Parametrically Ellipses and Circles Usually we let t range from 0 to 2π and parametrize the ellipse by setting If b = a, we have a circle. We can parametrize the circle by setting x 2 + y 2 = a 2

33 Curves given Parametrically Hyperbolas Take a, b > 0. The functions x(t) = a cosh t, y(t) = b sinh t satisfy the identity ( ) yt ( ) 2 2 = xt a b Since x(t) = a cosh t > 0 for all t, as t ranges over the set of real numbers, the point (x(t), y(t)) traces out the right branch of the hyperbola x a y b 2 2 = 1 2 2

34 Tangents to Curves Given Parametrically Let C be a curve parametrized by the functions x = x(t), y = y(t) defined on some interval I. We will assume that I is an open interval and the parametrizing functions are differentiable. Since a parametrized curve can intersect itself, at a point of C there can be (i) one tangent, (ii) two or more tangents, or (iii) no tangent at all. To make sure that there is at least one tangent line at each point of C, we will make the additional assumption that

35 Tangents to Curves Given Parametrically

36 Arc Length and Speed Figure represents a curve C parametrized by a pair of functions x = x(t), y = y(t) t [a, b]. We will assume that the functions are continuously differentiable on [a, b] (have first derivatives which are continuous on [a, b]). We want to determine the length of C.

37 Arc Length and Speed The length of the path C traced out by a pair of continuously differentiable functions is given by the formula x = x(t), y = y(t) t [a, b]

38 Arc Length and Speed Suppose now that C is the graph of a continuously differentiable function y = f (x), x [a, b]. We can parametrize C by setting Since (10.7.1) gives Replacing t by x, we can write: x(t) = t, y(t) = f (t) t [a, b]. x (t) = 1 and y (t) = f (t), b ( ) = 1+ ( ) 2 L C f t dt a

39 Arc Length and Speed The Geometric Significance of dx/ds and dy/ds

40 Arc Length and Speed Speed Along a Plane Curve So far we have talked about speed only in connection with straight-line motion. How can we calculate the speed of an object that moves along a curve? Imagine an object moving along some curved path. Suppose that (x(t), y(t)) gives the position of the object at time t. The distance traveled by the object from time zero to any later time t is simply the length of the path up to time t: t ( ) ( ) ( ) s t = x u + y u du The time rate of change of this distance is what we call the speed of the object. Denoting the speed of the object at time t by ν(t), we have

41 The Area Of A Surface Of Revolution; The Centroid Of A Curve; Pappus's Theorem On Surface Area The Area of a Surface of Revolution

42 The Area Of A Surface Of Revolution; The Centroid Of A Curve; Pappus's Theorem On Surface Area

43 The Area Of A Surface Of Revolution; The Centroid Of A Curve; Pappus's Theorem On Surface Area Centroid of a Curve We can locate the centroid of a curve from the following principles, which we take from physics. Principle 1: Symmetry. If a curve has an axis of symmetry, then the centroid somewhere along that axis. ( x, y) Principle 2: Additivity. If a curve with length L is broken up into a finite number of pieces with arc lengths Δs 1,..., Δs n and centroids x, y,, x, y, then ( ) ( ) 1 1 xl = x s + + x s and yl = y s + + y s 1 1 n n 1 1 n n n n lies

44 The Area Of A Surface Of Revolution; The Centroid Of A Curve; Pappus's Theorem On Surface Area

45 The Area Of A Surface Of Revolution; The Centroid Of A Curve; Pappus's Theorem On Surface Area

Distance and Midpoint Formula 7.1

Distance and Midpoint Formula 7.1 Distance and Midpoint Formula 7.1 Distance Formula d ( x - x ) ( y - y ) 1 1 Example 1 Find the distance between the points (4, 4) and (-6, -). Example Find the value of a to make the distance = 10 units

More information

Circles. Example 2: Write an equation for a circle if the enpoints of a diameter are at ( 4,5) and (6, 3).

Circles. Example 2: Write an equation for a circle if the enpoints of a diameter are at ( 4,5) and (6, 3). Conics Unit Ch. 8 Circles Equations of Circles The equation of a circle with center ( hk, ) and radius r units is ( x h) ( y k) r. Example 1: Write an equation of circle with center (8, 3) and radius 6.

More information

3. A( 2,0) and B(6, -2), find M 4. A( 3, 7) and M(4,-3), find B. 5. M(4, -9) and B( -10, 11) find A 6. B(4, 8) and M(-2, 5), find A

3. A( 2,0) and B(6, -2), find M 4. A( 3, 7) and M(4,-3), find B. 5. M(4, -9) and B( -10, 11) find A 6. B(4, 8) and M(-2, 5), find A Midpoint and Distance Formula Class Work M is the midpoint of A and B. Use the given information to find the missing point. 1. A(4, 2) and B(3, -8), find M 2. A(5, 7) and B( -2, -9), find M 3. A( 2,0)

More information

Conic Sections. Geometry - Conics ~1~ NJCTL.org. Write the following equations in standard form.

