Maple Projects to Accompany
Calculus, Fifth Edition
By James Stewart

Mike Penna, Indiana University - Purdue University, Indianapolis

Preface

The computer projects presented below are based on Maple Release 9.5, and meant to accompany Calculus, Fifth Edition, by James Stewart. The objectives of these projects are discussed here, and some suggestions for their usage are discussed here.

In the interest of letting these projects speak for themselves, this Preface will be mercifully short. However, it cannot be so short that I ignore thanking Bob Pirtle and Stacy Green at Thomson Learning for their guidance and support.


First Steps

The projects in this chapter serve as background for material to be covered later. Included in these projects are two Maple tutorials that must be covered. The remaining projects can be done at the beginning of the term or postponed until they are needed. Whichever approach you take, you may wish to select projects from later chapters first and then return to this chapter when you have a better idea of the background material you'll want (or need) to cover.

  • A Brief Maple Tutorial: Part 1, Arithmetic and Simple Algebra
  • A Brief Maple Tutorial: Part 2, Functions of a Single Variable and Graphing
    These two projects are extremely important and must be done.

  • Factoring Polynomials
    This is a very brief review. It does not use Maple: all computations can be done by hand.

  • Solving Equations with Maple
    While adequate illustration of how to solve various equations has (hopefully) been integrated into these projects, a more focused discussion may be desired. This is particularly the case for equations that involve trig functions, for solving them is not always easy or simple, and many interesting examples that involve (solving) them arise in applications of differentiation.

  • Variation and Proportion
    This is a brief review that anticipates applications of Calculus that involve, for example, force and work (Hooke's Law), and differential equations (the logistic equation and Newton's Law of Cooling). This project does not use Maple: all computations can be done by hand.

  • Complex Numbers
    While most undergraduate Calculus courses focus on the real numbers, there are topics for which knowing something about the complex numbers is important; this background is for those topics. This project does not use Maple: all computations can be done by hand.

  • Calculus, Maple, and Complex Numbers
    Consider covering this project if Maple is to be used in conjunction with examples that involve rational exponents. By choosing examples carefully, one can avoid contact with the complex numbers; doing so, however, can be extremely limiting. By using functions such as surd (which is discussed in this project), and a little more thought about mathematics and coding in Maple, one can broaden the scope of examples considerably.

  • Expressions and Functions
    There are major differences between how expressions and functions are treated in Maple, and this project discusses these differences.


    A Preview of Calculus

  • "Why Worry About Proving Things?"
    This project is aimed at motivating the need for proof in Calculus. It encourages experimentation, and it involves some elementary Maple programming by the student.


    Chapter 1: Functions and Models

  • Variations in the Graph of a Function
  • Variations in the Graph of the Sine Function
    These projects review how shifting and scaling affect the graph of a function. The first project uses a polynomial function, and the second uses the sine function.

  • Amplitude, Period, and Phase Angle
    This is a quick review of the amplitude, period, and phase angle of f(x) = A sin (Bx+C).

  • Function Addition
    This is just a simple project that discusses function addition. It also illustrates the use of Maple procedures and lists.

  • Linear Interpolation
    This project is for the benefit of students who need to know something about linear interpolation: many pre-Calculus texts and courses do not cover this topic. This project does not use Maple: all computations can be done by hand.


    Chapter 2: Limits and Rates of Change

  • Predicting the Slope of a Tangent Line
  • Predicting the Slope of a Tangent Line
    These projects have the student predict the slope of the tangent line to a function at a point by examining the slopes of secant lines. The first project computes limits as h approaches 0, and the second as x approaches a.

  • Linear Approximations
    This project is aimed at applying the derivative and tangent line early (before discussing differentials).

  • Guessing Limits Numerically
    In this project numerical data is created, and from it a limit is to be guessed.

  • The Definition of Limit
  • The Definition of Limit
    These projects are aimed at showing that deltas exist for given epsilons. These projects are slight variations of each other.

  • One-Sided Limits
    This project is aimed at guessing one-sided limits.

  • The Intermediate Value Theorem
    A simple Intermediate Value Theorem project.

  • The Intermediate Value Theorem and Graphing
  • The Intermediate Value Theorem and Graphing
    These projects discuss the Intermediate Value Theorem in the context of pre-Calculus graphing, and they also lay the foundation for the later discussion of graphing. The first of these projects does not involve vertical asymptotes but the second does.

