A Maple Package for Solving and Displaying
|
|
- Sabina Stewart
- 6 years ago
- Views:
Transcription
1 Classroom Tips and Techniques: A Package for Graphing Solution Sets of Nonlinear Inequalities Introduction Robert J. Lopez Emeritus Professor of Mathematics and Maple Fellow Maplesoft, a division of Waterloo Maple Inc., 2006 Maple's inequal command in the plots package will graph the feasible region for a set of linear inequalities. Even though Maple's solve command can solve some nonlinear inequalities exactly, Maple does not have specific tools for graphing solution sets of nonlinear inequalities. As we show in Example 6, the contourplot command (or even the plot3d command) can be used to graph the feasible region for an inequality of the form Considerably more effort must be exerted to graph the points satisfying several such inequalities. And although Maple can shade the region between a curve and the horizontal axis, it cannot easily shade the region between two arbitrary curves. In October 2005 the Maple Applications Center ( acquired A Maple Package for Solving and Displaying
2 Inequalities Robert Ipanaqué Chero National University of Piura an application that addresses several of these Maple shortcomings. You may view the original application here: By example, some taken from the author's work itself, this month's column will illustrate the usercontributed InequalityGraphics package and its applicability to classroom tasks. This package provides two commands, namely, inequalityplot and complexinequalityplot. The inequalityplot command will graph the two-dimensional feasible region for a set of nonlinear inequalities in two variables, that is, inequalities of the form Consequently, by plotting the solution set of the inequalities and the region between arbitrary plane curves can be shaded. The complexinequalityplot command will plot the solution set of one or more inequalities of the form where is a real-valued function and the functions are complex-valued functions of the complex variable In essence, however, the inequality is nothing more than because of the definition of. excludedcolor = colorname feasiblecolor = colorname linespoints = positive integer feasiblepoints = positive integer thickness = positive integer color = colorname Color of the non-feasible region Color of the feasible region Number of points plotted along curves separati Number of points computed inside feasible reg Thickness of curves separating feasible and non Color of curves separating feasible and non-fea Table 1 Options for the inequalityplot and complexinequalityplot commands in the InequalityGraphics pack To make it easier for perusing the functionality of the package, a working version of the the code for the InequalityGraphics package is embedded in Initializations section, below. Too often the process of installing a third-party package is a hurdle that prevents a user from trying out the new commands. The downloaded code for the package ( contains the three files applications. hdb, applications.ind, and applications.lib. Dropping these files into the LIB folder of a Maple 10 installation will give permanent access to the package. The name "applications" can be changed to
3 something more meaningful, like InequalityGraphics, with no ill effects. The ".ind" and ".lib" files provide the mathematical functionality of the package, whereas the ".hdb" file provides help pages without which the command of the package will still function. The one help page provided by the hdb file can be accessed with the command?inequals; We leave these matters to the reader whose interest in the package is sufficiently sparked by this article. The code available from the Application Center colors the non-feasible region white, by default, whereas the code embedded in this article colors it black. Hence, in this article we have explicitly used the option excludedcolor = white. Initializations restart; with(plots): interface(warnlevel=0): InequalityGraphics := module() export inequalityplot, complexinequalityplot; option package; inequalityplot:=proc() local ineq,ineqset,xx,yy,phaseselection,phase1,phase2, phases,var1, var2,xmin,xmax,ymin,ymax,dx,dy,jumpprecision, findjump,reg,c,p,x1,y1,p1list,p2list,aux1,aux2,jump1, jump2,eq,cc,ops,fc,ec,ls,fb,fp,lp,llss; if nargs<3 then ERROR(`expecting 3 arguments, got 0. This commmand also incorporates the options: (1) exludedcolor=yellow, (2) feasiblecolor=red, (3) linespoints=350, (4) feasiblepoints=15 and (5) thickness=1. Examples: <1> inequalityplot(x^2+y^2<1,x=-1..1,y=-1..1), <2> inequalityplot(x^2+y^2<1,x=-1..1,y=-1..1, excludedcolor=white,feasiblecolor=cyan)`) fi: ineq:=`if`(whattype(args[1])=set,convert(args[1],`and`),args
