GRAPHING IN POLAR COORDINATES SYMMETRY

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "GRAPHING IN POLAR COORDINATES SYMMETRY"

Transcription

1 GRAPHING IN POLAR COORDINATES SYMMETRY Recall from Algebra and Calculus I that the concept of symmetry was discussed using Cartesian equations. Also remember that there are three types of symmetry - y-axis, x- axis, and origin. Do you recall how we could test the functions for symmetry? If not, here are the tests. 1. A graph has symmetry with respect to the y-axis if, whenever (x, y) is on the graph, so is the point (-x, y). 2. A graph has symmetry with respect to the origin if, whenever (x, y) is on the graph, so is the point (-x, -y). 3. A graph has symmetry with respect to the x-axis if, whenever (x, y) is on the graph, so is the point (x, -y). The big question is how do we test for symmetry of an equation in polar coordinates? Let us look at the following diagrams to determine the answer to this question. x-axis symmetry y-axis symmetry

2 symmetry about the origin So here are the symmetry tests for polar graphs. 1. Symmetry about the x-axis: If the point (r, ) lies on the graph, then the point (r, - ) or (-r, - ) also lies on the graph. 2. Symmetry about the y-axis: If the point (r, ) lies on the graph, then the point (r, - ) or (-r, - ) also lies on the graph. 3. Symmetry about the origin: If the point (r, ) lies on the graph, then the point (- r, ) or (r, + ) also lies on the graph. EXAMPLE 1: Identify the symmetries of the curve r = cos and then sketch the graph. (r, - ) r = cos (- ) r = cos (Remember that cosine is an even function.) x-axis symmetry: yes (-r, - ) -r = cos (- ) -r = cos r = -2-2 cos y-axis symmetry: no (-r, ) -r = cos r = -2-2 cos symmetry with respect to the origin: no Now, let us compare our findings with the graph of this function.

3 Notice that the graph's only symmetry is with respect to the x- axis, and this is what we determine with our testing. EXAMPLE 2: Identify the symmetries of the curve r = 2 + sin and then sketch the graph. (r, - ) r = 2 + sin (- ) r = 2 -sin (Remember that sine is an odd function.) x-axis symmetry: no (-r, - ) -r = 2 + sin (- ) -r = 2 - sin r = -2 + sin y-axis symmetry: no Is this a correct answer? No! Let us look at the graph of these two functions on the same coordinate axis.

4 r = 2 + sin is the purple graph r = sin is the teal graph We have the same graph, but they start in different places. Therefore, this function does have y-axis symmetry. Sometimes it is best to look at the graph of the polar function instead of trusting algebraic manipulation. EXAMPLE 3: Identify the symmetries of the curve r 2 = cos and then sketch the graph. (r, - ) r 2 = cos (- ) r 2 = cos (Remember that cosine is an even function.) x-axis symmetry: yes (-r, - ) (-r) 2 = cos (- ) r 2 = cos y-axis symmetry: yes (-r, ) (-r) 2 = cos r 2 = cos symmetry with respect to the origin: yes Now, let us compare our findings with the graph of this function.

5 Yes this graph does fit the results that we received from algebraic manipulation. SLOPES Now let us look at how to determine the slope of a polar curve r = f ( ). Remember that the slope of any curve is given by dy/ dx not dr/ d, so we will have to derive out the formula for dy/dx. Let x = r cos = f ( ) cos and y = r sin = f ( ) sin. If f is a differentiable function of, then so is x and y. When dx/ d 0, we can find dy/ dx from the parametric formula. EXAMPLE 4: Find the slope of the curve r = -1 + sin at. Now evaluate dy/dx at.

6 EXAMPLE 5: Find the slope of the curve r = cos 2 at / 2. Now evaluate dy/ dx at / 2. FINDING POINTS WHERE POLAR GRAPHS INTERSECT There are two types of intersection points. They are (1) simultaneous, and (2) non-simultaneous. Here is how your find both types of points. To find the simultaneous intersection points, set the two equations equal to each other and solve for. To find the non-simultaneous intersection points, graph both equations and determine where the graphs cross each other. EXAMPLE 6: Find the points of intersection (both types) of the pair of curves r = 1 + sin and r = 1 - sin. SIMULTANEOUS INTERSECTION POINTS 1 + sin = 1 - sin 2sin = 0 = 0 and = When = 0, then r = 1 + sin 0 = 1 (1, 0).

