MATRIX TRANSFORMATIONS

Size: px
Start display at page:

Download "MATRIX TRANSFORMATIONS"

Transcription

1 CHAPTER 5. MATRIX TRANSFORMATIONS INSTITIÚID TEICNEOLAÍOCHTA CHEATHARLACH INSTITUTE OF TECHNOLOGY CARLOW MATRIX TRANSFORMATIONS Matri Transformations Definition Let A and B be sets. A function f : A B from A to B assigns to each element a of A a (unique) element f(a) of B. The set A on which the function is defined is referred to as the domain of the function f : A B. The set B into which the domain is mapped b f is referred to as the codomain of the function f. A B a f(a) f : A B For man common functions the domain and codomain are the sets of real numbers. concerned with functions for which the domain and codomain are vectors. We are

2 CHAPTER 5. MATRIX TRANSFORMATIONS Definition Let A and B be vectors and if f is a function with domain A and codomain B, then we sa that f is a transformation from A to B or that maps A to B, which we denote b writing f : A B In the special case where A = B, the transformation is also called and operator on V. We seek a transformation that maps the column vector in R n into the column vector in R n b multipling b the matri A. We call this a matri transformation and we denote it b T A : R n R n R n R n T A () T A : R n R n In shorthand, we have T A () = [T ] The matri transformation T A is called multiplication b A, and the matri A is called the standard matri for the transformation. The following theorem lists the four basic properties of matri transformations that follow from properties of matri multiplication.

3 3 CHAPTER 5. MATRIX TRANSFORMATIONS 3 Theorem For ever matri A, the matri transformation T A : R n R m has the following properties for all vectors u and v in R n and for ever scalar k: i T A () = ii T A (k u) = kt A ( u) iii T A ( u + v) = T A ( u) + T A ( v) iv T A ( u v) = T A ( u) T A ( v) The geometric effect of a matri transformation T A : R n R m is to map each vector (point) in R n into a vector (point) in R m. Remark There is a wa of finding the standard matri A for a matri transformation from R n to R m b considering the effect of that transformation on the standard basis vectors for R n. The standard basis vectors for R are e = (, ) and e = (, ). The standard basis vectors for R 3 are e = (,, ), e = (,, ) and e 3 = (,, ). To eplain the idea, in general, suppose that the standard matri A is unknown and that e, e,..., e n are the standard basis vectors for R n. transformation T A are Suppose also that the images of these vectors under the T A (e ) = Ae, T A (e ) = Ae,..., T A (e n ) = Ae n We can write that A = [ ] T A (e ) T A (e )... T A (e n ) In summar, we have the following procedure for finding the standard matri for a matri transformation:

4 4 CHAPTER 5. MATRIX TRANSFORMATIONS 4 Step : Find the images of the standard basis vectors e, e,..., e n for R n. in column form. Step : Construct the matri that has the images obtained in Step as its successive columns. This matri is the standard matri for the transformation.. Reflection Operators Some of the most basic matri operators on R 3 are those that map each point into its smmetric image about a fied plane. These are called reflection operators. The following diagrams show the standard matrices for the reflections about the coordinate planes in R 3. In each case the standard matri was obtained b finding the images of the standard basis vectors, converting those images to column vectors, and then using those column vectors as successive column of the standard matri. The following matrices will perform orthogonal reflections ABOUT the plane, the z plane and the z plane respectivel. Reflection about the z plane: T (,, z) = (,, z) z (,, z) T () (,, z) T (e ) = T (,, ) = (,, ) T (e ) = T (,, ) = (,, ) T (e 3 ) = T (,, ) = (,, ) T =

5 5 CHAPTER 5. MATRIX TRANSFORMATIONS 5 Reflection about the z plane: T (,, z) = (,, z) z (,, z) T () (,, z) T (e ) = T (,, ) = (,, ) T (e ) = T (,, ) = (,, ) T (e 3 ) = T (,, ) = (,, ) T = Reflection about the plane: T (,, z) = (,, z) z T () (,, z) (,, z) T (e ) = T (,, ) = (,, ) T (e ) = T (,, ) = (,, ) T (e 3 ) = T (,, ) = (,, ) T =

6 6 CHAPTER 5. MATRIX TRANSFORMATIONS 6. Projection Operators Matri operators on R 3 that map each point into its orthogonal projection on a fied plane are called projection operators (or more precisel, orthogonal projection operators). The following are the standard matrices for the orthogonal projections on the coordinate aes in R 3. Again, in each case the standard matri was obtained b finding the images of the standard basis vectors, converting those images to column vectors, and then using those column vectors as successive column of the standard matri. Orthogonal Projection onto the z plane: T (,, z) = (,, z) z (,, z) T () (,, z) T (e ) = T (,, ) = (,, ) T (e ) = T (,, ) = (,, ) T (e 3 ) = T (,, ) = (,, ) T = The following matrices will perform orthogonal projection ONTO the plane, the z plane and the z plane respectivel. A = A = A =

7 7 CHAPTER 5. MATRIX TRANSFORMATIONS 7 Eample To determine the coordinates of the point (,, 3) after a reflection in the z plane, we have T A () = [T ] =. 3 = 3 i.e., (,, 3) (,, 3)

8 8 CHAPTER 5. MATRIX TRANSFORMATIONS 8.3 Rotation Operators Matri Operators on on R and R 3 that move points along circular arcs are called rotation operators. Let us consider how to find the standard matri for the rotation operator T : R R that moves points anti-clockwise about the origin through an angle θ. e ( sin θ, cos θ) (cos θ, sin θ) θ θ e The images of the standard basis vectors in R are T (e ) = T (, ) = (cos θ, sin θ) T (e ) = T (, ) = ( sin θ, cos θ) Hence, the standard matri for this transformation is ( cos θ sin θ A = sin θ cos θ ) In keeping with common notation, we represent this operator b R θ, i.e., ( ) cos θ sin θ R θ = sin θ cos θ

9 9 CHAPTER 5. MATRIX TRANSFORMATIONS 9 (w, w ) w (, ) θ Note In the plane, anti-clockwise angles are positive and clockwise angles are negative. The rotation matri for a clockwise rotation of θ radians can be obtained b replacing θ b θ. Note that cos( θ) = cos θ and sin( θ) = sin θ. After simplification this ield ( cos θ sin θ R θ = sin θ cos θ ) A rotational operator in R 3 is a matri operator that rotates each vector in R 3 about some rotation ais through fied angle θ. A rotation of vectors in R 3 is usuall described in relation to a ra emanating from the origin, called the ais of rotation. As a vector revolves around the ais of rotation, it sweeps out some portion of a cone. The angle of rotation, which is measured in the base of the cone, is described as clockwise or anti-clockwise in relation to a viewpoint that is along the ais of rotation looking towards the origin. z (w, w, w 3 ) w θ (,, z) Ais of Rotation