4 [1]): xx:=args[2]:yy:=args[3]: if whattype(args[1])=set then ineqset:=args[1] elif whattype(ineq)=`<` or whattype(ineq)=`>` or whattype(ineq)=`<=` or whattype(ineq)=`>=` then ineqset:={ineq} else ineqset:=convert(ineq,set) fi: var1:=lhs(xx):var2:=lhs(yy): if not(depends(ineq,var1) and depends(ineq,var2)) then ERROR(ineq,`doesn't depend on`,var1,`and`,var2) fi: ops:=[args[4..nargs]]: if not hasoption(ops,feasiblecolor,fc,'ops') then fc:=color(rgb,.70,.70,1.0) fi: if not hasoption(ops,excludedcolor,ec,'ops') then ec:=color(rgb,.20,.20,.20) fi: if not hasoption(ops,feasiblepoints,fp,'ops') then fp:=15 fi: if not hasoption(ops,linespoints,lp,'ops') then lp:=350 fi: phaseselection:=unapply(piecewise(ineq,1,2),var1,var2): phase1:=(a,b)->1:phase2:=(a,b)->0:phases:=[phase1,phase2]: xmin:=lhs(rhs(xx)): xmax:=rhs(rhs(xx)): ymin:=lhs(rhs(yy)): ymax:=rhs(rhs(yy)): dx:=(xmax-xmin)/(fp-1): dy:=(ymax-ymin)/(fp-1): jumpprecision:=.02*min(dx,dy): findjump:=(l,r)-> `if`(evalm((l-r)&*(l-r))<jumpprecision^2,.5*(l+r), `if`(phaseselection(op(l))=phaseselection(op(.5*(l+r))), findjump(.5*(l+r),r),findjump(l,.5*(l+r)) )
5 ): reg:=[]: for y1 from ymin by dy while y1<=ymax-dy do for x1 from xmin by dx while x1<=xmax-dx do c(1):=[x1,y1]: c(2):=[x1+dx,y1]: c(3):=[x1+dx,y1+dy]: c(4):=[x1,y1+dy]: p(1):=phaseselection(op(c(1))): p(2):=phaseselection(op(c(2))): p(3):=phaseselection(op(c(3))): p(4):=phaseselection(op(c(4))): if p(1)=p(2) and p(2)=p(3) and p(3)=p(4) then reg:=[op(reg), POLYGONS( [seq(`if`(phases[p(1)](op(c(n)))=1,c(n),null), n=1..4)], op(convert(color=fc,plotoptions)))] else p1list:=[1]: if p(2)=p(1) then p1list:=[op(p1list),2]: if p(3)=p(1) then p1list:=[op(p1list),3] fi fi: if p(4)=p(1) then p1list:=[4,op(p1list)]: if p(3)=p(1) then p1list:=[3,op(p1list)] fi fi: p2list:=convert({1,2,3,4} minus {op(p1list)},list): jump1:=findjump(c(p1list[nops(p1list)]), c(p2list[1])): jump2:=findjump(c(p2list[nops(p2list)]), c(p1list[1])): aux1:=[jump2,op(map(c,p1list)),jump1]:
6 aux2:=[jump1,op(map(c,p2list)),jump2]: reg:=[op(reg), POLYGONS( [seq( `if`(phases[p(1)](op(aux1[i]))=1,aux1[i],null), i=1..nops(aux1))], op(convert(color=fc,plotoptions))), POLYGONS( [seq( `if`(phases[p(p2list[1])](op(aux2[i]))=1,aux2[i], NULL), i=1..nops(aux2))], op(convert(color=fc,plotoptions))) ] fi od od: ls:=(is)-> piecewise( whattype(is)=`<` or whattype(is)=`>`,2, whattype(is)=`<=` or whattype(is)=`>=`,1 ): llss:=map(ls,[op(ineqset)]): eq:= `if`(whattype(ineqset)=set, [seq(convert(ineqset[i],equality),i=1..nops(ineqset))], [convert(ineq,equality)] ): cc:=[seq(implicitplot(eq[i],xx,yy,color=black,numpoints=lp), i=1..nops(eq))]: PLOT(op(reg), seq( CURVES(op(op(cc[i])[1]),LINESTYLE(llss[i])),i=1..nops(cc) ), op(convert(ops,plotoptions)), POLYGONS([[xmin,ymin],[xmin,ymax],[xmax,ymax], [xmax,ymin],[xmin,ymin]], op(convert(color=ec,plotoptions))), SCALING(CONSTRAINED),STYLE(PATCHNOGRID)):
7 end proc; complexinequalityplot:=proc() local ineq,var,revar,imvar,ops; if nargs<2 then ERROR(`expecting 2 arguments, got 0. This commmand also incorporates the options: (1) exludedcolor=yellow, (2) feasiblecolor=red, (3) linespoints=350, (4) feasiblepoints=15, (5) linescolor=black and (6) thickness=1. Examples: <1> complexinequalityplot(abs(z)<1,z=(-1..1,-1..1) ), <2> complexinequalityplot(abs(z)<1,z=(-1..1,-1..1), excludedcolor=white,feasiblecolor=cyan)`) fi: var:=lhs(args[2]): Revar:=rhs(args[2])[1]: Imvar:=rhs(args[2])[2]: ineq:=subs(var=x+i*y,args[1]): ops:=args[3..nargs]: inequalityplot(ineq,x=revar,y=imvar,ops) end proc; end module: with(inequalitygraphics); Examples Example 0
8 When working with the Laplace transform, the convolution product of and is defined by = The convolution of the functions and (3.1.1) (3.1.2) is difficult to obtain by the definition because of the complex nature of the implied conditionals in both The integrand of the convolution integral is nonzero on the solution set of the inequalities (3.1.3) Figure 0.1, created with Maple's built-in inequal command from the plots package, displays this set in red. (This command will graph only the solution set for linear inequalities.)
9 t x Figure 0.1 In red, the solution set for { } For each fixed, the integration is over the intersection of the line and the red parallelogram in Figure 0.1. Thus, we have
10 a result extremely difficult to fathom without Figure 0.1. Example 1 A graph of the solution set of the inequality (3.2.1) drawn for the square appears in Figure y x Figure 1.1 Solution set of in yellow graphed
11 Example 2 A graph of the solution set of the inequalities and appears in Figure 2.1. (3.3.1) (3.3.2) Figure 2.1 Solution set of {, } shown in gray Example 3 A graph of the solution set of the inequalities and (3.4.1) (3.4.2) (3.4.3)
12 appears in Figure 3.1. Note that more than two inequalities are enclosed in set braces in the inequalityplot command. Figure 3.1 Solution set of shown in gray Example 4 Figure 4.1 compares the solution sets of and the first on the left, the second on the right.