7 When =, then r = 1 + sin = 1 (1, ). NON-SIMULTANEOUS INTERSECTION POINTS Let us graph both equations on the same axis. r = 1 + sin is in purple r = 1 - sin is in teal Notice that the graphs cross each other at the point (0, 0), so this is the non-simultaneous intersection point. This is the only one. EXAMPLE 7: Find the points of intersection (both types) of the pair of curves r = cos and r = 1 - cos. SIMULTANEOUS INTERSECTION POINTS

8 NON-SIMULTANEOUS INTERSECTION POINTS r = cos is in purple r = 1 - cos is in teal The graphs cross each other at the origin, so the only nonsimultaneous intersection point is (0, 0). EXAMPLE 8: Find the points of intersection (both types) of the pair of curves r 2 = cos 2 and r 2 = sin 2. SIMULTANEOUS INTERSECTION POINTS

9 NON-SIMULTANEOUS INTERSECTION POINTS r 2 = sin 2 is in purple r 2 = cos 2 is in teal The only non-simultaneous intersection point for these two graphs is the origin, (0, 0). I have discussed three major topics in this set of supplemental notes. The first was how to determine the symmetry of a polar graph. When looking at some examples, we concluded that we would sometimes have to look at the graph of the equation. The use of symmetry will be important when we start to determine the area inside the curve. The second topic that I discussed is the slope of a polar curve. This is an application of the derivative of a parametric curve. Finally, I talked about how to find the two types of intersection points. This will be useful when we start to determine the area between two curves

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

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

More information

This function is symmetric with respect to the y-axis, so I will let - /2 /2 and multiply the area by 2.

This function is symmetric with respect to the y-axis, so I will let - /2 /2 and multiply the area by 2. INTEGRATION IN POLAR COORDINATES One of the main reasons why we study polar coordinates is to help us to find the area of a region that cannot easily be integrated in terms of x. In this set of notes,

More information

Apr 23, 2015. Calculus with Algebra and Trigonometry II Lecture 23Final Review: Apr Curve 23, 2015 sketching 1 / and 19pa

Apr 23, 2015. Calculus with Algebra and Trigonometry II Lecture 23Final Review: Apr Curve 23, 2015 sketching 1 / and 19pa Calculus with Algebra and Trigonometry II Lecture 23 Final Review: Curve sketching and parametric equations Apr 23, 2015 Calculus with Algebra and Trigonometry II Lecture 23Final Review: Apr Curve 23,

More information

Calculus II MAT 146 Integration Applications: Area Between Curves

Calculus II MAT 146 Integration Applications: Area Between Curves Calculus II MAT 46 Integration Applications: Area Between Curves A fundamental application of integration involves determining the area under a curve for some interval on the x- or y-axis. In a previous

More information

Assignment 5 Math 101 Spring 2009

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)

More information

POLAR COORDINATES DEFINITION OF POLAR COORDINATES

POLAR COORDINATES DEFINITION OF POLAR COORDINATES POLAR COORDINATES DEFINITION OF POLAR COORDINATES Before we can start working with polar coordinates, we must define what we will be talking about. So let us first set us a diagram that will help us understand

More information

Applications of Integration Day 1

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

More information

Math 181 Spring 2007 HW 1 Corrected

Math 181 Spring 2007 HW 1 Corrected Math 181 Spring 2007 HW 1 Corrected February 1, 2007 Sec. 1.1 # 2 The graphs of f and g are given (see the graph in the book). (a) State the values of f( 4) and g(3). Find 4 on the x-axis (horizontal axis)

More information

Exam 1 Sample Question SOLUTIONS. y = 2x

Exam 1 Sample Question SOLUTIONS. y = 2x Exam Sample Question SOLUTIONS. Eliminate the parameter to find a Cartesian equation for the curve: x e t, y e t. SOLUTION: You might look at the coordinates and notice that If you don t see it, we can

More information

Name Calculus AP Chapter 7 Outline M. C.