10 CHAPTER 5. MATRIX TRANSFORMATIONS The following three matrices will rotate an vector anti-clockwise about the, and z aes through an angle of θ respectivel. For each of these rotations one of the components is unchanged and the relationship between the other components can be derived b the same procedure used to derive rotational matrices in R. About the -ais, -ais and z-ais respectivel, we have R θ = R θ = R z θ = cos θ sin θ sin θ cos θ cos θ sin θ sin θ cos θ cos θ sin θ sin θ cos θ The following three matrices will rotate an vector clockwise about the, and z aes through an angle of θ respectivel. About the -ais, -ais and z-ais respectivel, we have R θ = R θ = R z θ = cos θ sin θ sin θ cos θ cos θ sin θ sin θ cos θ cos θ sin θ sin θ cos θ

11 CHAPTER 5. MATRIX TRANSFORMATIONS Eample Using a matri transformation rotate the triangle ABC whose vertices are the points A(,, ), B(,, ), C(,, ) through 6 o anti-clockwise about the z-ais as follows: A β B α γ C of 6 o Firstl, the following matri will rotate an vector anti-clockwise about the z-aes through an angle Now R 6 o = R 6 o = [T ] = R 6 o = [T ] = R 6 o = [T ] = cos 6 o sin 6 o sin 6 o cos 6 o = = = = Finall, we have A ( 366, 366, ), B ( 34, 3, ), C ( 366, 366, ).

12 CHAPTER 5. MATRIX TRANSFORMATIONS Remark Consider the matri A = This matri represents a clockwise rotation about the z-ais through 9 o. Notice that the inverse matri is A = which is an anti-clockwise rotation about the z-ais through 9 o. It is eas to see here that the inverse matri is simpl the transpose of the original matri A. This is ver common in computer graphics and an matri with this propert is called an orthogonal matri. Definition A square matri A is an orthogonal matri if A = A t. This can alwas be tested b multipling A.A t and see if the result is the identit matri. Notice the the three clockwise rotational matrices R θ,r θ and R z θ are each orthogonal since multipling each matri b its transpose will ield I the identit matri. So, for eample, R θ.(r θ) t = cos θ sin θ sin θ cos θ cos θ sin θ sin θ cos θ = This can be confirmed for the matri R θ and R z θ in a similar wa. When an orthogonal matri is used to rotate vectors, it will keep the lengths of the vectors preserved as will the angle between the vectors. So an orthogonal matri ma represent a rotation.

13 3 CHAPTER 5. MATRIX TRANSFORMATIONS 3 Eample Consider the matri A = To show that A is orthogonal, note that a square matri A is an orthogonal matri if A = A t. Pre-multipling both sides b A ield A.A = I = A.A t Now A.A t = =.4 Some Eercises Eercise Using a matri transformation rotate the triangle ABC whose vertices are the points A(3, 4, 8), B(, 3, 5), C(, 6, 7) through 3 o anti-clockwise about the -ais. Eercise Using a matri transformation rotate the cube whose corners are the points A(, 5, ), B(4, 5, ), C(,, ), D(4,, ) E(, 5, 4), F (4, 5, 4), G(,, 4), H(4,, 4) through 45 o anti-clockwise about the -ais.

14 4 CHAPTER 5. MATRIX TRANSFORMATIONS 4.5 Dilations and Contractions If k is a non-negative scalar, then the operator T () = k in R or R 3 has the effect of increasing or decreasing the length of each vector b a factor of k. If k the operator is called a contraction with factor k, and if k > it is called a dilution with factor k. If k =, then T is the identit operator and can be regarded as either a contraction or a dilation. (, ) (, k) k > (, ) (k, ) k (, ) (, k) (, ) (k, ) The following matrices will perform a contraction or a dilation with factor k in R. A = ( k k ) The following matrices will perform a contraction or a dilation with factor k in R 3.

15 5 CHAPTER 5. MATRIX TRANSFORMATIONS 5 A = k k k In a dilation or contraction in R or R 3, all coordinates are multiplied b a factor k. If onl one of the coordinates is multiplied b b a factor k, then the resulting operator is called an epansion or compression with factor k. Consider an epansion or compression with factor k in the -direction. k > (, ) (, ) (, ) (k, ) k (, ) (, ) (, ) (k, ) The following matrices will perform a epansion or compression with factor k in R in the -direction. A = ( k )

16 6 CHAPTER 5. MATRIX TRANSFORMATIONS 6 Consider an epansion or compression with factor k in the -direction. (, ) (, k) k > (, ) (, ) k (, ) (, k) (, ) (, ) The following matrices will perform a epansion or compression with factor k in R in the -direction. A = ( k ) The following matrices will perform a epansion or compression with factor k in R 3 in the -direction, in the -direction and in the z-direction respectivel. A = A = A = k k k

17 7 CHAPTER 5. MATRIX TRANSFORMATIONS 7.6 Properties of Matri Transformations We seek to show that if several matri transformations are performed in succession, then the same result can be obtained b a single matri transformation that is chosen appropriatel. Firstl, we recall the definition of a composite function. Let A, B and C be sets, let f : A B be a function A to B, let g : B C be a function from B to C. Then there is a function g f : A C obtained b composing the functions f and g. This function is defined at each element a of A b the formula (g f)(a) = g(f(a)) In other words, in order to appl the composition function g f to an element a of A, we first appl the function f to the element a, and then we appl the function g to the resulting element f(a) of B to obtain an element g(f(a)) of C. A B C f g a f(a) g(f(a)) gof Remark Note that g f denotes the composition function f followed b g. The functions are specified in this order (which ma at first seem odd) in order that (g f)(a) = g(f(a)) for all elements a of the domain A of the function f. Eample Let f : R R defined b f() = and let g : R R defined b g() = + 5. Now (g f)() = g(f()) = g( ) = + 5. Also (f g)() = f(g()) = f( + 5) = Eercise Let f : R R defined b f() = ( + ) and let g : R R defined b g() = sin. Determine the composite functions g f and f g.