13
14 Figure 4.1 In gray, the solution set of in (a) and the solution set of in (b) Example 5 The set of points that satisfy either (3.6.1) (3.6.2) is graphed in Figure 5.1.
15 Figure 5.1 Gray points satisfy either Example 6 Drawn with the inequaltyplot command, Figure 6.1 shows the solution set of the inequality where is given by (3.7.1)
16 Figure 6.1 Solution set for drawn in gray Figure 6.2, drawn with the plot3d command, shows the surface for As such, it represents that part of the surface we can identify as being above the plane. Rotating the graph so that it is viewed from a point sufficiently high on the -axis gives a sketch of the feasible region similar to Figure 6.1.
17 (a)
18 (b) Figure 6.2 Graph of in (a); same graph rotated and viewed from above in (b) Figure 6.3, drawn with Maple's contourplot command, shows the graph of the curve determined by the equation. This is the boundary inside or outside of which will be satisfied.
19 y x Figure 6.3 Curve determined by solution of the equation The filled option in Maple's contourplot command shades in a specified color range according to the lowest and highest values along contours. Figure 6.4 is a contour plot of in which the color range is from white to gray, with white being assigned to any contours on which the values of are lower than the one specified for the single contour to be drawn. This specified contour value is zero, so any contour corresponding to a higher value will be shaded gray.
20 Figure 6.4 Shaded contour plot of Black curve is the single contour Gray Example 7 To shade the region between graphs of the curves and use the inequalityplot command as shown in Figure 7.1. Note that an endpoint cannot be a symbol such as or.
21 Figure 7.1 Region between and shaded gray Example 8 Nonlinear inequalities actually arise in freshman calculus when students first meet a rigorous ( ) definition of the limit: Formal Definition of Limit The real number is said to be the limit of at if, for any the inequality holds At the heart of this definition is the inequality that we would like to solve for. For example, to show that solve the inequality
22 (3.9.1) for negative. The solution set is shown in Figure 8.1 where is allowed to be both positive and Figure 8.1 Solution set for the inequality?? shaded in gray Although Maple cannot solve the inequality?? for it can solve the equation under the assumption that The solutions, namely (3.9.2)
23 are graphed in Figure 8.2 where the colors indicate which solutions assume real values. Compare Figure 8.2 to Figure epsilon Figure 8.2 Graph of the analytic solution for As with Example 6, the outline of the feasible region can be obtained with Maple's implicitplot command; the feasible region, with contourplot, as shown in Figure 8.3.
24 (a) e d 0
25 Figure 8.3 In (a), outline of feasible region via implicitplot; in (b), feasible region via contourplot Surprisingly, the feasible region in Figure 8.1 is not symmetric. Figure 8.4 shows the feasible region for much smaller values of and suggests the region might be more symmetric closer to. In fact, it suggests that might be well approximated by a linear function.
26 e 0 d Figure 8.4 Solution set for the inequality?? for "small" values of One way to find an approximation for when is small is to solve for a series solution about. This can be done in Maple via where we removed the absolute value function and created two equivalent equations. In this case we obtained exact solutions and not the series expansions and (3.9.5)
27 This suggests a hypothesis tested in Figure 8.5 where the lines are drawn in green, is drawn in black, and is drawn in red. 0 e Figure 8.5 The lines in green, in black, and in red Alternatively, we let Maple deduce true (3.9.
28 The function (3.9.9) is continuous from the right at so that The governing inequality is a graphical solution of which is seen in Figure 8.6. Figure 8.6 Shaded gray, the feasible region for the inequality For the inequality is satisfied for any positive as Figure 8.6 suggests, and the following calculation shows.
29 (3.9.10) Example 9 Drawn with the complexinequalityplot command, the solution set for the inequality where, appears as the shaded region in Figure 9.1. Figure 9.1 Solution set for shaded in gray The substitution
30 leads to (3.10.1) (3.10.2) and hence, Figure 9.2 drawn with the inequalityplot command. Figure 9.2 Solution set for (3.10.2) shaded in gray The advantages of the complexinequalityplot command over the inequalityplot command for inequalities in the complex variable are obvious, and need no comment. Example 10
31 Like the inequalityplot command, the complexinequalityplot command will graph the feasible region for a set of nonlinear complex inequalities. For example, Figure 10.1 shows the set of points simultaneously satisfying the inequalities and (3.11.1) (3.11.2) Figure 10.1 Solution set of shaded in gray Legal Notice: The copyright for this application is owned by Maplesoft. The application is intended to demonstrate the use of Maple to solve a particular problem. It has been made available for product evaluation purposes only and may not be used in any other context without the express permission of Maplesoft.