Conic Sections. Geometry - Conics ~1~ NJCTL.org. Write the following equations in standard form. Conic Sections Midpoint and Distance Formula M is the midpoint of A and B. Use the given information to find the missing point. 1. A(, 2) and B(3, -), find M 2. A(5, 7) and B( -2, -), find M 3. A( 2,0)

More information

3. A( 2,0) and B(6, -2), find M 4. A( 3, 7) and M(4,-3), find B. 5. M(4, -9) and B( -10, 11) find A 6. B(4, 8) and M(-2, 5), find A

3. A( 2,0) and B(6, -2), find M 4. A( 3, 7) and M(4,-3), find B. 5. M(4, -9) and B( -10, 11) find A 6. B(4, 8) and M(-2, 5), find A Midpoint and Distance Formula Class Work M is the midpoint of A and B. Use the given information to find the missing point. 1. A(, 2) and B(3, -8), find M 2. A(5, 7) and B( -2, -), find M (3. 5, 3) (1.

More information

The Distance Formula. The Midpoint Formula

The Distance Formula. The Midpoint Formula Math 120 Intermediate Algebra Sec 9.1: Distance Midpoint Formulas The Distance Formula The distance between two points P 1 = (x 1, y 1 ) P 2 = (x 1, y 1 ), denoted by d(p 1, P 2 ), is d(p 1, P 2 ) = (x

More information

Introduction to Computer Graphics (Lecture No 07) Ellipse and Other Curves

Introduction to Computer Graphics (Lecture No 07) Ellipse and Other Curves Introduction to Computer Graphics (Lecture No 07) Ellipse and Other Curves 7.1 Ellipse An ellipse is a curve that is the locus of all points in the plane the sum of whose distances r1 and r from two fixed

More information

Math 190 (Calculus II) Final Review

Math 190 (Calculus II) Final Review Math 90 (Calculus II) Final Review. Sketch the region enclosed by the given curves and find the area of the region. a. y = 7 x, y = x + 4 b. y = cos ( πx ), y = x. Use the specified method to find the

More information

10.1 Review of Parametric Equations

10.1 Review of Parametric Equations 10.1 Review of Parametric Equations Recall that often, instead of representing a curve using just x and y (called a Cartesian equation), it is more convenient to define x and y using parametric equations

More information

Honors Precalculus Chapter 8 Summary Conic Sections- Parabola

Honors Precalculus Chapter 8 Summary Conic Sections- Parabola Honors Precalculus Chapter 8 Summary Conic Sections- Parabola Definition: Focal length: y- axis P(x, y) Focal chord: focus Vertex x-axis directrix Focal width/ Latus Rectum: Derivation of equation of parabola:

More information

+ 4 Ex: y = v = (1, 4) x = 1 Focus: (h, k + ) = (1, 6) L.R. = 8 units We can have parabolas that open sideways too (inverses) x = a (y k) 2 + h

+ 4 Ex: y = v = (1, 4) x = 1 Focus: (h, k + ) = (1, 6) L.R. = 8 units We can have parabolas that open sideways too (inverses) x = a (y k) 2 + h Unit 7 Notes Parabolas: E: reflectors, microphones, (football game), (Davinci) satellites. Light placed where ras will reflect parallel. This point is the focus. Parabola set of all points in a plane that

More information

CIRCLES: #1. What is an equation of the circle at the origin and radius 12?

CIRCLES: #1. What is an equation of the circle at the origin and radius 12? 1 Pre-AP Algebra II Chapter 10 Test Review Standards/Goals: E.3.a.: I can identify conic sections (parabola, circle, ellipse, hyperbola) from their equations in standard form. E.3.b.: I can graph circles

More information

Chapter 1 Analytic geometry in the plane

Chapter 1 Analytic geometry in the plane 3110 General Mathematics 1 31 10 General Mathematics For the students from Pharmaceutical Faculty 1/004 Instructor: Dr Wattana Toutip (ดร.ว ฒนา เถาว ท พย ) Chapter 1 Analytic geometry in the plane Overview:

More information

8.6 Translate and Classify Conic Sections

8.6 Translate and Classify Conic Sections 8.6 Translate and Classify Conic Sections Where are the symmetric lines of conic sections? What is the general 2 nd degree equation for any conic? What information can the discriminant tell you about a

More information

Find the rectangular coordinates for each of the following polar coordinates:

Find the rectangular coordinates for each of the following polar coordinates: WORKSHEET 13.1 1. Plot the following: 7 3 A. 6, B. 3, 6 4 5 8 D. 6, 3 C., 11 2 E. 5, F. 4, 6 3 Find the rectangular coordinates for each of the following polar coordinates: 5 2 2. 4, 3. 8, 6 3 Given the

More information

Precalculus Conic Sections Unit 6. Parabolas. Label the parts: Focus Vertex Axis of symmetry Focal Diameter Directrix

Precalculus Conic Sections Unit 6. Parabolas. Label the parts: Focus Vertex Axis of symmetry Focal Diameter Directrix PICTURE: Parabolas Name Hr Label the parts: Focus Vertex Axis of symmetry Focal Diameter Directrix Using what you know about transformations, label the purpose of each constant: y a x h 2 k It is common