Name Calculus AP Chapter 7 Outline M. C. Name Calculus AP Chapter 7 Outline M. C. A. AREA UNDER A CURVE: a. If y = f (x) is continuous and non-negative on [a, b], then the area under the curve of f from a to b is: A = f (x) dx b. If y = f (x)

More information

Lengths in Polar Coordinates

Lengths in Polar Coordinates Lengths in Polar Coordinates Given a polar curve r = f (θ), we can use the relationship between Cartesian coordinates and Polar coordinates to write parametric equations which describe the curve using

More information

Chapter 11 - Curve Sketching. Lecture 17. MATH10070 - Introduction to Calculus. maths.ucd.ie/modules/math10070. Kevin Hutchinson.

Chapter 11 - Curve Sketching. Lecture 17. MATH10070 - Introduction to Calculus. maths.ucd.ie/modules/math10070. Kevin Hutchinson. Lecture 17 MATH10070 - Introduction to Calculus maths.ucd.ie/modules/math10070 Kevin Hutchinson 28th October 2010 Z Chain Rule (I): If y = f (u) and u = g(x) dy dx = dy du du dx Z Chain rule (II): d dx

More information

Solutions to Final Practice Problems

Solutions to Final Practice Problems s to Final Practice Problems Math March 5, Change the Cartesian integral into an equivalent polar integral and evaluate: I 5 x 5 5 x ( x + y ) dydx The domain of integration for this integral is D {(x,

More information

Section 1.8 Coordinate Geometry

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

More information

10 Polar Coordinates, Parametric Equations

10 Polar Coordinates, Parametric Equations Polar Coordinates, Parametric Equations ½¼º½ ÈÓÐ Ö ÓÓÖ Ò Ø Coordinate systems are tools that let us use algebraic methods to understand geometry While the rectangular (also called Cartesian) coordinates

More information

Lecture 8 : Coordinate Geometry. The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 20

Lecture 8 : Coordinate Geometry. The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 20 Lecture 8 : Coordinate Geometry The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 0 distance on the axis and give each point an identity on the corresponding

More information

1.2 GRAPHS OF EQUATIONS. Copyright Cengage Learning. All rights reserved.

1.2 GRAPHS OF EQUATIONS. Copyright Cengage Learning. All rights reserved. 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

More information

Graphs of Polar Equations

Graphs of Polar Equations Graphs of Polar Equations In the last section, we learned how to graph a point with polar coordinates (r, θ). We will now look at graphing polar equations. Just as a quick review, the polar coordinate

More information

Visualizing Differential Equations Slope Fields. by Lin McMullin

Visualizing Differential Equations Slope Fields. by Lin McMullin Visualizing Differential Equations Slope Fields by Lin McMullin The topic of slope fields is new to the AP Calculus AB Course Description for the 2004 exam. Where do slope fields come from? How should

More information

1.7 Cylindrical and Spherical Coordinates

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

More information

IB Practice Exam: 11 Paper 2 Zone 1 90 min, Calculator Allowed. Name: Date: Class:

IB Practice Exam: 11 Paper 2 Zone 1 90 min, Calculator Allowed. Name: Date: Class: IB Math Standard Level Year 2: May 11, Paper 2, TZ 1 IB Practice Exam: 11 Paper 2 Zone 1 90 min, Calculator Allowed Name: Date: Class: 1. The following diagram shows triangle ABC. AB = 7 cm, BC = 9 cm

More information

Tangent and normal lines to conics

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

More information

Engineering Math II Spring 2015 Solutions for Class Activity #2

Engineering Math II Spring 2015 Solutions for Class Activity #2 Engineering Math II Spring 15 Solutions for Class Activity # Problem 1. Find the area of the region bounded by the parabola y = x, the tangent line to this parabola at 1, 1), and the x-axis. Then find

More information

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

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

More information

lim lim x l 1 sx 1 and so the line x 1 is a vertical asymptote. f x 2xsx 1 x 2 1 (2sx 1) x 1

lim lim x l 1 sx 1 and so the line x 1 is a vertical asymptote. f x 2xsx 1 x 2 1 (2sx 1) x 1 SECTION 3.4 CURE SKETCHING 3.4 CURE SKETCHING EXAMPLE A Sketch the graph of f x. sx A. Domain x x 0 x x, B. The x- and y-intercepts are both 0. C. Symmetry: None D. Since x l sx there is no horizontal