18 8 CHAPTER 5. MATRIX TRANSFORMATIONS 8 Now, consider the following matri transformations T A : R n R n T B : R n R n The composition of T A with T B will be denoted b T B T A Note that T B T A denotes the composition transformation T A followed b T B, i.e., the transformation T A is performed first. This composition of transformations is defined as (T B T A )() = [T B ][T A ] R n R n R n T T T () T (T ()) T T Eample Let T A : R R such that T () = A denote a matri transformation with standard matri A. To find the standard matri for the operator T : R R that first reflects about the line =, then rotates through an angle of 8 o anti-clockwise about the origin, we proceed as follows. Firstl, the standard matri A for the operator can be epressed as the composition A = T T

19 9 CHAPTER 5. MATRIX TRANSFORMATIONS 9 where T is the reflection about the line = and T rotates through an angle of 8 o anti-clockwise about the origin. The standard matrices for these operations are [T ] = [T ] = ( ) ( ) thus, it follows, that the standard matri A = T T is [T ][T ] = ( ) (. ) = ( ) Therefore A = ( ) Eample Let T A : R 3 R 3 such that T () = A denote a matri transformation with standard matri A. To find the standard matri for the operator T : R 3 R 3 that first rotates a vector anti-clockwise about the z-ais through an angle θ, then reflects the resulting vector about the z-plane, and then projects that vector orthogonall onto the -plane, we proceed as follows. Firstl, the standard matri A for the operator can be epressed as the composition A = T 3 T T where T is the rotation about the z-ais, T is the reflection about the z-plane, and T 3 is the orthogonal projection n the -plane. The standard matrices for these operations are [T ] = [T ] = cos θ sin θ sin θ cos θ [T 3 ] =

20 CHAPTER 5. MATRIX TRANSFORMATIONS thus, it follows, that the standard matri A = T 3 T T is Therefore [T 3 ][T ][T ] = A =.. cos θ sin θ sin θ cos θ cos θ sin θ sin θ cos θ Eample Let T A : R 3 R 3 such that T () = A denote a matri transformation with standard matri A. To find the standard matri for the operator T : R 3 R 3 that first rotates a vector anti-clockwise about the -ais through an angle 3 o, then rotates the resulting vector anti-clockwise about the z-ais through an angle of 3 o, and then contracts that vector b a factor of k = 4, we proceed as follows. Firstl, the standard matri A for the operator can be epressed as the composition A = T 3 T T where T is the rotation anti-clockwise about the -ais through an angle 3 o, T is the rotation anti-clockwise about the z-ais through an angle 3 o, and T 3 contraction b a factor of k = 4. The standard matrices for these operations are [T ] = [T ] = 3 4 [T 3 ] = 4 4 thus, it follows, that the standard matri A = T 3 T T is

21 CHAPTER 5. MATRIX TRANSFORMATIONS [T 3 ][T ][T ] = Therefore A = Eercise Let T A : R 3 R 3 such that T () = A denote a matri transformation with standard matri A. Find the standard matri for the stated composition in R 3. i A reflection about the -plane, followed b a reflection about the z-plane, followed b an orthogonal projection on the z-plane. ii A rotation of 7 o anti-clockwise about the -ais, followed b a rotation of 9 o anti-clockwise about the -ais, followed b a rotation of 8 o anti-clockwise about the z-ais.

VECTORS IN THREE DIMENSIONS

VECTORS IN THREE DIMENSIONS 1 CHAPTER 2. BASIC TRIGONOMETRY 1 INSTITIÚID TEICNEOLAÍOCHTA CHEATHARLACH INSTITUTE OF TECHNOLOGY CARLOW VECTORS IN THREE DIMENSIONS 1 Vectors in Two Dimensions A vector is an object which has magnitude

More information

Some linear transformations on R 2 Math 130 Linear Algebra D Joyce, Fall 2013

Some linear transformations on R 2 Math 130 Linear Algebra D Joyce, Fall 2013 Some linear transformations on R 2 Math 3 Linear Algebra D Joce, Fall 23 Let s look at some some linear transformations on the plane R 2. We ll look at several kinds of operators on R 2 including reflections,

More information

Two conventions for coordinate systems. Left-Hand vs Right-Hand. x z. Which is which?

Two conventions for coordinate systems. Left-Hand vs Right-Hand. x z. Which is which? walters@buffalo.edu CSE 480/580 Lecture 2 Slide 3-D Transformations 3-D space Two conventions for coordinate sstems Left-Hand vs Right-Hand (Thumb is the ais, inde is the ais) Which is which? Most graphics

More information

LESSON 35: EIGENVALUES AND EIGENVECTORS APRIL 21, (1) We might also write v as v. Both notations refer to a vector.

LESSON 35: EIGENVALUES AND EIGENVECTORS APRIL 21, (1) We might also write v as v. Both notations refer to a vector. LESSON 5: EIGENVALUES AND EIGENVECTORS APRIL 2, 27 In this contet, a vector is a column matri E Note 2 v 2, v 4 5 6 () We might also write v as v Both notations refer to a vector (2) A vector can be man

More information

APPENDIX D Rotation and the General Second-Degree Equation

APPENDIX D Rotation and the General Second-Degree Equation APPENDIX D Rotation and the General Second-Degree Equation Rotation of Aes Invariants Under Rotation After rotation of the - and -aes counterclockwise through an angle, the rotated aes are denoted as the

More information

Identifying second degree equations

Identifying second degree equations Chapter 7 Identifing second degree equations 71 The eigenvalue method In this section we appl eigenvalue methods to determine the geometrical nature of the second degree equation a 2 + 2h + b 2 + 2g +

More information

Homework Notes Week 6

Homework Notes Week 6 Homework Notes Week 6 Math 24 Spring 24 34#4b The sstem + 2 3 3 + 4 = 2 + 2 + 3 4 = 2 + 2 3 = is consistent To see this we put the matri 3 2 A b = 2 into reduced row echelon form Adding times the first

More information

Affine transformations

Affine transformations Reading Optional reading: Affine transformations Brian Curless CSE 557 Autumn 207 Angel and Shreiner: 3., 3.7-3. Marschner and Shirle: 2.3, 2.4.-2.4.4, 6..-6..4, 6.2., 6.3 Further reading: Angel, the rest

More information

10.2 The Unit Circle: Cosine and Sine

10.2 The Unit Circle: Cosine and Sine 0. The Unit Circle: Cosine and Sine 77 0. The Unit Circle: Cosine and Sine In Section 0.., we introduced circular motion and derived a formula which describes the linear velocit of an object moving on

More information

Mathematics 309 Conic sections and their applicationsn. Chapter 2. Quadric figures. ai,j x i x j + b i x i + c =0. 1. Coordinate changes

Mathematics 309 Conic sections and their applicationsn. Chapter 2. Quadric figures. ai,j x i x j + b i x i + c =0. 1. Coordinate changes Mathematics 309 Conic sections and their applicationsn Chapter 2. Quadric figures In this chapter want to outline quickl how to decide what figure associated in 2D and 3D to quadratic equations look like.