More information

Conic Sections Session 3: Hyperbola

Conic Sections Session 3: Hyperbola Conic Sections Session 3: Hyperbola Toh Pee Choon NIE Oct 2017 Toh Pee Choon (NIE) Session 3: Hyperbola Oct 2017 1 / 16 Problem 3.1 1 Recall that an ellipse is defined as the locus of points P such that

More information

ALGEBRA 2 X. Final Exam. Review Packet

ALGEBRA 2 X. Final Exam. Review Packet ALGEBRA X Final Exam Review Packet Multiple Choice Match: 1) x + y = r a) equation of a line ) x = 5y 4y+ b) equation of a hyperbola ) 4) x y + = 1 64 9 c) equation of a parabola x y = 1 4 49 d) equation

More information

PARAMETRIC EQUATIONS AND POLAR COORDINATES

PARAMETRIC EQUATIONS AND POLAR COORDINATES 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES PARAMETRIC EQUATIONS & POLAR COORDINATES 10.5 Conic Sections In this section, we will learn: How to derive standard equations for conic sections. CONIC SECTIONS

More information

Chapter 11 Parametric Equations, Polar Curves, and Conic Sections

Chapter 11 Parametric Equations, Polar Curves, and Conic Sections Chapter 11 Parametric Equations, Polar Curves, and Conic Sections ü 11.1 Parametric Equations Students should read Sections 11.1-11. of Rogawski's Calculus [1] for a detailed discussion of the material

More information

Pre-Calculus Final Exam Review Name: May June Use the following schedule to complete the final exam review.

Pre-Calculus Final Exam Review Name: May June Use the following schedule to complete the final exam review. Pre-Calculus Final Exam Review Name: May June 2015 Use the following schedule to complete the final exam review. Homework will be checked in every day. Late work will NOT be accepted. Homework answers

More information

Geometry and Motion, MA 134 Week 1

Geometry and Motion, MA 134 Week 1 Geometry and Motion, MA 134 Week 1 Mario J. Micallef Spring, 2007 Warning. These handouts are not intended to be complete lecture notes. They should be supplemented by your own notes and, importantly,

More information

Chapter 9. Conic Sections and Analytic Geometry. 9.2 The Hyperbola. Copyright 2014, 2010, 2007 Pearson Education, Inc.

Chapter 9. Conic Sections and Analytic Geometry. 9.2 The Hyperbola. Copyright 2014, 2010, 2007 Pearson Education, Inc. Chapter 9 Conic Sections and Analytic Geometry 9. The Hyperbola Copyright 014, 010, 007 Pearson Education, Inc. 1 Objectives: Locate a hyperbola s vertices and foci. Write equations of hyperbolas in standard

More information

MATH Final Review

MATH Final Review MATH 1592 - Final Review 1 Chapter 7 1.1 Main Topics 1. Integration techniques: Fitting integrands to basic rules on page 485. Integration by parts, Theorem 7.1 on page 488. Guidelines for trigonometric

More information

9.1 Circles and Parabolas. Copyright Cengage Learning. All rights reserved.

9.1 Circles and Parabolas. Copyright Cengage Learning. All rights reserved. 9.1 Circles and Parabolas Copyright Cengage Learning. All rights reserved. What You Should Learn Recognize a conic as the intersection of a plane and a double-napped cone. Write equations of circles in

More information

January 21, 2018 Math 9. Geometry. The method of coordinates (continued). Ellipse. Hyperbola. Parabola.

January 21, 2018 Math 9. Geometry. The method of coordinates (continued). Ellipse. Hyperbola. Parabola. January 21, 2018 Math 9 Ellipse Geometry The method of coordinates (continued) Ellipse Hyperbola Parabola Definition An ellipse is a locus of points, such that the sum of the distances from point on the

More information

Calculus III. George Voutsadakis 1. LSSU Math 251. Lake Superior State University. 1 Mathematics and Computer Science

Calculus III. George Voutsadakis 1. LSSU Math 251. Lake Superior State University. 1 Mathematics and Computer Science Calculus III George Voutsadakis 1 1 Mathematics and Computer Science Lake Superior State University LSSU Math 251 George Voutsadakis (LSSU) Calculus III January 2016 1 / 76 Outline 1 Parametric Equations,

More information

School of Distance Education UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION. B Sc Mathematics. (2011 Admission Onwards) IV Semester.

School of Distance Education UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION. B Sc Mathematics. (2011 Admission Onwards) IV Semester. School of Dtance Education UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc Mathematics 0 Admsion Onwards IV Semester Core Course CALCULUS AND ANALYTIC GEOMETRY QUESTION BANK The natural logarithm

More information

Convert the equation to the standard form for an ellipse by completing the square on x and y. 3) 16x y 2-32x - 150y = 0 3)

Convert the equation to the standard form for an ellipse by completing the square on x and y. 3) 16x y 2-32x - 150y = 0 3) Math 370 Exam 5 Review Name Graph the ellipse and locate the foci. 1) x 6 + y = 1 1) foci: ( 15, 0) and (- 15, 0) Objective: (9.1) Graph Ellipses Not Centered at the Origin Graph the ellipse. ) (x + )

More information

2. Determine the domain of the function. Verify your result with a graph. f(x) = 25 x 2

2. Determine the domain of the function. Verify your result with a graph. f(x) = 25 x 2 29 April PreCalculus Final Review 1. Find the slope and y-intercept (if possible) of the equation of the line. Sketch the line: y = 3x + 13 2. Determine the domain of the function. Verify your result with

More information

1 x. II. CHAPTER 2: (A) Graphing Rational Functions: Show Asymptotes using dotted lines, Intercepts, Holes(Coordinates, if any.)

