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3 Pages

### tcu11_appa3

Course: MATH 1240, Spring 2010
School: Westminster UT
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Word Count: 511

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AM Page 4100AWL/Thomas_APPp001-034 8/27/04 6:49 7 A.3 Commonly Occurring Limits AP-7 A.3 Commonly Occurring Limits This appendix verifies limits (4)(6) in Theorem 5 of Section 11.1. Limit 4: If x 6 1, lim x n = 0 We need to show that to each P 7 0 there corresn: ponds an integer N so large that x n 6 P for all n greater than N. Since P1&gt;n : 1, while Copyright 2005 Pearson Education, Inc.,...

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AM Page 4100AWL/Thomas_APPp001-034 8/27/04 6:49 7 A.3 Commonly Occurring Limits AP-7 A.3 Commonly Occurring Limits This appendix verifies limits (4)(6) in Theorem 5 of Section 11.1. Limit 4: If x 6 1, lim x n = 0 We need to show that to each P 7 0 there corresn: ponds an integer N so large that x n 6 P for all n greater than N. Since P1>n : 1, while Copyright 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley 4100AWL/Thomas_APPp001-034 8/27/04 6:49 AM Page 8 AP-8 Appendices x 6 1, there exists an integer N for which P1>N 7 x . In other words, xN = x N 6 P. This is the integer we seek because, if x 6 1, then x n 6 x N for all n 7 N . Combining (1) and (2) produces x n 6 P for all n 7 N , concluding the proof. xn Limit 5: For any number x, lim A 1 + n B = e x n: (2) (1) Let n x an = a1 + n b . Then x x ln an = ln a1 + n b = n ln a1 + n b : x , as we can see by the following application of lHpitals Rule, in which we differentiate with respect to n: lns 1 + x> n d x lim n ln a1 + n b = lim 1> n n: q n: q a n = lim n: q Apply Theorem 4, Section 11.1, with s x d = e x to conclude that x a1 + b n = an = e ln an : e x . xn = 0 Since n! x n x n xn , n! n! n! n x 1 b # a- 2 b 1 + x> n n x = lim = x. n : q 1 + x> n - 1> n2 Limit 6: For any number x, lim n: all we need to show is that x n> n! : 0. We can then apply the Sandwich Theorem for Sequences (Section 11.1, Theorem 2) to conclude that x n> n! : 0. The first step in showing that x n> n! : 0 is to choose an integer M 7 x , so that s x > M d 6 1. By Limit 4, just proved, we then have s x > M dn : 0. We then restrict our attention to values of n 7 M . For these values of n, we can write x n x n = # ## # n! 12 M s M + 1 ds M + 2 d # # n ('''''')''''''* s n - M d factors n x n x nM M MM x = = a b. M! M M!M n M!M n - M Copyright 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley 4100AWL/Thomas_APPp001-034 8/27/04 6:50 AM Page 9 A.3 Commonly Occurring Limits AP-9 Thus, 0 n x n MM x a b. n! M! M Now, the constant M M> M! does not change as n increases. Thus the Sandwich Theorem tells us that x n> n! : 0 because s x > M dn : 0. Copyright 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley
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Westminster UT - MATH - 1240
4100AWL/Thomas_APPp001-0348/27/046:50 AMPage 9A.4 Theory of the Real NumbersAP-9A.4Theory of the Real NumbersA rigorous development of calculus is based on properties of the real numbers. Many results about functions, derivatives, and integrals wo
Westminster UT - MATH - 1240
4100AWL/Thomas_APPp001-0348/27/046:50 AMPage 12AP-12AppendicesA.5Complex NumbersComplex numbers are expressions of the form a + ib , where a and b are real numbers and i is a symbol for 1 - 1 . Unfortunately, the words real and imaginary have conn
Westminster UT - MATH - 1240
4100AWL/Thomas_APPp001-0348/27/046:50 AMPage 22AP-22AppendicesA.6The Distributive Law for Vector Cross ProductsIn this appendix, we prove the Distributive Law u * sv + wd = u * v + u * w which is Property 2 in Section 12.4. Proof To derive the Dis
Westminster UT - MATH - 1240
4100AWL/Thomas_APPp001-0348/27/046:50 AMPage 23A.7 The Mixed Derivative Theorem and the Increment TheoremAP-23A.7The Mixed Derivative Theorem and the Increment TheoremThis appendix derives the Mixed Derivative Theorem (Theorem 2, Section 14.3) and
Westminster UT - MATH - 1240