More information

Affine transformations. Brian Curless CSE 557 Fall 2014

Affine transformations. Brian Curless CSE 557 Fall 2014 Affine transformations Brian Curless CSE 557 Fall 2014 1 Reading Required: Shirle, Sec. 2.4, 2.7 Shirle, Ch. 5.1-5.3 Shirle, Ch. 6 Further reading: Fole, et al, Chapter 5.1-5.5. David F. Rogers and J.

More information

KEY IDEAS. Chapter 1 Function Transformations. 1.1 Horizontal and Vertical Translations Pre-Calculus 12 Student Workbook MHR 1

KEY IDEAS. Chapter 1 Function Transformations. 1.1 Horizontal and Vertical Translations Pre-Calculus 12 Student Workbook MHR 1 Chapter Function Transformations. Horizontal and Vertical Translations A translation can move the graph of a function up or down (vertical translation) and right or left (horizontal translation). A translation

More information

UNCORRECTED SAMPLE PAGES. 3Quadratics. Chapter 3. Objectives

UNCORRECTED SAMPLE PAGES. 3Quadratics. Chapter 3. Objectives Chapter 3 3Quadratics Objectives To recognise and sketch the graphs of quadratic polnomials. To find the ke features of the graph of a quadratic polnomial: ais intercepts, turning point and ais of smmetr.

More information

we must pay attention to the role of the coordinate system w.r.t. which we perform a tform

we must pay attention to the role of the coordinate system w.r.t. which we perform a tform linear SO... we will want to represent the geometr of points in space we will often want to perform (rigid) transformations to these objects to position them translate rotate or move them in an animation

More information

Affine transformations

Affine transformations Reading Required: Affine transformations Brian Curless CSE 557 Fall 2009 Shirle, Sec. 2.4, 2.7 Shirle, Ch. 5.-5.3 Shirle, Ch. 6 Further reading: Fole, et al, Chapter 5.-5.5. David F. Rogers and J. Alan

More information

MAT 1275: Introduction to Mathematical Analysis. Graphs and Simplest Equations for Basic Trigonometric Functions. y=sin( x) Function

MAT 1275: Introduction to Mathematical Analysis. Graphs and Simplest Equations for Basic Trigonometric Functions. y=sin( x) Function MAT 275: Introduction to Mathematical Analsis Dr. A. Rozenblum Graphs and Simplest Equations for Basic Trigonometric Functions We consider here three basic functions: sine, cosine and tangent. For them,

More information

Analytic Geometry in Three Dimensions

Analytic Geometry in Three Dimensions Analtic Geometr in Three Dimensions. The Three-Dimensional Coordinate Sstem. Vectors in Space. The Cross Product of Two Vectors. Lines and Planes in Space The three-dimensional coordinate sstem is used

More information

15. Eigenvalues, Eigenvectors

15. Eigenvalues, Eigenvectors 5 Eigenvalues, Eigenvectors Matri of a Linear Transformation Consider a linear ( transformation ) L : a b R 2 R 2 Suppose we know that L and L Then c d because of linearit, we can determine what L does

More information

Systems of Linear Equations: Solving by Graphing

Systems of Linear Equations: Solving by Graphing 8.1 Sstems of Linear Equations: Solving b Graphing 8.1 OBJECTIVE 1. Find the solution(s) for a set of linear equations b graphing NOTE There is no other ordered pair that satisfies both equations. From

More information

we must pay attention to the role of the coordinate system w.r.t. which we perform a tform

we must pay attention to the role of the coordinate system w.r.t. which we perform a tform linear SO... we will want to represent the geometr of points in space we will often want to perform (rigid) transformations to these objects to position them translate rotate or move them in an animation

More information

9.2. Cartesian Components of Vectors. Introduction. Prerequisites. Learning Outcomes

9.2. Cartesian Components of Vectors. Introduction. Prerequisites. Learning Outcomes Cartesian Components of Vectors 9.2 Introduction It is useful to be able to describe vectors with reference to specific coordinate sstems, such as the Cartesian coordinate sstem. So, in this Section, we

More information

Trigonometry Outline

Trigonometry Outline Trigonometr Outline Introduction Knowledge of the content of this outline is essential to perform well in calculus. The reader is urged to stud each of the three parts of the outline. Part I contains the

More information

1 HOMOGENEOUS TRANSFORMATIONS

1 HOMOGENEOUS TRANSFORMATIONS HOMOGENEOUS TRANSFORMATIONS Purpose: The purpose of this chapter is to introduce ou to the Homogeneous Transformation. This simple 4 4 transformation is used in the geometr engines of CAD sstems and in

More information

11.4 Polar Coordinates

11.4 Polar Coordinates 11. Polar Coordinates 917 11. Polar Coordinates In Section 1.1, we introduced the Cartesian coordinates of a point in the plane as a means of assigning ordered pairs of numbers to points in the plane.