1 x. II. CHAPTER 2: (A) Graphing Rational Functions: Show Asymptotes using dotted lines, Intercepts, Holes(Coordinates, if any.) FINAL REVIEW-014: Before using this review guide be sure to study your test and quizzes from this year. The final will contain big ideas from the first half of the year (chapters 1-) but it will be focused

More information

Math 2412 Final Exam Review

Math 2412 Final Exam Review Math 41 Final Exam Review MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Factor and simplify the algebraic expression. 1) (x + 4) /5 - (x + 4) 1/5

More information

Algebra 2 Unit 9 (Chapter 9)

Algebra 2 Unit 9 (Chapter 9) Algebra Unit 9 (Chapter 9) 0. Spiral Review Worksheet 0. Find verte, line of symmetry, focus and directri of a parabola. (Section 9.) Worksheet 5. Find the center and radius of a circle. (Section 9.3)

More information

HHS Pre-Calculus Reference Book

HHS Pre-Calculus Reference Book HHS Pre-Calculus Reference Book Purpose: To create a reference book to review topics for your final exam and to prepare you for Calculus. Instructions: Students are to compose a reference book containing

More information

Exercise. Exercise 1.1. MA112 Section : Prepared by Dr.Archara Pacheenburawana 1

Exercise. Exercise 1.1. MA112 Section : Prepared by Dr.Archara Pacheenburawana 1 MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 1 Exercise Exercise 1.1 1 8 Find the vertex, focus, and directrix of the parabola and sketch its graph. 1. x = 2y 2 2. 4y +x 2 = 0 3. 4x 2 =

More information

Conic Sections in Polar Coordinates

Conic Sections in Polar Coordinates Conic Sections in Polar Coordinates MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Introduction We have develop the familiar formulas for the parabola, ellipse, and hyperbola

More information

Math Conic Sections

Math Conic Sections Math 114 - Conic Sections Peter A. Perry University of Kentucky April 13, 2017 Bill of Fare Why Conic Sections? Parabolas Ellipses Hyperbolas Shifted Conics Goals of This Lecture By the end of this lecture,

More information

Parametric Curves You Should Know

Parametric Curves You Should Know Parametric Curves You Should Know Straight Lines Let a and c be constants which are not both zero. Then the parametric equations determining the straight line passing through (b d) with slope c/a (i.e.

More information

8.2 Graphs of Polar Equations

8.2 Graphs of Polar Equations 8. Graphs of Polar Equations Definition: A polar equation is an equation whose variables are polar coordinates. One method used to graph a polar equation is to convert the equation to rectangular form.

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 3 2, 5 2 C) - 5 2

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 3 2, 5 2 C) - 5 2 Test Review (chap 0) Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. ) Find the point on the curve x = sin t, y = cos t, -

More information

MATH 1020 WORKSHEET 12.1 & 12.2 Vectors in the Plane

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.

More information

1. Graph each of the given equations, state the domain and range, and specify all intercepts and symmetry. a) y 3x

1. Graph each of the given equations, state the domain and range, and specify all intercepts and symmetry. a) y 3x MATH 94 Final Exam Review. Graph each of the given equations, state the domain and range, and specify all intercepts and symmetry. a) y x b) y x 4 c) y x 4. Determine whether or not each of the following

More information

CONIC SECTIONS TEST FRIDAY, JANUARY 5 TH

CONIC SECTIONS TEST FRIDAY, JANUARY 5 TH CONIC SECTIONS TEST FRIDAY, JANUARY 5 TH DAY 1 - CLASSIFYING CONICS 4 Conics Parabola Circle Ellipse Hyperbola DAY 1 - CLASSIFYING CONICS GRAPHICALLY Parabola Ellipse Circle Hyperbola DAY 1 - CLASSIFYING

More information

REVIEW OF KEY CONCEPTS

REVIEW OF KEY CONCEPTS REVIEW OF KEY CONCEPTS 8.1 8. Equations of Loci Refer to the Key Concepts on page 598. 1. Sketch the locus of points in the plane that are cm from a circle of radius 5 cm.. a) How are the lines y = x 3

More information

Learning Objectives for Math 166

Learning Objectives for Math 166 Learning Objectives for Math 166 Chapter 6 Applications of Definite Integrals Section 6.1: Volumes Using Cross-Sections Draw and label both 2-dimensional perspectives and 3-dimensional sketches of the

More information

The region enclosed by the curve of f and the x-axis is rotated 360 about the x-axis. Find the volume of the solid formed.