4100AWL/Thomas_APPp001-0348/27/046:50 AMPage 28AP-28AppendicesA.8P u QThe Area of a Parallelograms Projection on a PlaneThis appendix proves the result needed in Section 16.5 that s u * v d # p is the area of the projection of the parallelogram w
Westminster UT - MATH - 1240
4100AWL/Thomas_APPp001-0348/27/046:50 AMPage 29A.9 Basic Algebra, Geometry, and Trigonometry FormulasAP-29A.9Basic Algebra, Geometry, and Trigonometry FormulasAlgebraArithmetic Operations as b + c d = ab + ac, a c ad + bc += , b d bd Laws of Sign
Westminster UT - CS - 390
/ / / /import import import import import import importjavax.swing.*; java.awt.*; javax.imageio.ImageIO; java.awt.image.*; java.awt.event.*; java.io.*; raytracer.world.World;public class RayTracerGUI extends JPanel implements Runnable cfw_ private priv
Westminster UT - CS - 390
package raytracer.cameras; import raytracer.utility.Point2D; import raytracer.utility.Point3D; import raytracer.utility.RGBColor; import raytracer.utility.Ray; import raytracer.utility.Vector3D; import raytracer.world.ViewPlane; import raytracer.world.Wor
Westminster UT - CS - 390
package raytracer.cameras; import raytracer.utility.*; import raytracer.world.*; public class Pinhole extends Camera cfw_ private float private float d; / view plane distance zoom; / zoom factor/ -default constructor public Pinhole()cfw_ super(); d = 500
Westminster UT - CS - 390
package raytracer.geometricObjects; import raytracer.utility.*; import import import import import import java.util.Vector; java.io.BufferedReader; java.io.FileNotFoundException; java.io.FileReader; java.util.Scanner; java.util.Random;public class Compou
Westminster UT - CS - 390
package raytracer.geometricObjects; import raytracer.utility.*; public class FlatMeshTriangle extends MeshTriangle cfw_ / -constructor public FlatMeshTriangle() cfw_ super(); / -constructor public FlatMeshTriangle (Mesh mesh_ptr, int i0, int i1, int i2)
Westminster UT - CS - 390
package raytracer.geometricObjects; import raytracer.utility.RGBColor; import raytracer.utility.Ray; import raytracer.utility.ShadeRec; /* this file contains the definition of the class GeometricObject * */ public abstract class GeometricObject cfw_ prote
Westminster UT - CS - 390
package raytracer.geometricObjects; import raytracer.utility.*; / / / / Copyright (C) Kevin Suffern 2000-2007. This C+ code is for non-commercial purposes only. This C+ code is licensed under the GNU General Public License Version 2. See the file COPYING.
Westminster UT - CS - 390
package raytracer.geometricObjects; import raytracer.utility.*; public class Plane extends GeometricObject cfw_ private Point3D a; private Normal n; rays / point through which plane passes / normal to the plane / for shadows and secondaryprivate static f
Westminster UT - CS - 390
package raytracer.geometricObjects; import raytracer.utility.*; public class Sphere extends GeometricObject cfw_ private Point3D center; / center coordinates as a point private double radius; / the radius private static final double kEpsilon = 0.001; / fo
Westminster UT - CS - 390
package raytracer.geometricObjects; import raytracer.utility.*; /this is the triangle discussed in Section 19.3 public class Triangle extends GeometricObject cfw_ private Point3D v0, v1, v2; private Normal normal; / -constructor public Triangle()cfw_ supe
Westminster UT - CS - 390
package raytracer.tracers; import raytracer.utility.*; import raytracer.world.World; public class MultipleObjects extends Tracer cfw_ / -default constructor public MultipleObjects() cfw_ super(); / -constructor public MultipleObjects(World world) cfw_ su
Westminster UT - CS - 390
package raytracer.tracers; import raytracer.utility.*; import raytracer.world.World; public class SingleSphere extends Tracer cfw_ / -default constructor public SingleSphere() cfw_ super(); / -constructor public SingleSphere(World world)cfw_ super(world)
Westminster UT - CS - 390
package raytracer.tracers; import raytracer.utility.Ray; import raytracer.utility.RGBColor; import raytracer.world.World; /This is the declaration of the base class Tracer /The tracer classes have no copy constructor, assignment operator. or clone functio
Westminster UT - CS - 390