32
Classroom Tips and Techniques: Branch Cuts for a Product of Two Square-Roots
Classroom Tips and Techniques: Branch Cuts for a Product of Two Square-Roots Robert J. Lopez Emeritus Professor of Mathematics and Maple Fellow Maplesoft Introduction Naive simplification of to results
More informationClassroom Tips and Techniques: Interactive Plotting of Points on a Curve
Classroom Tips and Techniques: Interactive Plotting of Points on a Curve Robert J. Lopez Emeritus Professor of Mathematics and Maple Fellow Maplesoft Introduction Recently, I needed to draw points on the
More informationClassroom Tips and Techniques: Solving Algebraic Equations by the Dragilev Method
Classroom Tips and Techniques: Solving Algebraic Equations by the Dragilev Method Introduction Robert J. Lopez Emeritus Professor of Mathematics and Maple Fellow Maplesoft On February 13, 2013, Markian
More informationClassroom Tips and Techniques: Nonlinear Curve Fitting. Robert J. Lopez Emeritus Professor of Mathematics and Maple Fellow Maplesoft
Classroom Tips and Techniques: Nonlinear Curve Fitting Robert J. Lopez Emeritus Professor of Mathematics and Maple Fellow Maplesoft Introduction I was recently asked for help in fitting a nonlinear curve
More informationClassroom Tips and Techniques: Stepwise Solutions in Maple - Part 2 - Linear Algebra
Introduction Classroom Tips and Techniques: Stepwise Solutions in Maple - Part 2 - Linear Algebra Robert J. Lopez Emeritus Professor of Mathematics and Maple Fellow Maplesoft In the preceding article Stepwise
More informationClassroom Tips and Techniques: Maple Meets Marden's Theorem. Robert J. Lopez Emeritus Professor of Mathematics and Maple Fellow Maplesoft
Introduction Classroom Tips and Techniques: Maple Meets Marden's Theorem Robert J. Lopez Emeritus Professor of Mathematics and Maple Fellow Maplesoft The statement of Marden's theorem in Table 1 is taken
More informationClassroom Tips and Techniques: Drawing a Normal and Tangent Plane on a Surface
Classroom Tips and Techniques: Drawing a Normal and Tangent Plane on a Surface Robert J. Lopez Emeritus Professor of Mathematics and Maple Fellow Maplesoft Introduction A question posted to MaplePrimes
More informationClassroom Tips and Techniques: The Lagrange Multiplier Method
Classroom Tips and Techniques: The Lagrange Multiplier Method Initializations Robert J. Lopez Emeritus Professor of Mathematics and Maple Fellow Maplesoft, a division of Waterloo Maple Inc., 2006 Introduction
More informationVisualizing Regions of Integration in 2-D Cartesian Coordinates
Classroom Tips and Techniques: Visualizing Regions of Integration Robert J. Lopez Emeritus Professor of Mathematics and Maple Fellow Maplesoft Introduction Five of the new task templates in Maple 14 are
More informationIntroduction. Classroom Tips and Techniques: The Lagrange Multiplier Method
Classroom Tips and Techniques: The Lagrange Multiplier Method Robert J. Lopez Emeritus Professor of Mathematics and Maple Fellow Maplesoft Introduction The typical multivariate calculus course contains
More informationClassroom Tips and Techniques: Plotting Curves Defined Parametrically
Classroom Tips and Techniques: Plotting Curves Defined Parametrically Robert J. Lopez Emeritus Professor of Mathematics and Maple Fellow Maplesoft Introduction If the vector representation of a curve is
More informationIntegration by Parts in Maple
Classroom Tips and Techniques: Integration by Parts in Maple Robert J. Lopez Emeritus Professor of Mathematics and Maple Fellow Maplesoft Introduction Maple implements "integration by parts" with two different
More informationClassroom Tips and Techniques: Least-Squares Fits. Robert J. Lopez Emeritus Professor of Mathematics and Maple Fellow Maplesoft
Introduction Classroom Tips and Techniques: Least-Squares Fits Robert J. Lopez Emeritus Professor of Mathematics and Maple Fellow Maplesoft The least-squares fitting of functions to data can be done in
More informationClassroom Tips and Techniques: Notational Devices for ODEs
Classroom Tips and Techniques: Notational Devices for ODEs Introduction Robert J. Lopez Emeritus Professor of Mathematics and Maple Fellow Maplesoft One of the earliest notational devices relevant to differential
More informationChapter 1. Linear Equations and Straight Lines. 2 of 71. Copyright 2014, 2010, 2007 Pearson Education, Inc.
Chapter 1 Linear Equations and Straight Lines 2 of 71 Outline 1.1 Coordinate Systems and Graphs 1.4 The Slope of a Straight Line 1.3 The Intersection Point of a Pair of Lines 1.2 Linear Inequalities 1.5
More informationConstrained Optimization with Calculus. Background Three Big Problems Setup and Vocabulary
Constrained Optimization with Calculus Background Three Big Problems Setup and Vocabulary Background Information In unit 3, you learned about linear programming, in which all constraints and the objective
More information(Refer Slide Time: 00:02:02)
Computer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture - 20 Clipping: Lines and Polygons Hello and welcome everybody to the lecture
More informationTopic 6: Calculus Integration Volume of Revolution Paper 2
Topic 6: Calculus Integration Standard Level 6.1 Volume of Revolution Paper 1. Let f(x) = x ln(4 x ), for < x
More informationSection 18-1: Graphical Representation of Linear Equations and Functions
Section 18-1: Graphical Representation of Linear Equations and Functions Prepare a table of solutions and locate the solutions on a coordinate system: f(x) = 2x 5 Learning Outcome 2 Write x + 3 = 5 as
More informationIn this chapter, we will investigate what have become the standard applications of the integral:
Chapter 8 Overview: Applications of Integrals Calculus, like most mathematical fields, began with trying to solve everyday problems. The theory and operations were formalized later. As early as 70 BC,
More informationIn other words, we want to find the domain points that yield the maximum or minimum values (extrema) of the function.