More information

Table of Contents. Module 1

Table of Contents. Module 1 Table of Contents Module 1 11 Order of Operations 16 Signed Numbers 1 Factorization of Integers 17 Further Signed Numbers 13 Fractions 18 Power Laws 14 Fractions and Decimals 19 Introduction to Algebra

More information

14.1 Systems of Linear Equations in Two Variables

14.1 Systems of Linear Equations in Two Variables 86 Chapter 1 Sstems of Equations and Matrices 1.1 Sstems of Linear Equations in Two Variables Use the method of substitution to solve sstems of equations in two variables. Use the method of elimination

More information

Properties of Limits

Properties of Limits 33460_003qd //04 :3 PM Page 59 SECTION 3 Evaluating Limits Analticall 59 Section 3 Evaluating Limits Analticall Evaluate a it using properties of its Develop and use a strateg for finding its Evaluate

More information

Polynomial and Rational Functions

Polynomial and Rational Functions Polnomial and Rational Functions Figure -mm film, once the standard for capturing photographic images, has been made largel obsolete b digital photograph. (credit film : modification of work b Horia Varlan;

More information

Strain Transformation and Rosette Gage Theory

Strain Transformation and Rosette Gage Theory Strain Transformation and Rosette Gage Theor It is often desired to measure the full state of strain on the surface of a part, that is to measure not onl the two etensional strains, and, but also the shear

More information

STUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE. Functions & Graphs

STUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE. Functions & Graphs STUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE Functions & Graphs Contents Functions and Relations... 1 Interval Notation... 3 Graphs: Linear Functions... 5 Lines and Gradients... 7 Graphs: Quadratic

More information

Math 323 Exam 2 - Practice Problem Solutions. 2. Given the vectors a = 1,2,0, b = 1,0,2, and c = 0,1,1, compute the following:

Math 323 Exam 2 - Practice Problem Solutions. 2. Given the vectors a = 1,2,0, b = 1,0,2, and c = 0,1,1, compute the following: Math 323 Eam 2 - Practice Problem Solutions 1. Given the vectors a = 2,, 1, b = 3, 2,4, and c = 1, 4,, compute the following: (a) A unit vector in the direction of c. u = c c = 1, 4, 1 4 =,, 1+16+ 17 17

More information

6. Linear transformations. Consider the function. f : R 2 R 2 which sends (x, y) (x, y)

6. Linear transformations. Consider the function. f : R 2 R 2 which sends (x, y) (x, y) Consider the function 6 Linear transformations f : R 2 R 2 which sends (x, ) (, x) This is an example of a linear transformation Before we get into the definition of a linear transformation, let s investigate

More information

2D Geometric Transformations. (Chapter 5 in FVD)

2D Geometric Transformations. (Chapter 5 in FVD) 2D Geometric Transformations (Chapter 5 in FVD) 2D geometric transformation Translation Scaling Rotation Shear Matri notation Compositions Homogeneous coordinates 2 2D Geometric Transformations Question:

More information

Eigenvectors and Eigenvalues 1

Eigenvectors and Eigenvalues 1 Ma 2015 page 1 Eigenvectors and Eigenvalues 1 In this handout, we will eplore eigenvectors and eigenvalues. We will begin with an eploration, then provide some direct eplanation and worked eamples, and

More information

All parabolas through three non-collinear points

All parabolas through three non-collinear points ALL PARABOLAS THROUGH THREE NON-COLLINEAR POINTS 03 All parabolas through three non-collinear points STANLEY R. HUDDY and MICHAEL A. JONES If no two of three non-collinear points share the same -coordinate,

More information

Solutions to the Exercises of Chapter 4

Solutions to the Exercises of Chapter 4 Solutions to the Eercises of Chapter 4 4A. Basic Analtic Geometr. The distance between (, ) and (4, 5) is ( 4) +( 5) = 9+6 = 5 and that from (, 6) to (, ) is ( ( )) +( 6 ( )) = ( + )=.. i. AB = (6 ) +(

More information

2: Distributions of Several Variables, Error Propagation

2: Distributions of Several Variables, Error Propagation : Distributions of Several Variables, Error Propagation Distribution of several variables. variables The joint probabilit distribution function of two variables and can be genericall written f(, with the

More information

Vectors in Two Dimensions

Vectors in Two Dimensions Vectors in Two Dimensions Introduction In engineering, phsics, and mathematics, vectors are a mathematical or graphical representation of a phsical quantit that has a magnitude as well as a direction.

More information

Reading. 4. Affine transformations. Required: Watt, Section 1.1. Further reading:

Reading. 4. Affine transformations. Required: Watt, Section 1.1. Further reading: Reading Required: Watt, Section.. Further reading: 4. Affine transformations Fole, et al, Chapter 5.-5.5. David F. Rogers and J. Alan Adams, Mathematical Elements for Computer Graphics, 2 nd Ed., McGraw-Hill,

More information

5.6. Differential equations

5.6. Differential equations 5.6. Differential equations The relationship between cause and effect in phsical phenomena can often be formulated using differential equations which describe how a phsical measure () and its derivative

More information

MAT389 Fall 2016, Problem Set 2

MAT389 Fall 2016, Problem Set 2 MAT389 Fall 2016, Problem Set 2 Circles in the Riemann sphere Recall that the Riemann sphere is defined as the set Let P be the plane defined b Σ = { (a, b, c) R 3 a 2 + b 2 + c 2 = 1 } P = { (a, b, c)

More information

Worksheet #1. A little review.

Worksheet #1. A little review. Worksheet #1. A little review. I. Set up BUT DO NOT EVALUATE definite integrals for each of the following. 1. The area between the curves = 1 and = 3. Solution. The first thing we should ask ourselves

More information

Matrices. VCE Maths Methods - Unit 2 - Matrices

Matrices. VCE Maths Methods - Unit 2 - Matrices Matrices Introduction to matrices Addition & subtraction Scalar multiplication Matri multiplication The unit matri Matri division - the inverse matri Using matrices - simultaneous equations Matri transformations

More information

Prepared by: M. S. KumarSwamy, TGT(Maths) Page

Prepared by: M. S. KumarSwamy, TGT(Maths) Page Prepared b: M. S. KumarSwam, TGT(Maths) Page - 77 - CHAPTER 4: DETERMINANTS QUICK REVISION (Important Concepts & Formulae) Determinant a b If A = c d, then determinant of A is written as A = a b = det

More information

Matrices. VCE Maths Methods - Unit 2 - Matrices

Matrices. VCE Maths Methods - Unit 2 - Matrices Matrices Introduction to matrices Addition subtraction Scalar multiplication Matri multiplication The unit matri Matri division - the inverse matri Using matrices - simultaneous equations Matri transformations

More information

Computer Graphics: 2D Transformations. Course Website:

Computer Graphics: 2D Transformations. Course Website: Computer Graphics: D Transformations Course Website: http://www.comp.dit.ie/bmacnamee 5 Contents Wh transformations Transformations Translation Scaling Rotation Homogeneous coordinates Matri multiplications

More information

2. Jan 2010 qu June 2009 qu.8

2. Jan 2010 qu June 2009 qu.8 C3 Functions. June 200 qu.9 The functions f and g are defined for all real values of b f() = 4 2 2 and g() = a + b, where a and b are non-zero constants. (i) Find the range of f. [3] Eplain wh the function