package raytracer.utility; public class public public public public Constants cfw_ static final static final static final static final double double double double TWO_PI PI_ON_180 invPI invTWO_PI = = = = Math.PI*2.; Math.PI/180.; 1./Math.PI; 0.5*Math.PI;
Westminster UT - CS - 390
package raytracer.utility; /* * this file contains the declaration of the class Matrix * Matrix is a 4 x 4 square matrix that is used to represent affine transformations * we don't need a general m x n matrix * */ public class Matrix cfw_ public double[][
Westminster UT - CS - 390
package raytracer.utility; import java.util.Vector; import raytracer.geometricObjects.MeshTriangle; /* * / / / /Copyright (C) Kevin Suffern 2000-2007. This C+ code is for non-commercial purposes only. This C+ code is licensed under the GNU General Public
Westminster UT - CS - 390
package raytracer.utility; public class Normal cfw_ public double x, y, z;/ - default constructor public Normal() cfw_ x = y = z = 0; / - constructor public Normal(double a) cfw_ x = y = z = a; / - constructor public Normal(double a, double b, double c
Westminster UT - CS - 390
package raytracer.utility; /2D points are used to store sample points public class Point2D cfw_ public float x, y; / - default constructor public Point2D() cfw_ x = 0.f; y = 0.f; / - constructor public Point2D(float arg)cfw_ x = arg; y = arg; / - constr
Westminster UT - CS - 390
package raytracer.utility; public class Point3D cfw_ public double x, y, z; / - default constructor public Point3D()cfw_ x = y = z = 0; / - constructor public Point3D(double a)cfw_ x = y = z = a; / - constructor public Point3D(double a, double b, double
Westminster UT - CS - 390
package raytracer.utility; public class Ray cfw_ public Point3D public Vector3D o; d; / origin / direction/ - default constructor public Ray () cfw_ o = new Point3D(0.0); d = new Vector3D(0.0, 0.0, 1.0); / -constructor public Ray (Point3D origin, Vector
Westminster UT - CS - 390
package raytracer.utility; public class public public public public RGBColor cfw_ float r, g, b; static final RGBColor black = new RGBColor(0f); static final RGBColor white = new RGBColor(1f); static final RGBColor red = new RGBColor(1.0f, 0.0f, 0.0f);/
Westminster UT - CS - 390
package raytracer.utility; import raytracer.world.World; /There is no default constructor as the World reference has to be initialised /There is also no assignment operator as we don't want to assign the world anywhere /The copy constructor only copies th
Westminster UT - CS - 390
package raytracer.utility; public class Vector3D cfw_ public double x, y, z; / - default constructor public Vector3D() cfw_ x = y = z = 0.0; / - constructor public Vector3D(double a) cfw_ x = y = z = a; / - constructor public Vector3D(double a, double b
Westminster UT - CS - 390
package raytracer.world; /import raytracer.utility.Normal; public class ViewPlane cfw_ public int image resolution public int resolution public float size public int samples per pixel public float correction factor public float the gamma correction factor
Westminster UT - CS - 390
package raytracer.world; import raytracer.utility.*; import raytracer.geometricObjects.*; import raytracer.tracers.*; import raytracer.cameras.Camera; import raytracer.cameras.Pinhole; import java.util.Vector; import java.awt.*; /This file contains the de
Westminster UT - CS - 390
/ This file contains the definition the ViewPlane class # #include &quot;ViewPlane.h&quot; / - default constructor V ViewPlane:ViewPlane(void) : hres(400), vres(400), s(1.0), gamma(1.0), inv_gamma(1.0), show_out_of_gamut(false) cfw_ cfw_ / - copy constructor ViewPl
Westminster UT - CS - 390
#ifndef _VIEW_PLANE_ # #define _VIEW_PLANE_ /- class ViewPlane class ViewPlane cfw_ public: int image resolution int resolution float size float correction factor float the gamma correction factor bool RGBColor out of gamuthres; vres; s; gamma; inv_gamma
Westminster UT - CS - 390
/ this file contains the definition of the World class # #include &quot;wxraytracer.h&quot; #include &quot;World.h&quot; # #include &quot;Constants.h&quot; / / geometric objects #include &quot;Plane.h&quot; # #include &quot;Sphere.h&quot; / / tracers #include &quot;SingleSphere.h&quot; # #include &quot;MultipleObjects.