More information

The Derivative and the Tangent Line Problem. The Tangent Line Problem

The Derivative and the Tangent Line Problem. The Tangent Line Problem The Derivative and the Tangent Line Problem Calculus grew out of four major problems that European mathematicians were working on during the seventeenth century. 1. The tangent line problem 2. The velocity

More information

Conic Sections in Cartesian and Polar Coordinates

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

More information

Quadratic Functions. Teachers Teaching with Technology. Scotland T 3. Symmetry of Graphs. Teachers Teaching with Technology (Scotland)

Quadratic Functions. Teachers Teaching with Technology. Scotland T 3. Symmetry of Graphs. Teachers Teaching with Technology (Scotland) Teachers Teaching with Technology (Scotland) Teachers Teaching with Technology T 3 Scotland Quadratic Functions Symmetry of Graphs Teachers Teaching with Technology (Scotland) QUADRATIC FUNCTION Aim To

More information

1. AREA BETWEEN the CURVES

1. AREA BETWEEN the CURVES 1 The area between two curves The Volume of the Solid of revolution (by slicing) 1. AREA BETWEEN the CURVES da = {( outer function ) ( inner )} dx function b b A = da = [y 1 (x) y (x)]dx a a d d A = da

More information

2 Applications of Integration

2 Applications of Integration Brian E. Veitch 2 Applications of Integration 2.1 Area between curves In this section we are going to find the area between curves. Recall that the integral can represent the area between f(x) and the

More information

c sigma & CEMTL

c sigma & CEMTL c sigma & CEMTL Foreword The Regional Centre for Excellence in Mathematics Teaching and Learning (CEMTL) is collaboration between the Shannon Consortium Partners: University of Limerick, Institute of Technology,

More information

Cork Institute of Technology. CIT Mathematics Examination, Paper 2 Sample Paper A

Cork Institute of Technology. CIT Mathematics Examination, Paper 2 Sample Paper A Cork Institute of Technology CIT Mathematics Examination, 2015 Paper 2 Sample Paper A Answer ALL FIVE questions. Each question is worth 20 marks. Total marks available: 100 marks. The standard Formulae

More information

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

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

More information

Plotting Polar Curves We continue to study the plotting of polar curves. Recall the family of cardioids shown last time.

Plotting Polar Curves We continue to study the plotting of polar curves. Recall the family of cardioids shown last time. Plotting Polar Curves We continue to study the plotting of polar curves. Recall the family of cardioids shown last time. r = 1 cos(θ) r = 1 + cos(θ) r = 1 + sin(θ) r = 1 sin(θ) Now let us look at a similar

More information

2 Unit Bridging Course Day 2 Linear functions II: Finding equations

2 Unit Bridging Course Day 2 Linear functions II: Finding equations 1 / 38 2 Unit Bridging Course Day 2 Linear functions II: Finding equations Clinton Boys 2 / 38 Finding equations of lines If we have the information of (i) the gradient of a line (ii) the coordinates of

More information

Intersections of Polar Curves

Intersections of Polar Curves Intersections of Polar Curves The purpose of this supplement is to find a method for determining where graphs of polar equations intersect each other. Let s start with a fairly straightforward example.

More information

AP CALCULUS AB 2008 SCORING GUIDELINES

AP CALCULUS AB 2008 SCORING GUIDELINES AP CALCULUS AB 2008 SCORING GUIDELINES Question 1 Let R be the region bounded by the graphs of y = sin( π x) and y = x 4 x, as shown in the figure above. (a) Find the area of R. (b) The horizontal line

More information

Student Activity: To investigate an ESB bill

Student Activity: To investigate an ESB bill Student Activity: To investigate an ESB bill Use in connection with the interactive file, ESB Bill, on the Student s CD. 1. What are the 2 main costs that contribute to your ESB bill? 2. a. Complete the

More information

PURE MATHEMATICS AM 27

PURE MATHEMATICS AM 27 AM SYLLABUS (013) PURE MATHEMATICS AM 7 SYLLABUS 1 Pure Mathematics AM 7 Syllabus (Available in September) Paper I(3hrs)+Paper II(3hrs) 1. AIMS To prepare students for further studies in Mathematics and