The region enclosed by the curve of f and the x-axis is rotated 360 about the x-axis. Find the volume of the solid formed. Section A ln. Let g() =, for > 0. ln Use the quotient rule to show that g ( ). 3 (b) The graph of g has a maimum point at A. Find the -coordinate of A. (Total 7 marks) 6. Let h() =. Find h (0). cos 3.

More information

( ) ( ) ( ) ( ) Given that and its derivative are continuous when, find th values of and. ( ) ( )

( ) ( ) ( ) ( ) Given that and its derivative are continuous when, find th values of and. ( ) ( ) 1. The piecewise function is defined by where and are constants. Given that and its derivative are continuous when, find th values of and. When When of of Substitute into ; 2. Using the substitution, evaluate

More information

Chapter 9. Conic Sections and Analytic Geometry. 9.3 The Parabola. Copyright 2014, 2010, 2007 Pearson Education, Inc.

Chapter 9. Conic Sections and Analytic Geometry. 9.3 The Parabola. Copyright 2014, 2010, 2007 Pearson Education, Inc. Chapter 9 Conic Sections and Analytic Geometry 9.3 The Parabola Copyright 014, 010, 007 Pearson Education, Inc. 1 Objectives: Graph parabolas with vertices at the origin. Write equations of parabolas in

More information

b = 2, c = 3, we get x = 0.3 for the positive root. Ans. (D) x 2-2x - 8 < 0, or (x - 4)(x + 2) < 0, Therefore -2 < x < 4 Ans. (C)

b = 2, c = 3, we get x = 0.3 for the positive root. Ans. (D) x 2-2x - 8 < 0, or (x - 4)(x + 2) < 0, Therefore -2 < x < 4 Ans. (C) SAT II - Math Level 2 Test #02 Solution 1. The positive zero of y = x 2 + 2x is, to the nearest tenth, equal to (A) 0.8 (B) 0.7 + 1.1i (C) 0.7 (D) 0.3 (E) 2.2 ± Using Quadratic formula, x =, with a = 1,

More information

3.4 Conic sections. Such type of curves are called conics, because they arise from different slices through a cone

3.4 Conic sections. Such type of curves are called conics, because they arise from different slices through a cone 3.4 Conic sections Next we consider the objects resulting from ax 2 + bxy + cy 2 + + ey + f = 0. Such type of curves are called conics, because they arise from different slices through a cone Circles belong

More information

(a) Write down the value of q and of r. (2) Write down the equation of the axis of symmetry. (1) (c) Find the value of p. (3) (Total 6 marks)

(a) Write down the value of q and of r. (2) Write down the equation of the axis of symmetry. (1) (c) Find the value of p. (3) (Total 6 marks) 1. Let f(x) = p(x q)(x r). Part of the graph of f is shown below. The graph passes through the points ( 2, 0), (0, 4) and (4, 0). (a) Write down the value of q and of r. (b) Write down the equation of

More information

AP Calculus (BC) Chapter 10 Test No Calculator Section. Name: Date: Period:

AP Calculus (BC) Chapter 10 Test No Calculator Section. Name: Date: Period: AP Calculus (BC) Chapter 10 Test No Calculator Section Name: Date: Period: Part I. Multiple-Choice Questions (5 points each; please circle the correct answer.) 1. The graph in the xy-plane represented

More information

Logs and Exponential functions e, ln, solving exponential functions, solving log and exponential equations, properties of logs

Logs and Exponential functions e, ln, solving exponential functions, solving log and exponential equations, properties of logs Page 1 AM1 Final Exam Review Packet TOPICS Complex Numbers, Vectors, and Parametric Equations Change back and forth from and to polar and rectangular forms. Raise a term in polar form to a power (DeMoivre).

More information

HW - Chapter 10 - Parametric Equations and Polar Coordinates

HW - Chapter 10 - Parametric Equations and Polar Coordinates Berkeley City College Due: HW - Chapter 0 - Parametric Equations and Polar Coordinates Name Parametric equations and a parameter interval for the motion of a particle in the xy-plane are given. Identify

More information

Pre Calculus Gary Community School Corporation Unit Planning Map

Pre Calculus Gary Community School Corporation Unit Planning Map UNIT/TIME FRAME STANDARDS Functions and Graphs (6 weeks) PC.F.1: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities,

More information

Things You Should Know Coming Into Calc I

Things You Should Know Coming Into Calc I Things You Should Know Coming Into Calc I Algebraic Rules, Properties, Formulas, Ideas and Processes: 1) Rules and Properties of Exponents. Let x and y be positive real numbers, let a and b represent real

More information

AP PHYSICS SUMMER ASSIGNMENT

AP PHYSICS SUMMER ASSIGNMENT AP PHYSICS SUMMER ASSIGNMENT There are two parts of the summer assignment, both parts mirror the course. The first part is problem solving, where there are 14 math problems that you are given to solve

More information

10.3 Parametric Equations. 1 Math 1432 Dr. Almus

10.3 Parametric Equations. 1 Math 1432 Dr. Almus Math 1432 DAY 39 Dr. Melahat Almus almus@math.uh.edu OFFICE HOURS (212 PGH) MW12-1:30pm, F:12-1pm. If you email me, please mention the course (1432) in the subject line. Check your CASA account for Quiz