Westminster UT - CS - 390
#ifndef _WORLD_ # #define _WORLD_ / This file contains the declaration of the class World / The World class does not have a copy constructor or an assignment operator, for the followign reasons: / 1 There's no need to copy construct or assign the World /
Westminster UT - CS - 390
#ifndef _CONSTANTS_ # #define _CONSTANTS_ #include &lt;stdlib.h&gt; # #include &quot;RGBColor.h&quot; const const const const const double double double double double PI TWO_PI PI_ON_180 invPI invTWO_PI kEpsilon kHugeValue = 3.1415926535897932384; = 6.2831853071795864769
Westminster UT - CS - 390
#ifndef _MATHS_ # #define _MATHS_ inline double m max(double x0, double x1); inline double max(double x0, double x1) cfw_ return(x0 &gt; x1) ? x0 : x1); #endif
Westminster UT - CS - 390
/ This file contains the definition of the class Matrix # #include &quot;Matrix.h&quot; / - default constructor / / a default matrix is an identity matrix M Matrix:Matrix(void) cfw_ for (int x = 0; x &lt; 4; x+) for (int y = 0; y &lt; 4; y+) cfw_ if (x = y) m[x][y] = 1.0
Westminster UT - CS - 390
#ifndef _MATRIX_ # #define _MATRIX_ / this file contains the declaration of the class Matrix / Matrix is a 4 x 4 square matrix that is used to represent affine transformations / / we don't need a general m x n matrix / /- class Matrix c class Matrix cfw_
Westminster UT - CS - 390
/ This file contains the defintion of the class Normal # #include &lt;math.h&gt; # #include &quot;Normal.h&quot; / / - default constructor Normal:Normal(void) : x(0.0), y(0.0), z(0.0) cfw_ / / - constructor Normal:Normal(double a) : x(a), y(a), z(a) cfw_ / / - constructo
Westminster UT - CS - 390
#ifndef _NORMAL_ # #define _NORMAL_ / / This file contains the declaration of the class Normal #include &quot;Matrix.h&quot; #include &quot;Vector3D.h&quot; # #include &quot;Point3D.h&quot; class Normal cfw_ p public: double p public: Normal(void); / default constructor Normal(double
Westminster UT - CS - 390
/ this file contains the definition of the class Point3D #include &lt;math.h&gt; # #include &quot;Point3D.h&quot; / / - default constructor Point3D:Point3D() :x(0), y(0), z(0) cfw_ cfw_ / / - constructor Point3D:Point3D(const double a) :x(a), y(a), z(a) cfw_ cfw_ / / - c
Westminster UT - CS - 390
#ifndef _POINT3D_ # #define _POINT3D_ / / This file contains the defintion of the class Point3D #include &quot;Matrix.h&quot; # #include &quot;Vector3D.h&quot; class Point3D cfw_ p public: d double x, y, z; Point3D(); / default constructor Point3D(const double a); / construc
Westminster UT - CS - 390
#include &quot;Ray.h&quot; / - default constructor Ray:Ray (void) : o(0.0), d(0.0, 0.0, 1.0) cfw_ / / - constructor Ray:Ray (const Point3D&amp; origin, const Vector3D&amp; dir) : o(origin), d(dir) cfw_ / - copy constructor Ray:Ray (const Ray&amp; ray) : o(ray.o), d(ray.d) cfw_
Westminster UT - CS - 390
#ifndef _RAY_ # #define _RAY_ #include &quot;Point3D.h&quot; # #include &quot;Vector3D.h&quot; class Ray cfw_ p public: Point3D Vector3D R Ray(void); R Ray(const Point3D&amp; origin, const Vector3D&amp; dir); R Ray(const Ray&amp; ray); R Ray&amp; operator= (const Ray&amp; rhs); ~Ray(void); ; #
Westminster UT - CS - 390
/ This file contains the definition of the class RGBColor # #include &lt;math.h&gt; # #include &quot;RGBColor.h&quot; / / - default constructor RGBColor:RGBColor(void) : r(0.0), g(0.0), b(0.0) cfw_ cfw_ / / - constructor RGBColor:RGBColor(float c) : r(c), g(c), b(c) cfw_
Westminster UT - CS - 390
#ifndef _RGB_COLOR_ # #define _RGB_COLOR_ / / This file contains the declaration of the class RGBColor / /- class RGBColor c class RGBColor cfw_ p public: float r, g, b; r p public: RGBColor(void); / default constructor RGBColor(float c); / constructor RG