More information

PURE MATHEMATICS AM 27

PURE MATHEMATICS AM 27 AM Syllabus (015): Pure Mathematics AM SYLLABUS (015) PURE MATHEMATICS AM 7 SYLLABUS 1 AM Syllabus (015): Pure Mathematics Pure Mathematics AM 7 Syllabus (Available in September) Paper I(3hrs)+Paper II(3hrs)

More information

x = y + 2, and the line

x = y + 2, and the line WS 8.: Areas between Curves Name Date Period Worksheet 8. Areas between Curves Show all work on a separate sheet of paper. No calculator unless stated. Multiple Choice. Let R be the region in the first

More information

Practice Problems for Exam 1 Math 140A, Summer 2014, July 2

Practice Problems for Exam 1 Math 140A, Summer 2014, July 2 Practice Problems for Exam 1 Math 140A, Summer 2014, July 2 Name: INSTRUCTIONS: These problems are for PRACTICE. For the practice exam, you may use your book, consult your classmates, and use any other

More information

3.3. Parabola and parallel lines

3.3. Parabola and parallel lines 3.3. Parabola and parallel lines This example is similar in spirit to the last, except this time the parabola is held fixed and a line of slope is moved parallel to itself. The objective is to see how

More information

Area and Arc Length in Polar Coordinates. Area of a Polar Region

Area and Arc Length in Polar Coordinates. Area of a Polar Region 46_5.qxd //4 :7 PM Page 79 SECTION.5 Area and Arc Length in Polar Coordinates 79 θ Section.5 r The area of a sector of a circle is A r. Figure.49 (a) β r = f( θ) α Area and Arc Length in Polar Coordinates

More information

Solutions to Homework 10

Solutions 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 information

MATH 105: Finite Mathematics 1-1: Rectangular Coordinates, Lines

MATH 105: Finite Mathematics 1-1: Rectangular Coordinates, Lines MATH 105: Finite Mathematics 1-1: Rectangular Coordinates, Lines Prof. Jonathan Duncan Walla Walla College Winter Quarter, 2006 Outline 1 Rectangular Coordinate System 2 Graphing Lines 3 The Equation of

More information

4.4 CURVE SKETCHING. there is no horizontal asymptote. Since sx 1 l 0 as x l 1 and f x is always positive, we have

4.4 CURVE SKETCHING. there is no horizontal asymptote. Since sx 1 l 0 as x l 1 and f x is always positive, we have SECTION 4.4 CURE SKETCHING 4.4 CURE SKETCHING EXAMPLE A Sketch the graph of f x. sx A. Domain x x 0 x x, B. The x- and y-intercepts are both 0. C. Symmetry: None D. Since x l sx there is no horizontal

More information

Section 10.4 Vectors

Section 10.4 Vectors Section 10.4 Vectors A vector is represented by using a ray, or arrow, that starts at an initial point and ends at a terminal point. Your textbook will always use a bold letter to indicate a vector (such

More information

We can use more sectors (i.e., decrease the sector s angle θ) to get a better approximation:

We can use more sectors (i.e., decrease the sector s angle θ) to get a better approximation: Section 1.4 Areas of Polar Curves In this section we will find a formula for determining the area of regions bounded by polar curves. To do this, wee again make use of the idea of approximating a region

More information

Fourier Series. 1. Full-range Fourier Series. ) + b n sin L. [ a n cos L )

Fourier Series. 1. Full-range Fourier Series. ) + b n sin L. [ a n cos L ) Fourier Series These summary notes should be used in conjunction with, and should not be a replacement for, your lecture notes. You should be familiar with the following definitions. A function f is periodic

More information

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

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

More information

Section 12.6: Directional Derivatives and the Gradient Vector

Section 12.6: Directional Derivatives and the Gradient Vector Section 26: Directional Derivatives and the Gradient Vector Recall that if f is a differentiable function of x and y and z = f(x, y), then the partial derivatives f x (x, y) and f y (x, y) give the rate

More information

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

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,

More information

MATHEMATICS (CLASSES XI XII)

MATHEMATICS (CLASSES XI XII) MATHEMATICS (CLASSES XI XII) General Guidelines (i) All concepts/identities must be illustrated by situational examples. (ii) The language of word problems must be clear, simple and unambiguous. (iii)