More information

MA 323 Geometric Modelling Course Notes: Day 07 Parabolic Arcs

MA 323 Geometric Modelling Course Notes: Day 07 Parabolic Arcs MA 323 Geometric Modelling Course Notes: Day 07 Parabolic Arcs David L. Finn December 9th, 2004 We now start considering the basic curve elements to be used throughout this course; polynomial curves and

More information

1. Which of the following defines a function f for which f ( x) = f( x) 2. ln(4 2 x) < 0 if and only if

1. Which of the following defines a function f for which f ( x) = f( x) 2. ln(4 2 x) < 0 if and only if . Which of the following defines a function f for which f ( ) = f( )? a. f ( ) = + 4 b. f ( ) = sin( ) f ( ) = cos( ) f ( ) = e f ( ) = log. ln(4 ) < 0 if and only if a. < b. < < < < > >. If f ( ) = (

More information

Semester 2 Final Review

Semester 2 Final Review Name: Date: Per: Unit 6: Radical Functions [1-6] Simplify each real expression completely. 1. 27x 2 y 7 2. 80m n 5 3. 5x 2 8x 3 y 6 3. 2m 6 n 5 5. (6x 9 ) 1 3 6. 3x 1 2 8x 3 [7-10] Perform the operation

More information

Chapter 10 Conics, Parametric Equations, and Polar Coordinates Conics and Calculus

Chapter 10 Conics, Parametric Equations, and Polar Coordinates Conics and Calculus Chapter 10 Conics, Parametric Equations, and Polar Coordinates 10.1 Conics and Calculus 1. Parabola A parabola is the set of all points x, y ( ) that are equidistant from a fixed line and a fixed point

More information

4.1 Analysis of functions I: Increase, decrease and concavity

4.1 Analysis of functions I: Increase, decrease and concavity 4.1 Analysis of functions I: Increase, decrease and concavity Definition Let f be defined on an interval and let x 1 and x 2 denote points in that interval. a) f is said to be increasing on the interval

More information

Conic Sections. Pre-Calculus Unit Completing the Square. Solve each equation by completing the square x 2 + 8x 10 = 0

Conic Sections. Pre-Calculus Unit Completing the Square. Solve each equation by completing the square x 2 + 8x 10 = 0 Pre-Calculus Unit 7 Conic Sections Name: 7.1 Completing the Square Solve each equation by completing the square. 1. x 2 + 4x = 21 6. x 2 5x 5 = 0 11. x 2 6x + 6 = 0 2. x 2 8x = 33 7. x 2 + 7x = 0 12. x

More information

PreCalculus. Curriculum (637 topics additional topics)

PreCalculus. Curriculum (637 topics additional topics) PreCalculus This course covers the topics shown below. Students navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence to meet curricular needs.

More information

GLOSSARY GLOSSARY. For commonly-used formulas, see inside back cover.

GLOSSARY GLOSSARY. For commonly-used formulas, see inside back cover. A abscissa (5) the first element of an ordered pair absolute maximum (171) a point that represents the maximum value a function assumes over its domain absolute minimum (171) a point that represents the

More information

Section 8.2: Integration by Parts When you finish your homework, you should be able to

Section 8.2: Integration by Parts When you finish your homework, you should be able to Section 8.2: Integration by Parts When you finish your homework, you should be able to π Use the integration by parts technique to find indefinite integral and evaluate definite integrals π Use the tabular

More information

Portable Assisted Study Sequence ALGEBRA IIB

Portable Assisted Study Sequence ALGEBRA IIB SCOPE This course is divided into two semesters of study (A & B) comprised of five units each. Each unit teaches concepts and strategies recommended for intermediate algebra students. The second half of

More information

OHSx XM521 Multivariable Differential Calculus: Homework Solutions 13.1

OHSx XM521 Multivariable Differential Calculus: Homework Solutions 13.1 OHSx XM521 Multivariable Differential Calculus: Homework Solutions 13.1 (37) If a bug walks on the sphere x 2 + y 2 + z 2 + 2x 2y 4z 3 = 0 how close and how far can it get from the origin? Solution: Complete

More information

MA 162 FINAL EXAM PRACTICE PROBLEMS Spring Find the angle between the vectors v = 2i + 2j + k and w = 2i + 2j k. C.

MA 162 FINAL EXAM PRACTICE PROBLEMS Spring Find the angle between the vectors v = 2i + 2j + k and w = 2i + 2j k. C. MA 6 FINAL EXAM PRACTICE PROBLEMS Spring. Find the angle between the vectors v = i + j + k and w = i + j k. cos 8 cos 5 cos D. cos 7 E. cos. Find a such that u = i j + ak and v = i + j + k are perpendicular.