1 The Lagrange multipliers is a mathematical method for performing constrained optimization of differentiable functions. Recall unconstrained optimization of differentiable functions, in which we want
More informationDynamics and Vibrations Mupad tutorial
Dynamics and Vibrations Mupad tutorial School of Engineering Brown University ENGN40 will be using Matlab Live Scripts instead of Mupad. You can find information about Live Scripts in the ENGN40 MATLAB
More informationThe base of a solid is the region in the first quadrant bounded above by the line y = 2, below by
Chapter 7 1) (AB/BC, calculator) The base of a solid is the region in the first quadrant bounded above by the line y =, below by y sin 1 x, and to the right by the line x = 1. For this solid, each cross-section
More informationQUADRATIC AND CUBIC GRAPHS
NAME SCHOOL INDEX NUMBER DATE QUADRATIC AND CUBIC GRAPHS KCSE 1989 2012 Form 3 Mathematics Working Space 1. 1989 Q22 P1 (a) Using the grid provided below draw the graph of y = -2x 2 + x + 8 for values
More informationLagrange Multipliers
Lagrange Multipliers Introduction and Goals: The goal of this lab is to become more familiar with the process and workings of Lagrange multipliers. This lab is designed more to help you understand the
More information6.5. SYSTEMS OF INEQUALITIES
6.5. SYSTEMS OF INEQUALITIES What You Should Learn Sketch the graphs of inequalities in two variables. Solve systems of inequalities. Use systems of inequalities in two variables to model and solve real-life
More informationAlaska Mathematics Standards Vocabulary Word List Grade 7
1 estimate proportion proportional relationship rate ratio rational coefficient rational number scale Ratios and Proportional Relationships To find a number close to an exact amount; an estimate tells
More information9.1 Parametric Curves
Math 172 Chapter 9A notes Page 1 of 20 9.1 Parametric Curves So far we have discussed equations in the form. Sometimes and are given as functions of a parameter. Example. Projectile Motion Sketch and axes,
More informationThe Three Dimensional Coordinate System
The Three-Dimensional Coordinate System The Three Dimensional Coordinate System You can construct a three-dimensional coordinate system by passing a z-axis perpendicular to both the x- and y-axes at the
More information(Refer Slide Time: 00:03:51)
Computer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture 17 Scan Converting Lines, Circles and Ellipses Hello and welcome everybody
More informationGraphical Analysis. Figure 1. Copyright c 1997 by Awi Federgruen. All rights reserved.
Graphical Analysis For problems with 2 variables, we can represent each solution as a point in the plane. The Shelby Shelving model (see the readings book or pp.68-69 of the text) is repeated below for
More informationCURVE SKETCHING EXAM QUESTIONS
CURVE SKETCHING EXAM QUESTIONS Question 1 (**) a) Express f ( x ) in the form ( ) 2 f x = x + 6x + 10, x R. f ( x) = ( x + a) 2 + b, where a and b are integers. b) Describe geometrically the transformations
More information1 Programs for double integrals
> restart; Double Integrals IMPORTANT: This worksheet depends on some programs we have written in Maple. You have to execute these first. Click on the + in the box below, then follow the directions you
More informationGraphing Techniques. Domain (, ) Range (, ) Squaring Function f(x) = x 2 Domain (, ) Range [, ) f( x) = x 2
Graphing Techniques In this chapter, we will take our knowledge of graphs of basic functions and expand our ability to graph polynomial and rational functions using common sense, zeros, y-intercepts, stretching
More informationComputer Graphics: 7-Polygon Rasterization, Clipping
Computer Graphics: 7-Polygon Rasterization, Clipping Prof. Dr. Charles A. Wüthrich, Fakultät Medien, Medieninformatik Bauhaus-Universität Weimar caw AT medien.uni-weimar.de Filling polygons (and drawing
More informationComputer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture - 14
Computer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture - 14 Scan Converting Lines, Circles and Ellipses Hello everybody, welcome again
More informationReview for Mastery Using Graphs and Tables to Solve Linear Systems
3-1 Using Graphs and Tables to Solve Linear Systems A linear system of equations is a set of two or more linear equations. To solve a linear system, find all the ordered pairs (x, y) that make both equations
More information(Section 6.2: Volumes of Solids of Revolution: Disk / Washer Methods)
(Section 6.: Volumes of Solids of Revolution: Disk / Washer Methods) 6.. PART E: DISK METHOD vs. WASHER METHOD When using the Disk or Washer Method, we need to use toothpicks that are perpendicular to
More informationLecture 5. If, as shown in figure, we form a right triangle With P1 and P2 as vertices, then length of the horizontal
Distance; Circles; Equations of the form Lecture 5 y = ax + bx + c In this lecture we shall derive a formula for the distance between two points in a coordinate plane, and we shall use that formula to
More informationSNAP Centre Workshop. Graphing Lines
SNAP Centre Workshop Graphing Lines 45 Graphing a Line Using Test Values A simple way to linear equation involves finding test values, plotting the points on a coordinate plane, and connecting the points.