More information

1.4 Exponential and logarithm graphs.

1.4 Exponential and logarithm graphs. 1.4 Exponential and logarithm graphs. Example 1. Recall that b = 2 a if and only if a = log 2 (b) That tells us that the functions f(x) = 2 x and g(x) = log 2 (x) are inverse functions. It also tells us

More information

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

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

More information

APPLICATIONS OF DIFFERENTIATION

APPLICATIONS OF DIFFERENTIATION 4 APPLICATIONS OF DIFFERENTIATION APPLICATIONS OF DIFFERENTIATION So far, we have been concerned with some particular aspects of curve sketching: Domain, range, and symmetry (Chapter 1) Limits, continuity,

More information

Section 6-4 Product Sum and Sum Product Identities

Section 6-4 Product Sum and Sum Product Identities 480 6 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS Section 6-4 Product Sum and Sum Product Identities Product Sum Identities Sum Product Identities Our work with identities is concluded by developing

More information

Section 10.7 Parametric Equations

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

More information

Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum

Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum UNIT I: The Hyperbolic Functions basic calculus concepts, including techniques for curve sketching, exponential and logarithmic

More information

Chapter 5: Trigonometric Functions of Real Numbers

Chapter 5: Trigonometric Functions of Real Numbers Chapter 5: Trigonometric Functions of Real Numbers 5.1 The Unit Circle The unit circle is the circle of radius 1 centered at the origin. Its equation is x + y = 1 Example: The point P (x, 1 ) is on the

More information

Inverse Trigonometric Functions

Inverse Trigonometric Functions Inverse Trigonometric Functions Inverse Sine Function Recall from Section.8 that, for a function to have an inverse function, it must be one-to-one--- that is, it must pass the Horizontal Line Test. From

More information

Calculating Areas Section 6.1

Calculating Areas Section 6.1 A B I L E N E C H R I S T I A N U N I V E R S I T Y Department of Mathematics Calculating Areas Section 6.1 Dr. John Ehrke Department of Mathematics Fall 2012 Measuring Area By Slicing We first defined

More information

1.5 ANALYZING GRAPHS OF FUNCTIONS. Copyright Cengage Learning. All rights reserved.

1.5 ANALYZING GRAPHS OF FUNCTIONS. Copyright Cengage Learning. All rights reserved. 1.5 ANALYZING GRAPHS OF FUNCTIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Use the Vertical Line Test for functions. Find the zeros of functions. Determine intervals on which

More information

Class Notes for MATH 2 Precalculus. Fall Prepared by. Stephanie Sorenson

Class Notes for MATH 2 Precalculus. Fall Prepared by. Stephanie Sorenson Class Notes for MATH 2 Precalculus Fall 2012 Prepared by Stephanie Sorenson Table of Contents 1.2 Graphs of Equations... 1 1.4 Functions... 9 1.5 Analyzing Graphs of Functions... 14 1.6 A Library of Parent

More information

Procedure In each case, draw and extend the given series to the fifth generation, then complete the following tasks:

Procedure In each case, draw and extend the given series to the fifth generation, then complete the following tasks: Math IV Nonlinear Algebra 1.2 Growth & Decay Investigation 1.2 B: Nonlinear Growth Introduction The previous investigation introduced you to a pattern of nonlinear growth, as found in the areas of a series

More information

10.1 Notes-Graphing Quadratics

10.1 Notes-Graphing Quadratics Name: Period: 10.1 Notes-Graphing Quadratics Section 1: Identifying the vertex (minimum/maximum), the axis of symmetry, and the roots (zeros): State the maximum or minimum point (vertex), the axis of symmetry,

More information

-axis -axis. -axis. at point.

-axis -axis. -axis. at point. Chapter 5 Tangent Lines Sometimes, a concept can make a lot of sense to us visually, but when we try to do some explicit calculations we are quickly humbled We are going to illustrate this sort of thing

More information

Teacher Questionnaire

Teacher Questionnaire Identification Label Teacher Name: Class Name: Teacher ID: Teacher Link # Teacher Questionnaire Advanced Mathematics International Association for

More information

AQA Level 2 Certificate FURTHER MATHEMATICS

AQA Level 2 Certificate FURTHER MATHEMATICS AQA Qualifications AQA Level 2 Certificate FURTHER MATHEMATICS Level 2 (8360) Our specification is published on our website (www.aqa.org.uk). We will let centres know in writing about any changes to the