More information

1. The positive zero of y = x 2 + 2x 3/5 is, to the nearest tenth, equal to

1. The positive zero of y = x 2 + 2x 3/5 is, to the nearest tenth, equal to SAT II - Math Level Test #0 Solution SAT II - Math Level Test No. 1. The positive zero of y = x + x 3/5 is, to the nearest tenth, equal to (A) 0.8 (B) 0.7 + 1.1i (C) 0.7 (D) 0.3 (E). 3 b b 4ac Using Quadratic

More information

Since x + we get x² + 2x = 4, or simplifying it, x² = 4. Therefore, x² + = 4 2 = 2. Ans. (C)

Since x + we get x² + 2x = 4, or simplifying it, x² = 4. Therefore, x² + = 4 2 = 2. Ans. (C) SAT II - Math Level 2 Test #01 Solution 1. x + = 2, then x² + = Since x + = 2, by squaring both side of the equation, (A) - (B) 0 (C) 2 (D) 4 (E) -2 we get x² + 2x 1 + 1 = 4, or simplifying it, x² + 2

More information

Conic Sections and Polar Graphing Lab Part 1 - Circles

Conic Sections and Polar Graphing Lab Part 1 - Circles MAC 1114 Name Conic Sections and Polar Graphing Lab Part 1 - Circles 1. What is the standard equation for a circle with center at the origin and a radius of k? 3. Consider the circle x + y = 9. a. What

More information

Further Pure Mathematics 3 GCE Further Mathematics GCE Pure Mathematics and Further Mathematics (Additional) A2 optional unit

Further Pure Mathematics 3 GCE Further Mathematics GCE Pure Mathematics and Further Mathematics (Additional) A2 optional unit Unit FP3 Further Pure Mathematics 3 GCE Further Mathematics GCE Pure Mathematics and Further Mathematics (Additional) A optional unit FP3.1 Unit description Further matrix algebra; vectors, hyperbolic

More information

Grade 11/12 Math Circles Conics & Applications The Mathematics of Orbits Dr. Shahla Aliakbari November 18, 2015

Grade 11/12 Math Circles Conics & Applications The Mathematics of Orbits Dr. Shahla Aliakbari November 18, 2015 Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 11/12 Math Circles Conics & Applications The Mathematics of Orbits Dr. Shahla Aliakbari November

More information

KEMATH1 Calculus for Chemistry and Biochemistry Students. Francis Joseph H. Campeña, De La Salle University Manila

KEMATH1 Calculus for Chemistry and Biochemistry Students. Francis Joseph H. Campeña, De La Salle University Manila KEMATH1 Calculus for Chemistry and Biochemistry Students Francis Joseph H Campeña, De La Salle University Manila February 9, 2015 Contents 1 Conic Sections 2 11 A review of the coordinate system 2 12 Conic

More information

CRASH COURSE IN PRECALCULUS

CRASH COURSE IN PRECALCULUS CRASH COURSE IN PRECALCULUS Shiah-Sen Wang The graphs are prepared by Chien-Lun Lai Based on : Precalculus: Mathematics for Calculus by J. Stuwart, L. Redin & S. Watson, 6th edition, 2012, Brooks/Cole

More information

Calculus I

Calculus I Calculus I 978-1-63545-038-5 To learn more about all our offerings Visit Knewton.com/highered Source Author(s) (Text or Video) Title(s) Link (where applicable) OpenStax Gilbert Strang, Massachusetts Institute

More information

Math 259 Winter Solutions to Homework # We will substitute for x and y in the linear equation and then solve for r. x + y = 9.

Math 259 Winter Solutions to Homework # We will substitute for x and y in the linear equation and then solve for r. x + y = 9. Math 59 Winter 9 Solutions to Homework Problems from Pages 5-5 (Section 9.) 18. We will substitute for x and y in the linear equation and then solve for r. x + y = 9 r cos(θ) + r sin(θ) = 9 r (cos(θ) +

More information

Chapter 9 Overview: Parametric and Polar Coordinates

Chapter 9 Overview: Parametric and Polar Coordinates Chapter 9 Overview: Parametric and Polar Coordinates As we saw briefly last year, there are axis systems other than the Cartesian System for graphing (vector coordinates, polar coordinates, rectangular

More information

MATH-1420 Review Concepts (Haugen)

MATH-1420 Review Concepts (Haugen) MATH-40 Review Concepts (Haugen) Unit : Equations, Inequalities, Functions, and Graphs Rational Expressions Determine the domain of a rational expression Simplify rational expressions -factor and then

More information

Brief answers to assigned even numbered problems that were not to be turned in

Brief answers to assigned even numbered problems that were not to be turned in Brief answers to assigned even numbered problems that were not to be turned in Section 2.2 2. At point (x 0, x 2 0) on the curve the slope is 2x 0. The point-slope equation of the tangent line to the curve

More information

Fall Exam 4: 8&11-11/14/13 - Write all responses on separate paper. Show your work for credit.

