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A Guide to Maple 8 & 9 Commands for Calculus I, II & III By Dr. Jennifer Bergne

A Guide to Maple 8 & 9 Commands for Calculus I, II & III By Dr. Jennifer Bergner Dr. Don Spickler Department of Mathematics and Computer Science Salisbury University Copyright ©2003 A Guide to Maple 8 & 9 Commands for Calculus I, II & III 1 Table of Contents Introduction...................................................................................................................4 Basic Commands & Concepts.......................................................................................5 Exact verses Approximate Solutions............................................................................5 Obtaining Previous Results, The Ditto Operator..........................................................5 Expressions.................................................................................................................6 Setting a Variable to a Value .......................................................................................7 Setting a Variable Back to a Variable ..........................................................................8 Defining a List and a Set .............................................................................................9 The seq Command..................................................................................................... 10 The evalf Command .................................................................................................. 12 The eval Command ................................................................................................... 13 The subs Command................................................................................................... 15 Mathematical Functions ............................................................................................ 16 Trigonometric and Hyperbolic Functions............................................................... 16 Exponential and Logarithmic Functions.................................................................17 Root Functions ...................................................................................................... 17 Other Functions..................................................................................................... 17 The solve Command.................................................................................................. 17 The fsolve Command ................................................................................................ 20 Functions ..................................................................................................................... 24 Defining a Function of a Single Variable................................................................... 24 Using the map Command .......................................................................................... 24 Defining Piecewise Functions ................................................................................... 25 Defining a Function of Several Variables .................................................................. 27 Limits, Differentiation, Integration & Sums.............................................................. 29 Evaluating Limits of Function of a Single Variable ................................................... 29 Evaluating Limits of Multivariable Functions............................................................ 30 Derivatives of Functions of a Single Variable............................................................ 32 Partial Derivatives of Multivariable Functions........................................................... 35 Integration of Single Variable Functions ................................................................... 39 Integration of Multivariable Functions....................................................................... 40 Finite and Infinite Sums ............................................................................................ 43 The taylor Command.................................................................................................46 Graphing ..................................................................................................................... 48 Two Dimensional Plots ............................................................................................. 48 The plot Command: Plotting Functions.................................................................. 48 The plot Command: Plotting Parametrically Defined Equations............................. 54 The implicitplot Command.................................................................................... 55 The contourplot Command .................................................................................... 57 A Guide to Maple 8 & 9 Commands for Calculus I, II & III 2 The gradplot Command ......................................................................................... 58 The fieldplot Command......................................................................................... 60 Infinity Plots.......................................................................................................... 61 The logplot and loglogplot Commands .................................................................. 62 The arrow command.............................................................................................. 63 The display command............................................................................................ 66 Three Dimensional Plots ........................................................................................... 66 The plot3d Command: Plotting Functions.............................................................. 66 The cylinderplot Command ................................................................................... 73 The sphereplot Command...................................................................................... 74 The arrow Command............................................................................................. 74 The fieldplot3d Command ..................................................................................... 77 The gradplot3d Command ..................................................................................... 78 The implicitplot3d Command ................................................................................ 79 The spacecurve Command..................................................................................... 80 The display Command........................................................................................... 81 Packages ...................................................................................................................... 83 The student Package.................................................................................................. 83 The changevar Command ...................................................................................... 84 The distance Command ......................................................................................... 85 The intercept Command ........................................................................................ 85 The leftbox, leftsum, rightbox, rightsum, middlebox and middlesum Commands .. 86 The midpoint Command........................................................................................ 89 The showtangent Command .................................................................................. 89 The trapezoid and simpson Commands.................................................................. 90 The Lineint Command........................................................................................... 91 The Student Package: Student[Calculus1] .................................................................92 The AntiderivativePlot Command ......................................................................... 92 The ApproximateInt Command ............................................................................. 93 The DerivativePlot Command ............................................................................... 98 The InversePlot Command .................................................................................... 99 The MeanValueTheorem Command .................................................................... 100 The NewtonsMethod Command .......................................................................... 101 The RiemannSum Command............................................................................... 103 The RollesTheorem Command ............................................................................ 103 The SurfaceOfRevolution Command................................................................... 103 The Tangent Command ....................................................................................... 105 The TaylorApproximation Command .................................................................. 