More information

Angles and Quadrants. Angle Relationships and Degree Measurement. Chapter 7: Trigonometry

Angles and Quadrants. Angle Relationships and Degree Measurement. Chapter 7: Trigonometry Chapter 7: Trigonometry Trigonometry is the study of angles and how they can be used as a means of indirect measurement, that is, the measurement of a distance where it is not practical or even possible

More information

MATHEMATICS Unit Pure Core 2

MATHEMATICS Unit Pure Core 2 General Certificate of Education June 2006 Advanced Subsidiary Examination MATHEMATICS Unit Pure Core 2 MPC2 Monday 22 May 2006 9.00 am to 10.30 am For this paper you must have: * an 8-page answer book

More information

Midterm Exam I, Calculus III, Sample A

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

More information

Applications of the Integral

Applications of the Integral Chapter 6 Applications of the Integral Evaluating integrals can be tedious and difficult. Mathematica makes this work relatively easy. For example, when computing the area of a region the corresponding

More information

Fourier Series Chapter 3 of Coleman

Fourier Series Chapter 3 of Coleman Fourier Series Chapter 3 of Coleman Dr. Doreen De eon Math 18, Spring 14 1 Introduction Section 3.1 of Coleman The Fourier series takes its name from Joseph Fourier (1768-183), who made important contributions

More information

Patterns, Equations, and Graphs. Section 1-9

Patterns, Equations, and Graphs. Section 1-9 Patterns, Equations, and Graphs Section 1-9 Goals Goal To use tables, equations, and graphs to describe relationships. Vocabulary Solution of an equation Inductive reasoning Review: Graphing in the Coordinate

More information

AP Calculus Project 4 - Curve Sketching

AP Calculus Project 4 - Curve Sketching AP Calculus Project 4 - Curve Sketching Name You will be in a group of 2 to 4. You will be given a series of s that must be accurately sketched with attention paid to roots, extrema, critical points, inflection

More information

1 Lecture 19: Implicit differentiation

1 Lecture 19: Implicit differentiation Lecture 9: Implicit differentiation. Outline The technique of implicit differentiation Tangent lines to a circle Examples.2 Implicit differentiation Suppose we have two quantities or variables x and y

More information

Unit Overview. Content Area: Math Unit Title: Functions and Their Graphs Target Course/Grade Level: Advanced Math Duration: 4 Weeks

Unit Overview. Content Area: Math Unit Title: Functions and Their Graphs Target Course/Grade Level: Advanced Math Duration: 4 Weeks Content Area: Math Unit Title: Functions and Their Graphs Target Course/Grade Level: Advanced Math Duration: 4 Weeks Unit Overview Description In this unit the students will examine groups of common functions

More information

AP Calculus BC 2013 Free-Response Questions

AP Calculus BC 2013 Free-Response Questions AP Calculus BC 013 Free-Response Questions About the College Board The College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity. Founded in

More information

Using The TI-Nspire Calculator in AP Calculus

Using The TI-Nspire Calculator in AP Calculus Using The TI-Nspire Calculator in AP Calculus (Version 3.0) You must be able to perform the following procedures on your calculator: 1. Plot the graph of a function within an arbitrary viewing window,

More information

MATH FINAL EXAMINATION - 3/22/2012

MATH FINAL EXAMINATION - 3/22/2012 MATH 22 - FINAL EXAMINATION - /22/22 Name: Section number: About this exam: Partial credit will be given on the free response questions. To get full credit you must show all of your work. This is a closed

More information

Level 2 Certificate Further MATHEMATICS

Level 2 Certificate Further MATHEMATICS Level 2 Certificate Further MATHEMATICS 83601 Paper 1 non-calculator Report on the Examination Specification 8360 June 2013 Version: 1.0 Further copies of this Report are available from aqa.org.uk Copyright

More information

RADIUS OF CURVATURE AND EVOLUTE OF THE FUNCTION y=f(x)

RADIUS OF CURVATURE AND EVOLUTE OF THE FUNCTION y=f(x) RADIUS OF CURVATURE AND EVOLUTE OF THE FUNCTION y=f( In introductory calculus one learns about the curvature of a function y=f( also about the path (evolute that the center of curvature traces out as x