  • The Bisection Method
    The Bisection Method is a consequence of the Intermediate Value Theorem, and an alternative to Newton's Method for solving equations. The Bisection Method doesn't converge as fast as Newton's Method, but it's very easy to program and it illustrates limiting behavior.


    Chapter 3: Derivatives

  • Graphing a Function and its Derivative using a Table
    In this project the student is asked to graph a function and its derivative by using a table.

  • Graphing a Function and its Derivative
    In this project, some points are plotted that lie on the graph of a function and its derivative, and the student is asked to connect the dots and numerically estimate the derivative at a point using the original function.

  • Differentiation
    The student is led through the computation of a derivative using the definition of derivative, and then asked to correlate the first three derivatives of a single function.

  • The First Derivative and Slope
  • The First Derivative and Slope
    In the first of these projects, the student works through the computation of a derivative using the definition of derivative, and then is asked to correlate the graph of a function with the graph of its derivative. In the second, the student is asked only to correlate the graph of a function with the graph of its derivative.

  • The Graph of a Function and the Graph of its Derivative
    The graphs of functions are given and the student is asked to draw the graph of their derivatives by hand.

  • The Graph of a Derivative
  • The Graph of a Derivative
    Given the graph of a derivative and an initial value, the student is asked to graph the original function. The functions in the second project present more of a challenge than those in the first.

  • Implicit Functions and Implicit Differentiation
  • Implicit Functions and Implicit Differentiation
    These projects illustrate the utility of implicit differentiation. They follow two different Maple approaches to implicit differentiation.

  • Linear Approximations and Differentials
    This project emphasizes the geometry of differentials.


    Chapter 4: Applications of Differentiation

    There are three different versions of several of the projects in this chapter: in each case one version involves a polynomial function, one involves a rational function, and one involves an algebraic function. These different versions are presented to accommodate different levels of courses and different levels of student abilities. In different versions of a given project, the Maple code is slightly different. In particular, working with algebraic functions may require working with the surd command (which is why the project Calculus, Maple, and Complex Numbers was suggested earlier.)

  • Maximum and Minimum Values on a Closed Interval
  • Maximum and Minimum Values on a Closed Interval
  • Maximum and Minimum Values on a Closed Interval
    The student is led through the process of finding the maximum and minimum values of a function on a closed interval. The difference between these projects is that the first addresses a polynomial function, the second a rational function, and the third an algebraic function with a fractional exponent.

  • The Mean Value Theorem
  • The Mean Value Theorem
    These address the Mean Value Theorem.

  • Finding Relative Extrema on an Open Interval: The First Derivative Test
  • Finding Relative Extrema on an Open Interval: The First Derivative Test
  • Finding Relative Extrema on an Open Interval: The First Derivative Test
    The student is led through the process of finding the relative extrema of a function on an open interval using the First Derivative Test. The difference between these is that the first addresses a polynomial function, the second a rational function, and the third an algebraic function with a fractional exponent.

  • The Second Derivative and Concavity
    This establishes the connection between the second derivative and concavity.

  • Finding Relative Extrema on an Open Interval: The Second Derivative Test
  • Finding Relative Extrema on an Open Interval: The Second Derivative Test
  • Finding Relative Extrema on an Open Interval: The Second Derivative Test
    The student is led through the process of finding the relative extrema of a function on an open interval using the Second Derivative Test. The difference between these is that the first addresses a polynomial function, the second a rational function, and the third an algebraic function with a fractional exponent.

  • Concavity and Inflection Points
  • Concavity and Inflection Points
  • Concavity and Inflection Points
    The student is led through a discussion of the concavity and inflection points of a function on an open interval. These projects anticipate a more complete coverage of graphing. The difference between these is that the first addresses a polynomial function, the second a rational function, and the third an algebraic function with a fractional exponent.

  • Critical Numbers and Inflection Points
  • Critical Numbers and Inflection Points
  • Critical Numbers and Inflection Points
    These projects have the student graph some functions with Maple and then label the critical points and the inflection points. These projects again anticipate a more complete coverage of graphing. The difference between these is that the first addresses a polynomial function, the second a rational function, and the third an algebraic function with a fractional exponent.