More informationMath Analysis Chapter 1 Notes: Functions and Graphs
Math Analysis Chapter 1 Notes: Functions and Graphs Day 6: Section 1-1 Graphs Points and Ordered Pairs The Rectangular Coordinate System (aka: The Cartesian coordinate system) Practice: Label each on the
More informationFunctions of Several Variables
Chapter 3 Functions of Several Variables 3.1 Definitions and Examples of Functions of two or More Variables In this section, we extend the definition of a function of one variable to functions of two or
More informationMath 1113 Notes - Functions Revisited
Math 1113 Notes - Functions Revisited Philippe B. Laval Kennesaw State University February 14, 2005 Abstract This handout contains more material on functions. It continues the material which was presented
More informationChanging Variables in Multiple Integrals
Changing Variables in Multiple Integrals 3. Examples and comments; putting in limits. If we write the change of variable formula as (x, y) (8) f(x, y)dx dy = g(u, v) dudv, (u, v) where (x, y) x u x v (9)
More informationA is any set of ordered pairs of real numbers. This is a set of ordered pairs of real numbers, so it is a.
Fry Texas A&M University!! Math 150!! Chapter 3!! Fall 2014! 1 Chapter 3A Rectangular Coordinate System A is any set of ordered pairs of real numbers. A relation can be finite: {(-3, 1), (-3, -1), (0,
More informationLinear Programming. L.W. Dasanayake Department of Economics University of Kelaniya
Linear Programming L.W. Dasanayake Department of Economics University of Kelaniya Linear programming (LP) LP is one of Management Science techniques that can be used to solve resource allocation problem
More informationSeptember 08, Graph y 2 =x. How? Is it a function? Function?
Graph y 2 =x How? Is it a function? Function? Section 1.3 Graphs of Functions Objective: Analyze the graphs of functions. Important Vocabulary Graph of a function The collection of ordered pairs ( x, f(x))
More informationSTEP Support Programme. Assignment 13
STEP Support Programme Assignment 13 Warm-up You probably already know that for a graph with gradient dy : if dy > 0 then the graph is increasing ; if dy < 0 then the graph is decreasing. The sign of the
More information1.1 - Functions, Domain, and Range
1.1 - Functions, Domain, and Range Lesson Outline Section 1: Difference between relations and functions Section 2: Use the vertical line test to check if it is a relation or a function Section 3: Domain
More informationEXTREME POINTS AND AFFINE EQUIVALENCE
EXTREME POINTS AND AFFINE EQUIVALENCE The purpose of this note is to use the notions of extreme points and affine transformations which are studied in the file affine-convex.pdf to prove that certain standard
More informationPolar Coordinates. 2, π and ( )
Polar Coordinates Up to this point we ve dealt exclusively with the Cartesian (or Rectangular, or x-y) coordinate system. However, as we will see, this is not always the easiest coordinate system to work
More informationSection 7.2 Volume: The Disk Method
Section 7. Volume: The Disk Method White Board Challenge Find the volume of the following cylinder: No Calculator 6 ft 1 ft V 3 1 108 339.9 ft 3 White Board Challenge Calculate the volume V of the solid
More informationTopic. Section 4.1 (3, 4)
Topic.. California Standards: 6.0: Students graph a linear equation and compute the x- and y-intercepts (e.g., graph x + 6y = ). They are also able to sketch the region defined by linear inequality (e.g.,
More informationMath Analysis Chapter 1 Notes: Functions and Graphs
Math Analysis Chapter 1 Notes: Functions and Graphs Day 6: Section 1-1 Graphs; Section 1- Basics of Functions and Their Graphs Points and Ordered Pairs The Rectangular Coordinate System (aka: The Cartesian
More informationENGI Parametric & Polar Curves Page 2-01
ENGI 3425 2. Parametric & Polar Curves Page 2-01 2. Parametric and Polar Curves Contents: 2.1 Parametric Vector Functions 2.2 Parametric Curve Sketching 2.3 Polar Coordinates r f 2.4 Polar Curve Sketching
More informationMATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED DETERMINING THE INTERSECTIONS USING THE GRAPHING CALCULATOR
FOM 11 T15 INTERSECTIONS & OPTIMIZATION PROBLEMS - 1 1 MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED 1) INTERSECTION = a set of coordinates of the point on the grid where two or more graphed lines touch
More information3.7. Vertex and tangent
3.7. Vertex and tangent Example 1. At the right we have drawn the graph of the cubic polynomial f(x) = x 2 (3 x). Notice how the structure of the graph matches the form of the algebraic expression. The
More informationSystems of Equations and Inequalities. Copyright Cengage Learning. All rights reserved.
5 Systems of Equations and Inequalities Copyright Cengage Learning. All rights reserved. 5.5 Systems of Inequalities Copyright Cengage Learning. All rights reserved. Objectives Graphing an Inequality Systems
More informationChapter - 2: Geometry and Line Generations
Chapter - 2: Geometry and Line Generations In Computer graphics, various application ranges in different areas like entertainment to scientific image processing. In defining this all application mathematics
More informationPolynomial and Rational Functions. Copyright Cengage Learning. All rights reserved.