More information

Higher Mathematics (2014 on) Expressions and Functions. Practice Unit Assessment B

Higher Mathematics (2014 on) Expressions and Functions. Practice Unit Assessment B Pegass Educational Publishing Higher Mathematics (014 on) Epressions and Functions Practice Unit Assessment B otes: 1. Read the question full before answering it.. Alwas show our working.. Check our paper

More information

INSTITIÚID TEICNEOLAÍOCHTA CHEATHARLACH INSTITUTE OF TECHNOLOGY CARLOW MATRICES

INSTITIÚID TEICNEOLAÍOCHTA CHEATHARLACH INSTITUTE OF TECHNOLOGY CARLOW MATRICES 1 CHAPTER 4 MATRICES 1 INSTITIÚID TEICNEOLAÍOCHTA CHEATHARLACH INSTITUTE OF TECHNOLOGY CARLOW MATRICES 1 Matrices Matrices are of fundamental importance in 2-dimensional and 3-dimensional graphics programming

More information

ES.1803 Topic 16 Notes Jeremy Orloff

ES.1803 Topic 16 Notes Jeremy Orloff ES803 Topic 6 Notes Jerem Orloff 6 Eigenalues, diagonalization, decoupling This note coers topics that will take us seeral classes to get through We will look almost eclusiel at 2 2 matrices These hae

More information

3.7 InveRSe FUnCTIOnS

3.7 InveRSe FUnCTIOnS CHAPTER functions learning ObjeCTIveS In this section, ou will: Verif inverse functions. Determine the domain and range of an inverse function, and restrict the domain of a function to make it one-to-one.

More information

Graphics Example: Type Setting

Graphics Example: Type Setting D Transformations Graphics Eample: Tpe Setting Modern Computerized Tpesetting Each letter is defined in its own coordinate sstem And positioned on the page coordinate sstem It is ver simple, m she thought,

More information

CS 335 Graphics and Multimedia. 2D Graphics Primitives and Transformation

CS 335 Graphics and Multimedia. 2D Graphics Primitives and Transformation C 335 Graphics and Multimedia D Graphics Primitives and Transformation Basic Mathematical Concepts Review Coordinate Reference Frames D Cartesian Reference Frames (a) (b) creen Cartesian reference sstems

More information

SEPARABLE EQUATIONS 2.2

SEPARABLE EQUATIONS 2.2 46 CHAPTER FIRST-ORDER DIFFERENTIAL EQUATIONS 4. Chemical Reactions When certain kinds of chemicals are combined, the rate at which the new compound is formed is modeled b the autonomous differential equation

More information

APPENDIXES. B Coordinate Geometry and Lines C. D Trigonometry E F. G The Logarithm Defined as an Integral H Complex Numbers I

APPENDIXES. B Coordinate Geometry and Lines C. D Trigonometry E F. G The Logarithm Defined as an Integral H Complex Numbers I APPENDIXES A Numbers, Inequalities, and Absolute Values B Coordinate Geometr and Lines C Graphs of Second-Degree Equations D Trigonometr E F Sigma Notation Proofs of Theorems G The Logarithm Defined as

More information

The details of the derivation of the equations of conics are com-

The details of the derivation of the equations of conics are com- Part 6 Conic sections Introduction Consider the double cone shown in the diagram, joined at the verte. These cones are right circular cones in the sense that slicing the double cones with planes at right-angles

More information

Outline. MA 138 Calculus 2 with Life Science Applications Linear Maps (Section 9.3) Graphical Representation of (Column) Vectors. Addition of Vectors

Outline. MA 138 Calculus 2 with Life Science Applications Linear Maps (Section 9.3) Graphical Representation of (Column) Vectors. Addition of Vectors MA 8 Calculus with Life Science Applications Linear Maps (Section 9) Alberto Corso albertocorso@ukedu Department of Mathematics Uniersit of Kentuck Wednesda, March 8, 07 Outline We mostl focus on matrices,

More information

Section 3.1. ; X = (0, 1]. (i) f : R R R, f (x, y) = x y

Section 3.1. ; X = (0, 1]. (i) f : R R R, f (x, y) = x y Paul J. Bruillard MATH 0.970 Problem Set 6 An Introduction to Abstract Mathematics R. Bond and W. Keane Section 3.1: 3b,c,e,i, 4bd, 6, 9, 15, 16, 18c,e, 19a, 0, 1b Section 3.: 1f,i, e, 6, 1e,f,h, 13e,

More information

Solutions to O Level Add Math paper

Solutions to O Level Add Math paper Solutions to O Level Add Math paper 4. Bab food is heated in a microwave to a temperature of C. It subsequentl cools in such a wa that its temperature, T C, t minutes after removal from the microwave,

More information

2.2 SEPARABLE VARIABLES

2.2 SEPARABLE VARIABLES 44 CHAPTER FIRST-ORDER DIFFERENTIAL EQUATIONS 6 Consider the autonomous DE 6 Use our ideas from Problem 5 to find intervals on the -ais for which solution curves are concave up and intervals for which

More information

m x n matrix with m rows and n columns is called an array of m.n real numbers

m x n matrix with m rows and n columns is called an array of m.n real numbers LINEAR ALGEBRA Matrices Linear Algebra Definitions m n matri with m rows and n columns is called an arra of mn real numbers The entr a a an A = a a an = ( a ij ) am am amn a ij denotes the element in the

More information

Vocabulary. The Pythagorean Identity. Lesson 4-3. Pythagorean Identity Theorem. Mental Math

Vocabulary. The Pythagorean Identity. Lesson 4-3. Pythagorean Identity Theorem. Mental Math Lesson 4-3 Basic Basic Trigonometric Identities Identities Vocabular identit BIG IDEA If ou know cos, ou can easil fi nd cos( ), cos(90º - ), cos(180º - ), and cos(180º + ) without a calculator, and similarl

More information

LESSON #42 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART 2 COMMON CORE ALGEBRA II

LESSON #42 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART 2 COMMON CORE ALGEBRA II LESSON #4 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART COMMON CORE ALGEBRA II You will recall from unit 1 that in order to find the inverse of a function, ou must switch and and solve for. Also,

More information

9.1 VECTORS. A Geometric View of Vectors LEARNING OBJECTIVES. = a, b

9.1 VECTORS. A Geometric View of Vectors LEARNING OBJECTIVES. = a, b vectors and POLAR COORDINATES LEARNING OBJECTIVES In this section, ou will: View vectors geometricall. Find magnitude and direction. Perform vector addition and scalar multiplication. Find the component

More information

5. Zeros. We deduce that the graph crosses the x-axis at the points x = 0, 1, 2 and 4, and nowhere else. And that s exactly what we see in the graph.