More information

Algebra. 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. 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 information

Year 11 - GCSE Higher Tier. Revision 2011

Year 11 - GCSE Higher Tier. Revision 2011 Year 11 - GCSE Higher Tier Revision 2011 Transformations Reflect the triangle in the line x = - 2 Translate this triangle 2 right and 3 up Reflect the triangle in the line y = 5 Rotate this shape 90º anticlockwise

More information

Roots and Coefficients of a Quadratic Equation Summary

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

More information

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

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

More information

Mathematics (Project Maths Phase 3)

Mathematics (Project Maths Phase 3) 2014. M329 Coimisiún na Scrúduithe Stáit State Examinations Commission Leaving Certificate Examination 2014 Mathematics (Project Maths Phase 3) Paper 1 Higher Level Friday 6 June Afternoon 2:00 4:30 300

More information

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 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 >

More information

Graphing Quadratic Functions

Graphing Quadratic Functions Problem 1 The Parabola Examine the data in L 1 and L to the right. Let L 1 be the x- value and L be the y-values for a graph. 1. How are the x and y-values related? What pattern do you see? To enter the

More information

You must have: Ruler graduated in centimetres and millimetres, protractor, compasses, pen, HB pencil, eraser, calculator. Tracing paper may be used.

You must have: Ruler graduated in centimetres and millimetres, protractor, compasses, pen, HB pencil, eraser, calculator. Tracing paper may be used. Write your name here Surname Other names Pearson Edexcel Certificate Pearson Edexcel International GCSE Mathematics A Paper 3H Centre Number Tuesday 6 January 2015 Afternoon Time: 2 hours Candidate Number

More information

GRAPHING (2 weeks) Main Underlying Questions: 1. How do you graph points?

GRAPHING (2 weeks) Main Underlying Questions: 1. How do you graph points? GRAPHING (2 weeks) The Rectangular Coordinate System 1. Plot ordered pairs of numbers on the rectangular coordinate system 2. Graph paired data to create a scatter diagram 1. How do you graph points? 2.

More information

Multivariable Calculus MA22S1. Dr Stephen Britton

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

More information

Senior Math Circles February 4, 2009 Conics I

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

More information

The graphs of f and g intersect at (0, 0) and one other point. Find that point: f(y) = g(y) y 2 4y 2y 2 6y = = 2y y 2. 2y(y 3) = 0

The graphs of f and g intersect at (0, 0) and one other point. Find that point: f(y) = g(y) y 2 4y 2y 2 6y = = 2y y 2. 2y(y 3) = 0 . Compute the area between the curves x y 4y and x y y. Let f(y) y 4y y(y 4). f(y) when y or y 4. Let g(y) y y y( y). g(y) when y or y. x 3 y? The graphs of f and g intersect at (, ) and one other point.

More information

Ordered Pairs. Graphing Lines and Linear Inequalities, Solving System of Linear Equations. Cartesian Coordinates System.

Ordered Pairs. Graphing Lines and Linear Inequalities, Solving System of Linear Equations. Cartesian Coordinates System. Ordered Pairs Graphing Lines and Linear Inequalities, Solving System of Linear Equations Peter Lo All equations in two variables, such as y = mx + c, is satisfied only if we find a value of x and a value

More information

Year 11 - GCSE Higher Tier. Revision Solutions

Year 11 - GCSE Higher Tier. Revision Solutions Year 11 - GCSE Higher Tier Revision 2011 - Solutions Transformations Reflect the triangle in the line x = - 2 Translate this triangle 2 right and 3 up Reflect the triangle in the line y = 5 Rotate this

More information

2.2 Derivative as a Function

2.2 Derivative as a Function 2.2 Derivative as a Function Recall that we defined the derivative as f (a) = lim h 0 f(a + h) f(a) h But since a is really just an arbitrary number that represents an x-value, why don t we just use x

More information

Exam 2 Review. 3. How to tell if an equation is linear? An equation is linear if it can be written, through simplification, in the form.

Exam 2 Review. 3. How to tell if an equation is linear? An equation is linear if it can be written, through simplification, in the form. Exam 2 Review Chapter 1 Section1 Do You Know: 1. What does it mean to solve an equation? To solve an equation is to find the solution set, that is, to find the set of all elements in the domain of the

More information