Fall Exam 4: 8&11-11/14/13 - Write all responses on separate paper. Show your work for credit. Math Fall - Exam : 8& - // - Write all responses on separate paper. Show your work for credit. Name (Print):. Convert the rectangular equation to polar coordinates and solve for r. (a) x + (y ) = 6 Solution:

More information

SOLUTIONS TO HOMEWORK ASSIGNMENT #2, Math 253

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

More information

QUESTION BANK ON. CONIC SECTION (Parabola, Ellipse & Hyperbola)

QUESTION BANK ON. CONIC SECTION (Parabola, Ellipse & Hyperbola) QUESTION BANK ON CONIC SECTION (Parabola, Ellipse & Hyperbola) Question bank on Parabola, Ellipse & Hyperbola Select the correct alternative : (Only one is correct) Q. Two mutually perpendicular tangents

More information

Find: sinθ. Name: Date:

Find: sinθ. Name: Date: Name: Date: 1. Find the exact value of the given trigonometric function of the angle θ shown in the figure. (Use the Pythagorean Theorem to find the third side of the triangle.) Find: sinθ c a θ a a =

More information

Conic section. Ans: c. Ans: a. Ans: c. Episode:43 Faculty: Prof. A. NAGARAJ. 1. A circle

Conic section. Ans: c. Ans: a. Ans: c. Episode:43 Faculty: Prof. A. NAGARAJ. 1. A circle Episode:43 Faculty: Prof. A. NAGARAJ Conic section 1. A circle gx fy c 0 is said to be imaginary circle if a) g + f = c b) g + f > c c) g + f < c d) g = f. If (1,-3) is the centre of the circle x y ax

More information

Polar Coordinates: Graphs

Polar Coordinates: Graphs Polar Coordinates: Graphs By: OpenStaxCollege The planets move through space in elliptical, periodic orbits about the sun, as shown in [link]. They are in constant motion, so fixing an exact position of

More information

SOLUTIONS TO SECOND PRACTICE EXAM Math 21a, Spring 2003

SOLUTIONS TO SECOND PRACTICE EXAM Math 21a, Spring 2003 SOLUTIONS TO SECOND PRACTICE EXAM Math a, Spring 3 Problem ) ( points) Circle for each of the questions the correct letter. No justifications are needed. Your score will be C W where C is the number of

More information

Ch 9/10/11/12 Exam Review

Ch 9/10/11/12 Exam Review Ch 9/0// Exam Review The vector v has initial position P and terminal point Q. Write v in the form ai + bj; that is, find its position vector. ) P = (4, 6); Q = (-6, -) Find the vertex, focus, and directrix

More information

Analytic Geometry MAT 1035

Analytic Geometry MAT 1035 Analytic Geometry MAT 035 5.09.04 WEEKLY PROGRAM - The first week of the semester, we will introduce the course and given a brief outline. We continue with vectors in R n and some operations including

More information

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

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

More information

Math 180 Chapter 10 Lecture Notes. Professor Miguel Ornelas

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

More information

A plane in which each point is identified with a ordered pair of real numbers (x,y) is called a coordinate (or Cartesian) plane.

A plane in which each point is identified with a ordered pair of real numbers (x,y) is called a coordinate (or Cartesian) plane. Coordinate Geometry Rene Descartes, considered the father of modern philosophy (Cogito ergo sum), also had a great influence on mathematics. He and Fermat corresponded regularly and as a result of their

More information

Algebra and Trigonometry

Algebra and Trigonometry Algebra and Trigonometry 978-1-63545-098-9 To learn more about all our offerings Visit Knewtonalta.com Source Author(s) (Text or Video) Title(s) Link (where applicable) OpenStax Jay Abramson, Arizona State

More information

Curriculum Scope & Sequence

Curriculum Scope & Sequence Book: Sullivan Pre-Calculus Enhanced with Graphing Utilities Subject/Grade Level: MATHEMATICS/HIGH SCHOOL Curriculum Scope & Sequence Course: PRE-CALCULUS CP/HONORS ***The goals and standards addressed

More information

9.5 Parametric Equations

9.5 Parametric Equations Date: 9.5 Parametric Equations Syllabus Objective: 1.10 The student will solve problems using parametric equations. Parametric Curve: the set of all points xy,, where on an interval I (called the parameter

More information

Precalculus Table of Contents Unit 1 : Algebra Review Lesson 1: (For worksheet #1) Factoring Review Factoring Using the Distributive Laws Factoring

Precalculus Table of Contents Unit 1 : Algebra Review Lesson 1: (For worksheet #1) Factoring Review Factoring Using the Distributive Laws Factoring Unit 1 : Algebra Review Factoring Review Factoring Using the Distributive Laws Factoring Trinomials Factoring the Difference of Two Squares Factoring Perfect Square Trinomials Factoring the Sum and Difference

More information

Revision Checklist. Unit FP3: Further Pure Mathematics 3. Assessment information

Revision Checklist. Unit FP3: Further Pure Mathematics 3. Assessment information Revision Checklist Unit FP3: Further Pure Mathematics 3 Unit description Further matrix algebra; vectors, hyperbolic functions; differentiation; integration, further coordinate systems Assessment information

More information

Pre-Calculus Mathematics Curriculum

Pre-Calculus Mathematics Curriculum Pre-Calculus Mathematics Curriculum First day introductions, materials, policies, procedures and Summer Exam (2 days) Unit 1 Estimated time frame for unit 1 Big Ideas Essential Question Competencies Lesson

More information