5. Zeros. We deduce that the graph crosses the x-axis at the points x = 0, 1, 2 and 4, and nowhere else. And that s exactly what we see in the graph. . Zeros Eample 1. At the right we have drawn the graph of the polnomial = ( 1) ( 2) ( 4). Argue that the form of the algebraic formula allows ou to see right awa where the graph is above the -ais, where

More information

SET-I SECTION A SECTION B. General Instructions. Time : 3 hours Max. Marks : 100

SET-I SECTION A SECTION B. General Instructions. Time : 3 hours Max. Marks : 100 General Instructions. All questions are compulsor.. This question paper contains 9 questions.. Questions - in Section A are ver short answer tpe questions carring mark each.. Questions 5- in Section B

More information

8.4 Inverse Functions

8.4 Inverse Functions Section 8. Inverse Functions 803 8. Inverse Functions As we saw in the last section, in order to solve application problems involving eponential unctions, we will need to be able to solve eponential equations

More information

14.3. Volumes of Revolution. Introduction. Prerequisites. Learning Outcomes

14.3. Volumes of Revolution. Introduction. Prerequisites. Learning Outcomes Volumes of Revolution 14.3 Introduction In this Section we show how the concept of integration as the limit of a sum, introduced in Section 14.1, can be used to find volumes of solids formed when curves

More information

Physically Based Rendering ( ) Geometry and Transformations

Physically Based Rendering ( ) Geometry and Transformations Phsicall Based Rendering (6.657) Geometr and Transformations 3D Point Specifies a location Origin 3D Point Specifies a location Represented b three coordinates Infinitel small class Point3D { public: Coordinate

More information

6 = 1 2. The right endpoints of the subintervals are then 2 5, 3, 7 2, 4, 2 9, 5, while the left endpoints are 2, 5 2, 3, 7 2, 4, 9 2.

6 = 1 2. The right endpoints of the subintervals are then 2 5, 3, 7 2, 4, 2 9, 5, while the left endpoints are 2, 5 2, 3, 7 2, 4, 9 2. 5 THE ITEGRAL 5. Approimating and Computing Area Preliminar Questions. What are the right and left endpoints if [, 5] is divided into si subintervals? If the interval [, 5] is divided into si subintervals,

More information

4 Strain true strain engineering strain plane strain strain transformation formulae

4 Strain true strain engineering strain plane strain strain transformation formulae 4 Strain The concept of strain is introduced in this Chapter. The approimation to the true strain of the engineering strain is discussed. The practical case of two dimensional plane strain is discussed,

More information

We could express the left side as a sum of vectors and obtain the Vector Form of a Linear System: a 12 a x n. a m2

We could express the left side as a sum of vectors and obtain the Vector Form of a Linear System: a 12 a x n. a m2 Week 22 Equations, Matrices and Transformations Coefficient Matrix and Vector Forms of a Linear System Suppose we have a system of m linear equations in n unknowns a 11 x 1 + a 12 x 2 + + a 1n x n b 1

More information

Transformations. Chapter D Transformations Translation

Transformations. Chapter D Transformations Translation Chapter 4 Transformations Transformations between arbitrary vector spaces, especially linear transformations, are usually studied in a linear algebra class. Here, we focus our attention to transformation

More information

Algebra/Pre-calc Review

Algebra/Pre-calc Review Algebra/Pre-calc Review The following pages contain various algebra and pre-calculus topics that are used in the stud of calculus. These pages were designed so that students can refresh their knowledge

More information

The first change comes in how we associate operators with classical observables. In one dimension, we had. p p ˆ

The first change comes in how we associate operators with classical observables. In one dimension, we had. p p ˆ VI. Angular momentum Up to this point, we have been dealing primaril with one dimensional sstems. In practice, of course, most of the sstems we deal with live in three dimensions and 1D quantum mechanics

More information

Which of the following expressions are monomials?

Which of the following expressions are monomials? 9 1 Stud Guide Pages 382 387 Polnomials The epressions, 6, 5a 2, and 10cd 3 are eamples of monomials. A monomial is a number, a variable, or a product of numbers and variables. An eponents in a monomial

More information

SECTION 8-7 De Moivre s Theorem. De Moivre s Theorem, n a Natural Number nth-roots of z

SECTION 8-7 De Moivre s Theorem. De Moivre s Theorem, n a Natural Number nth-roots of z 8-7 De Moivre s Theorem 635 B eactl; compute the modulus and argument for part C to two decimal places. 9. (A) 3 i (B) 1 i (C) 5 6i 10. (A) 1 i 3 (B) 3i (C) 7 4i 11. (A) i 3 (B) 3 i (C) 8 5i 12. (A) 3

More information

1.1 The Equations of Motion

1.1 The Equations of Motion 1.1 The Equations of Motion In Book I, balance of forces and moments acting on an component was enforced in order to ensure that the component was in equilibrium. Here, allowance is made for stresses which

More information

CS-184: Computer Graphics. Today

CS-184: Computer Graphics. Today CS-184: Computer Graphics Lecture #3: 2D Transformations Prof. James O Brien Universit of California, Berkele V2006-S-03-1.0 Toda 2D Transformations Primitive Operations Scale, Rotate, Shear, Flip, Translate

More information

MI-2 Prob. Set #4 DUE: Tuesday, Feb 15, 2011 Spring 11. 4) Write a matrix equation and give result:

MI-2 Prob. Set #4 DUE: Tuesday, Feb 15, 2011 Spring 11. 4) Write a matrix equation and give result: MI- Prob. Set #4 DUE: Tuesda, Feb 1, 011 Spring 11 1) Dr. Condie has sport coats, pairs of slacks, 4 shirts and 6 ties that he can choose to wear on an da. a) How man combinations of these four pieces

More information

RELATIONS AND FUNCTIONS through

RELATIONS AND FUNCTIONS through RELATIONS AND FUNCTIONS 11.1.2 through 11.1. Relations and Functions establish a correspondence between the input values (usuall ) and the output values (usuall ) according to the particular relation or