2 Polynomial and Rational Functions Copyright Cengage Learning. All rights reserved. 2.7 Graphs of Rational Functions Copyright Cengage Learning. All rights reserved. What You Should Learn Analyze and
More informationMATH 110 analytic geometry Conics. The Parabola
1 MATH 11 analytic geometry Conics The graph of a second-degree equation in the coordinates x and y is called a conic section or, more simply, a conic. This designation derives from the fact that the curve
More information1
Zeros&asymptotes Example 1 In an early version of this activity I began with a sequence of simple examples (parabolas and cubics) working gradually up to the main idea. But now I think the best strategy
More informationFunctions. Copyright Cengage Learning. All rights reserved.
Functions Copyright Cengage Learning. All rights reserved. 2.2 Graphs Of Functions Copyright Cengage Learning. All rights reserved. Objectives Graphing Functions by Plotting Points Graphing Functions with
More informationPolar (BC Only) They are necessary to find the derivative of a polar curve in x- and y-coordinates. The derivative
Polar (BC Only) Polar coordinates are another way of expressing points in a plane. Instead of being centered at an origin and moving horizontally or vertically, polar coordinates are centered at the pole
More informationIntegration. Edexcel GCE. Core Mathematics C4
Edexcel GCE Core Mathematics C Integration Materials required for examination Mathematical Formulae (Green) Items included with question papers Nil Advice to Candidates You must ensure that your answers
More informationOhio Tutorials are designed specifically for the Ohio Learning Standards to prepare students for the Ohio State Tests and end-ofcourse
Tutorial Outline Ohio Tutorials are designed specifically for the Ohio Learning Standards to prepare students for the Ohio State Tests and end-ofcourse exams. Math Tutorials offer targeted instruction,
More informationMat 241 Homework Set 7 Due Professor David Schultz
Mat 41 Homework Set 7 Due Professor David Schultz Directions: Show all algebraic steps neatly and concisely using proper mathematical symbolism When graphs and technology are to be implemented, do so appropriately
More informationLinear Programming. Meaning of Linear Programming. Basic Terminology
Linear Programming Linear Programming (LP) is a versatile technique for assigning a fixed amount of resources among competing factors, in such a way that some objective is optimized and other defined conditions
More informationOPTIMIZATION: Linear Programming
May 21, 2013 OPTIMIZATION: Linear Programming Linear programming (OPTIMIZATION) is the process of taking various linear inequalities (constraints) related to some situation, and finding the "best" value
More informationOpenGL Graphics System. 2D Graphics Primitives. Drawing 2D Graphics Primitives. 2D Graphics Primitives. Mathematical 2D Primitives.
D Graphics Primitives Eye sees Displays - CRT/LCD Frame buffer - Addressable pixel array (D) Graphics processor s main function is to map application model (D) by projection on to D primitives: points,
More informationSituation 3. Parentheses vs. Brackets. Colleen Foy
Situation 3 Parentheses vs. Brackets Colleen Foy Prompt: Students were finding the domain and range of various functions. Most of the students were comfortable with the set builder notation, but when asked
More informationMATHS METHODS QUADRATICS REVIEW. A reminder of some of the laws of expansion, which in reverse are a quick reference for rules of factorisation
MATHS METHODS QUADRATICS REVIEW LAWS OF EXPANSION A reminder of some of the laws of expansion, which in reverse are a quick reference for rules of factorisation a) b) c) d) e) FACTORISING Exercise 4A Q6ace,7acegi
More informationTHE UNIVERSITY OF AKRON Mathematics and Computer Science Article: Applications to Definite Integration
THE UNIVERSITY OF AKRON Mathematics and Computer Science Article: Applications to Definite Integration calculus menu Directory Table of Contents Begin tutorial on Integration Index Copyright c 1995 1998
More informationFinite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Lecture - 36
Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras Lecture - 36 In last class, we have derived element equations for two d elasticity problems
More informationIn this class, we addressed problem 14 from Chapter 2. So first step, we expressed the problem in STANDARD FORM:
In this class, we addressed problem 14 from Chapter 2. So first step, we expressed the problem in STANDARD FORM: Now that we have done that, we want to plot our constraint lines, so we can find our feasible
More informationSlide 1 / 96. Linear Relations and Functions
Slide 1 / 96 Linear Relations and Functions Slide 2 / 96 Scatter Plots Table of Contents Step, Absolute Value, Piecewise, Identity, and Constant Functions Graphing Inequalities Slide 3 / 96 Scatter Plots
More informationGraphing Linear Inequalities in Two Variables.
Many applications of mathematics involve systems of inequalities rather than systems of equations. We will discuss solving (graphing) a single linear inequality in two variables and a system of linear
More informationGRAPHING WORKSHOP. A graph of an equation is an illustration of a set of points whose coordinates satisfy the equation.