Westminster UT - CS - 390
/ this file contains the definition of the class ShadeRec / there is no default constructor as the World reference always has to be initialised / there is also no assignment operator as we don't want to assign the world / the copy constructor only copies
Westminster UT - CS - 390
#ifndef _SHADE_REC_ # #define _SHADE_REC_ / / this file contains the declaration of the class ShadeRec class World; is a reference / only need a forward class declaration as the World data member#include &quot;Point3D.h&quot; #include &quot;Normal.h&quot; # #include &quot;RGBCol
Westminster UT - CS - 390
/ This file contains the definition of the class Vector3D # #include &lt;math.h&gt; #include &quot;Vector3D.h&quot; #include &quot;Normal.h&quot; # #include &quot;Point3D.h&quot; / / - default constructor Vector3D:Vector3D(void) : x(0.0), y(0.0), z(0.0) cfw_ / / - constructor Vector3D:Vecto
Westminster UT - CS - 390
#ifndef _VECTOR_3D_ # #define _VECTOR_3D_ / / This file contains the defintion of the class Vector3D # #include &quot;Matrix.h&quot; class Normal; c class Point3D; / /- class Vector3D class Vector3D cfw_ p public: double p public: Vector3D(void); / default construc
Westminster UT - CS - 390
Westminster UT - CS - 390
Westminster UT - CS - 390
#include #include #include #include # #include&lt;wx/wx.h&gt; &lt;wx/dcbuffer.h&gt; &quot;wxraytracer.h&quot; &quot;main.xpm&quot; &quot;bg.xpm&quot;/*/ /* wxraytracerapp */ / /*/ BEGIN_EVENT_TABLE(wxraytracerapp, wxApp) E END_EVENT_TABLE() I IMPLEMENT_APP(wxraytracerapp) bool wxraytracerapp:On
Westminster UT - CS - 390
#ifndef _WXRAYTRACER_H_ # #define _WXRAYTRACER_H_ /* * Ray Tracer skeleton * * Author : Sverre Kvaale &lt;sverre@kvaale.com&gt; * Version: 0.8 * */ #include &lt;wx/wx.h&gt; #include &quot;World.h&quot; # #include &lt;vector&gt; u using namespace std; class class class c class wxrayt
Westminster UT - CS - 390
#include &quot;MultipleObjects.h&quot; # #include &quot;World.h&quot; / - default constructor MultipleObjects:MultipleObjects(void) : Tracer() cfw_ cfw_ / / - constructor MultipleObjects:MultipleObjects(World* _worldPtr) : Tracer(_worldPtr) cfw_ cfw_ / / - destructor M Multi
Westminster UT - CS - 390
#ifndef _MULTIPLE_OBJECTS_ # #define _MULTIPLE_OBJECTS_ # #include &quot;Tracer.h&quot; class MultipleObjects: public Tracer cfw_ p public: M MultipleObjects(void); M MultipleObjects(World* _world_ptr); v virtual ~ ~MultipleObjects(void); v virtual RGBColor trace_r
Westminster UT - CS - 390
#include &quot;SingleSphere.h&quot; #include &quot;World.h&quot; # #include &quot;ShadeRec.h&quot; / - default constructor SingleSphere:SingleSphere(void) : Tracer() cfw_ cfw_ / / - constructor SingleSphere:SingleSphere(World* _worldPtr) : Tracer(_worldPtr) cfw_ cfw_ / / - destructor
Westminster UT - CS - 390
#ifndef _SINGLE_SPHERE_ # #define _SINGLE_SPHERE_ # #include &quot;Tracer.h&quot; class SingleSphere: public Tracer cfw_ p public: S SingleSphere(void); S SingleSphere(World* _worldPtr); v virtual ~ ~SingleSphere(void); v virtual RGBColor trace_ray(const Ray&amp; ray)
Westminster UT - CS - 390
#include &quot;Tracer.h&quot; / - default constructor Tracer:Tracer(void) : world_ptr(NULL) cfw_ cfw_ / / - constructor Tracer:Tracer(World* _worldPtr) : world_ptr(_worldPtr) cfw_ cfw_/ / - destructor Tracer:~Tracer(void) cfw_ if (world_ptr) world_ptr = NULL; / /
Westminster UT - CS - 390
/ This is the declaration of the base class Tracer / The tracer classes have no copy constructor, assignment operator. or clone function because / of the world pointer, which should not be assigned or copy constructed / / See comments in the World.h file.