More information

Complex Numbers and Exponentials

Complex Numbers and Exponentials omple Numbers and Eponentials Definition and Basic Operations comple number is nothing more than a point in the plane. The sum and product of two comple numbers ( 1, 1 ) and ( 2, 2 ) is defined b ( 1,

More information

Introduction to 3D Game Programming with DirectX 9.0c: A Shader Approach

Introduction to 3D Game Programming with DirectX 9.0c: A Shader Approach Introduction to 3D Game Programming with DirectX 90c: A Shader Approach Part I Solutions Note : Please email to frank@moon-labscom if ou find an errors Note : Use onl after ou have tried, and struggled

More information

Trigonometric Functions

Trigonometric Functions Trigonometric Functions This section reviews radian measure and the basic trigonometric functions. C ' θ r s ' ngles ngles are measured in degrees or radians. The number of radians in the central angle

More information

UNCORRECTED. To recognise the rules of a number of common algebraic relations: y = x 1 y 2 = x

UNCORRECTED. To recognise the rules of a number of common algebraic relations: y = x 1 y 2 = x 5A galler of graphs Objectives To recognise the rules of a number of common algebraic relations: = = = (rectangular hperbola) + = (circle). To be able to sketch the graphs of these relations. To be able

More information

Section 4.1 Increasing and Decreasing Functions

Section 4.1 Increasing and Decreasing Functions Section.1 Increasing and Decreasing Functions The graph of the quadratic function f 1 is a parabola. If we imagine a particle moving along this parabola from left to right, we can see that, while the -coordinates

More information

Get Solution of These Packages & Learn by Video Tutorials on Matrices

Get Solution of These Packages & Learn by Video Tutorials on  Matrices FEE Download Stud Package from website: wwwtekoclassescom & wwwmathsbsuhagcom Get Solution of These Packages & Learn b Video Tutorials on wwwmathsbsuhagcom Matrices An rectangular arrangement of numbers

More information

Solutionbank C2 Edexcel Modular Mathematics for AS and A-Level

Solutionbank C2 Edexcel Modular Mathematics for AS and A-Level file://c:\users\buba\kaz\ouba\c_rev_a_.html Eercise A, Question Epand and simplify ( ) 5. ( ) 5 = + 5 ( ) + 0 ( ) + 0 ( ) + 5 ( ) + ( ) 5 = 5 + 0 0 + 5 5 Compare ( + ) n with ( ) n. Replace n by 5 and

More information

MA123, Chapter 1: Equations, functions and graphs (pp. 1-15)

MA123, Chapter 1: Equations, functions and graphs (pp. 1-15) MA123, Chapter 1: Equations, functions and graphs (pp. 1-15) Date: Chapter Goals: Identif solutions to an equation. Solve an equation for one variable in terms of another. What is a function? Understand

More information

8.1 Exponents and Roots

8.1 Exponents and Roots Section 8. Eponents and Roots 75 8. Eponents and Roots Before defining the net famil of functions, the eponential functions, we will need to discuss eponent notation in detail. As we shall see, eponents

More information

Algebra 1 Skills Needed for Success in Math

Algebra 1 Skills Needed for Success in Math Algebra 1 Skills Needed for Success in Math A. Simplifing Polnomial Epressions Objectives: The student will be able to: Appl the appropriate arithmetic operations and algebraic properties needed to simplif

More information

National Quali cations

National Quali cations National Quali cations AH08 X747/77/ Mathematics THURSDAY, MAY 9:00 AM :00 NOON Total marks 00 Attempt ALL questions. You may use a calculator. Full credit will be given only to solutions which contain

More information

Lecture 27: More on Rotational Kinematics

Lecture 27: More on Rotational Kinematics Lecture 27: More on Rotational Kinematics Let s work out the kinematics of rotational motion if α is constant: dω α = 1 2 α dω αt = ω ω ω = αt + ω ( t ) dφ α + ω = dφ t 2 α + ωo = φ φo = 1 2 = t o 2 φ

More information

What is Parametric Equation?

What is Parametric Equation? Chapter 13 Parametric Equation and Locus Wh the graph is so strange? Let s investigate a few points. t -5-4 -3 - -1 0 1 3 4 4 15 8 3 0-1 0 3 8 15-4.04-3.4-3.14 -.91-1.84 0 1.84.91 3.14 3.4 B plotting these

More information

On Range and Reflecting Functions About the Line y = mx

On Range and Reflecting Functions About the Line y = mx On Range and Reflecting Functions About the Line = m Scott J. Beslin Brian K. Heck Jerem J. Becnel Dept.of Mathematics and Dept. of Mathematics and Dept. of Mathematics and Computer Science Computer Science

More information

1.1 Laws of exponents Conversion between exponents and logarithms Logarithm laws Exponential and logarithmic equations 10

1.1 Laws of exponents Conversion between exponents and logarithms Logarithm laws Exponential and logarithmic equations 10 CNTENTS Algebra Chapter Chapter Chapter Eponents and logarithms. Laws of eponents. Conversion between eponents and logarithms 6. Logarithm laws 8. Eponential and logarithmic equations 0 Sequences and series.

More information

MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS PRELIMINARIES TO FEM

MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS PRELIMINARIES TO FEM B Course Content: A INTRODUCTION AND OVERVIEW Numerical method and Computer-Aided Engineering; Phsical problems; Mathematical models; Finite element method;. B Elements and nodes, natural coordinates,

More information

A Tutorial on Euler Angles and Quaternions

A Tutorial on Euler Angles and Quaternions A Tutorial on Euler Angles and Quaternions Moti Ben-Ari Department of Science Teaching Weimann Institute of Science http://www.weimann.ac.il/sci-tea/benari/ Version.0.1 c 01 17 b Moti Ben-Ari. This work

More information

Math 214 Spring problem set (a) Consider these two first order equations. (I) dy dx = x + 1 dy

Math 214 Spring problem set (a) Consider these two first order equations. (I) dy dx = x + 1 dy Math 4 Spring 08 problem set. (a) Consider these two first order equations. (I) d d = + d (II) d = Below are four direction fields. Match the differential equations above to their direction fields. Provide

More information

Cubic and quartic functions

Cubic and quartic functions 3 Cubic and quartic functions 3A Epanding 3B Long division of polnomials 3C Polnomial values 3D The remainder and factor theorems 3E Factorising polnomials 3F Sum and difference of two cubes 3G Solving

More information