GRAPHING WORKSHOP A graph of an equation is an illustration of a set of points whose coordinates satisfy the equation. The figure below shows a straight line drawn through the three points (2, 3), (-3,-2),
More informationINSTRUCTIONS FOR THE USE OF THE SUPER RULE TM
INSTRUCTIONS FOR THE USE OF THE SUPER RULE TM NOTE: All images in this booklet are scale drawings only of template shapes and scales. Preparation: Your SUPER RULE TM is a valuable acquisition for classroom
More informationSection 12.2: Quadric Surfaces
Section 12.2: Quadric Surfaces Goals: 1. To recognize and write equations of quadric surfaces 2. To graph quadric surfaces by hand Definitions: 1. A quadric surface is the three-dimensional graph of an
More informationGlossary Common Core Curriculum Maps Math/Grade 6 Grade 8
Glossary Common Core Curriculum Maps Math/Grade 6 Grade 8 Grade 6 Grade 8 absolute value Distance of a number (x) from zero on a number line. Because absolute value represents distance, the absolute value
More informationChapter 3A Rectangular Coordinate System
Fry Texas A&M University! Math 150! Spring 2015!!! Unit 4!!! 1 Chapter 3A Rectangular Coordinate System A is any set of ordered pairs of real numbers. The of the relation is the set of all first elements
More information6th Bay Area Mathematical Olympiad
6th Bay Area Mathematical Olympiad February 4, 004 Problems and Solutions 1 A tiling of the plane with polygons consists of placing the polygons in the plane so that interiors of polygons do not overlap,
More information8.2 Graph and Write Equations of Parabolas
8.2 Graph and Write Equations of Parabolas Where is the focus and directrix compared to the vertex? How do you know what direction a parabola opens? How do you write the equation of a parabola given the
More informationCore Mathematics 3 Functions
http://kumarmaths.weebly.com/ Core Mathematics 3 Functions Core Maths 3 Functions Page 1 Functions C3 The specifications suggest that you should be able to do the following: Understand the definition of
More information(Refer Slide Time: 0:32)
Digital Image Processing. Professor P. K. Biswas. Department of Electronics and Electrical Communication Engineering. Indian Institute of Technology, Kharagpur. Lecture-57. Image Segmentation: Global Processing
More informationAdvanced Operations Research Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras
Advanced Operations Research Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras Lecture 16 Cutting Plane Algorithm We shall continue the discussion on integer programming,
More informationExample 1: Give the coordinates of the points on the graph.
Ordered Pairs Often, to get an idea of the behavior of an equation, we will make a picture that represents the solutions to the equation. A graph gives us that picture. The rectangular coordinate plane,
More informationGraded Assignment 2 Maple plots
Graded Assignment 2 Maple plots The Maple part of the assignment is to plot the graphs corresponding to the following problems. I ll note some syntax here to get you started see tutorials for more. Problem
More informationChoose the file menu, and select Open. Input to be typed at the Maple prompt. Output from Maple. An important tip.
MAPLE Maple is a powerful and widely used mathematical software system designed by the Computer Science Department of the University of Waterloo. It can be used for a variety of tasks, such as solving
More informationQuadratic Equations Group Acitivity 3 Business Project Week #5
MLC at Boise State 013 Quadratic Equations Group Acitivity 3 Business Project Week #5 In this activity we are going to further explore quadratic equations. We are going to analyze different parts of the
More informationChapter 8: Applications of Definite Integrals
Name: Date: Period: AP Calc AB Mr. Mellina Chapter 8: Applications of Definite Integrals v v Sections: 8.1 Integral as Net Change 8.2 Areas in the Plane v 8.3 Volumes HW Sets Set A (Section 8.1) Pages
More informationRational functions, like rational numbers, will involve a fraction. We will discuss rational functions in the form:
Name: Date: Period: Chapter 2: Polynomial and Rational Functions Topic 6: Rational Functions & Their Graphs Rational functions, like rational numbers, will involve a fraction. We will discuss rational
More information9-1 GCSE Maths. GCSE Mathematics has a Foundation tier (Grades 1 5) and a Higher tier (Grades 4 9).
9-1 GCSE Maths GCSE Mathematics has a Foundation tier (Grades 1 5) and a Higher tier (Grades 4 9). In each tier, there are three exams taken at the end of Year 11. Any topic may be assessed on each of
More information2.2 Graphs Of Functions. Copyright Cengage Learning. All rights reserved.
2.2 Graphs Of Functions Copyright Cengage Learning. All rights reserved. Objectives Graphing Functions by Plotting Points Graphing Functions with a Graphing Calculator Graphing Piecewise Defined Functions
More informationCourse Number 432/433 Title Algebra II (A & B) H Grade # of Days 120
Whitman-Hanson Regional High School provides all students with a high- quality education in order to develop reflective, concerned citizens and contributing members of the global community. Course Number
More informationMAT 003 Brian Killough s Instructor Notes Saint Leo University
MAT 003 Brian Killough s Instructor Notes Saint Leo University Success in online courses requires self-motivation and discipline. It is anticipated that students will read the textbook and complete sample
More informationThe diagram above shows a sketch of the curve C with parametric equations
1. The diagram above shows a sketch of the curve C with parametric equations x = 5t 4, y = t(9 t ) The curve C cuts the x-axis at the points A and B. (a) Find the x-coordinate at the point A and the x-coordinate
More informationII. Linear Programming
II. Linear Programming A Quick Example Suppose we own and manage a small manufacturing facility that produced television sets. - What would be our organization s immediate goal? - On what would our relative
More informationMiddle School Math Course 3
Middle School Math Course 3 Correlation of the ALEKS course Middle School Math Course 3 to the Texas Essential Knowledge and Skills (TEKS) for Mathematics Grade 8 (2012) (1) Mathematical process standards.
More information