106 The VolumeOfRevolution Command .................................................................. 108 The Asymptotes Command ................................................................................. 110 The CriticalPoints Command............................................................................... 111 The ExtremePoints Command ............................................................................. 111 The InflectionPoints Command ........................................................................... 112 The Roots Command........................................................................................... 112 The linalg Package .................................................................................................. 113 The LinearAlgebra Package .................................................................................... 114 A Guide to Maple 8 & 9 Commands for Calculus I, II & III 3 The VectorCalculus Package................................................................................... 115 Defining a Vector................................................................................................ 116 The &x Operator and The CrossProduct Command ............................................ 117 The . Operator and The DotProduct Command .................................................... 117 The ArcLength Command ................................................................................... 118 The Binormal Command ..................................................................................... 119 The Curl Command ............................................................................................. 120 The Curvature Command .................................................................................... 120 The Del, Gradient and Nabla Commands............................................................. 122 The diff Command .............................................................................................. 123 The DirectionalDiff Command ............................................................................ 124 The Divergence Command .................................................................................. 125 The Flux Command............................................................................................. 126 The evalVF Command......................................................................................... 128 The int Command................................................................................................ 128 The Jacobian Command ...................................................................................... 131 The limit Command............................................................................................. 133 The LineInt Command......................................................................................... 133 The PathInt Command......................................................................................... 135 The PrincipalNormal Command .......................................................................... 137 The RadiusOfCurvature Command...................................................................... 138 The ScalarPotential Command............................................................................. 139 The SurfaceInt Command.................................................................................... 140 The TNBFrame Command .................................................................................. 140 The TangentLine Command ................................................................................ 142 The TangentPlane Command............................................................................... 142 The TangentVector Command............................................................................. 143 The Torsion Command........................................................................................ 144 The VectorPotential Command............................................................................ 145 The plots Package ................................................................................................... 146 The plottools Package ............................................................................................. 146 Animation.................................................................................................................. 150 Using Maple Help...................................................................................................... 153 A Guide to Maple 8 & 9 Commands for Calculus I, II & III 4 Introduction This manual is a quick reference guide to some of the Maple 8 and 9 commands that are pertinent to Calculus I, II and III. It is designed to show you the syntax and examples for specific Maple functions. It is not designed to show you how to put the commands together to solve problems or to explain the ins and outs of Maple programming. With this guide and your mathematical background you should be able to use Maple to help solve more difficult problems and explore both the theory and applications involved in the Calculus sequence. The guide is organized by topic and is not to be read as a textbook. Many of the examples that are given in the early sections use commands that are covered later in the guide. While this may cause some mild frustration when you are first learning Maple, once you have a little Maple under your belt you will find the extent of the examples quite helpful. Most of the material in this guide can also be found in Maple documentation and the Maple Help System. In fact, both of these other resources offer a more complete description of the command, its attributes and applicable options. One of the difficulties with both the written and electronic help systems is that they are too complete. Most descriptions in the help system explain the command to its fullest extent and offer rather advanced examples. This guide concentrates on the way you will most likely use the commands in a Calculus course and leaves the advanced topics alone. Curious readers are certainly welcome to explore the help system for further explanations of the commands given in this guide and many commands that we do not cover. A Guide to Maple 8 & 9 Commands for Calculus I, II & III 5 Basic Commands & Concepts Exact verses Approximate Solutions Maple is a computer algebra system, which means that it does symbol manipulation in the same way we do when solving a problem by hand. It will keep its answers in exact form whenever possible and in many cases you need to force it to give you an approximation. Maple has several built-in commands that will approximate a value, furthermore, there are some commands that have both exact and approximate forms. The approximate form usually has an f at the end of the command name. Obtaining Previous Results, The Ditto Operator In Maple, percentage signs are used to refer to previously computed expressions. Specifically, the % operator reevaluates the last expression computed, the %% operator reevaluates the second last expression computed, and the %%% operator reevaluates the third last expression computed. Be careful, the last expression is not always the one directly above the %, it is the last one done in the session. For example, if we execute the following commands in order, > x:=2; := x 2 > y:=3; := y 3 > z:=x+y; := z 5 Then the following ditto commands will return the following values. > %; 5 > %%; 3 > %%%; 2 If we execute the same commands in reverse order then the following ditto commands will return the following values. > %; A Guide to Maple 8 & 9 Commands for Calculus I, II & III 6 2 > %%; 3 > %%%; 5 Expressions The syntax for mathematical expressions in Maple is quite similar to other programs like Excel and to most graphing calculators. The main thing to watch is that you must always use an * to denote multiplication. Otherwise, it is simply + for addition, – for subtraction, * for multiplication, / for division, ^ for powers and we always use parentheses ( ) for grouping. Note that [ ] and { } have other uses in Maple. For example, > 3*x^2-2*x+7; − + 3 x2 2 x 7 > 4^(x-2); 4 ( ) − + 2 x Maple will also give you an error if the expression you input is in some way ambiguous. For example, > x^x^x; Error, `^` unexpected > x^(x^x); x ( ) xx > (x^x)^x; ( ) xx x As you know parentheses can make a big difference in the meaning of an expression. Note the difference in the following outputs. > (x^2-7*x+2)/(x+3); − + x2 7 x 2 + x 3 > x^2-7*x+2/x+3; A Guide to Maple 8 & 9 Commands for Calculus I, II & III 7 − + + x2 7 x 2 x 3 > x^2-7*x+2/(x+3); − + x2 7 x 2 + x 3 Setting a Variable to a Value To set a variable to a value you simply start the command with the variable name followed by := and then the value or expression. For example, to define the variable x to be the number 5 we would use, > x:=5; := x 5 To define the variable x to be the expression 7 2 − h we would use, uploads/Litterature/ maple-guide.pdf