  • Graphing Functions
  • Graphing Functions
  • Graphing Functions
    These projects each give a discussion and example of qualitative graphing, and then, given a blank graphing grid and data about a specific function, ask the student to graph. The difference between these is that the first addresses a polynomial function, the second a rational function, and the third an algebraic function with a fractional exponent.

  • Graphing Functions Again
  • Graphing Functions Again
  • Graphing Functions Again
    These projects are a follow-up to the previous projects: they just give a blank graphing grid and data about a specific function, and ask the student to graph. The difference between these is that the first addresses a polynomial function, the second a rational function, and the third an algebraic function with a fractional exponent.

  • Newton's Method: Part 1
  • Newton's Method: Part 2
  • Newton's Method
    These projects present a basic discussion of Newton's Method. A couple of approaches are possible: the first of these projects alone can be assigned, the first two of these projects can be assigned (for a more in-depth treatment), or just the third project (which attempts to compress the first two) can be assigned.

  • The Iteration Method
  • Iteration and Chaos
    The first of these projects presents a discussion of the Iteration Method and the second discusses iteration and chaos. The application requires student initiative.

  • Ballistics
  • Ballistics
  • Ballistics
  • Ballistics
    These are four variations on the classic problem of describing the motion of an object that is projected into the air. The first and second of these projects use English units, and the later two use metric units. The first and third specify the Maple code to be used, and the second and fourth leave writing the Maple code to the student. (Some students appreciate the freedom to do their own thing.)


    Chapter 5: Integrals

  • Integration: Riemann Sums
  • Integration: Riemann Sums
  • Integration: Riemann Sums
    These projects represent three different approaches to integration using Riemann sums. The second uses Maple's Student package, and the third asks the student to write some Maple code.

  • The Fundamental Theorem of Calculus
    This project discusses the Fundamental Theorem of Calculus, and emphasizes the definition of a function as an integral.

  • The Fundamental Theorem of Calculus
    This project describes an alternate proof of the Fundamental Theorem. It is a little off the beaten path, but it does anticipate Euler's Method and telescoping sums. This project does not use Maple: all computations can be done by hand.

  • The Fundamental Theorem and u-Substitution
    This project illustrates how Maple can be used to perform a u-substitution.


    Chapter 6: Applications of Integration

  • Areas of Type I and Type II Regions
    This is a basic "find the area" project.

  • Force and Work
    One of the major applications of Calculus is to problems that involve force and work. This project is preparation for those applications, and is aimed particularly at students who have not yet studied Physics. This project does not use Maple: all computations can be done by hand. (This project repeats a small part of the work on variation and proportion related to Hooke's Law covered in the Variation and Proportion project so that that project does not have to be covered.)

  • The Average Value of a Function
    This project discusses the average value of a function, and illustrates its geometric significance.


    Chapter 7: Inverse Functions

  • Exponentials and Logarithms
    This is a very brief review of pre-Calculus exponentials and logarithms in preparation for further development in Calculus: it emphasizes that prior to Calculus (ie, without introducing limits), exponentials are defined only for rational exponents. This project does not use Maple: all computations can be done by hand.

  • Inverse Functions
    This project is a brief review of pre-Calculus inverse functions.

  • Inverse Functions
  • Inverse Functions
  • Inverse Functions
    These projects are slightly different variations of each other. All three lead the student through checking the properties for inverse functions.

  • Logarithmic Identities and Inequalities
    This project graphically illustrates some logarithmic identities and a double logarithmic inequality. The double inequality is useful in series comparison tests that involve logarithms.

  • Exponential Growth and Decay
  • Exponential Functions
    These projects illustrate the graphs of several exponential functions.

  • Power Functions
    This project has the student graph and identify the graphs of several function of the form f(x) = ax, and also compare the graphs of f(x) = x^2, f(x) = 2^x, and f(x) = x^x.

  • Exponential and Logarithmic Functions
  • Exponentials, Logarithms, and Their Derivatives
    These projects have the student graph, and identify the graphs of, several exponential functions.

  • The Exponential Function and Interest
    In this project we examine the connection between the computation of compound interest and the exponential function, paying particular attention to domains.


    Chapter 8: Techniques of Integration

  • Techniques of Integration
    Maple is a very powerful integration tool. But getting the "right answer" to an integration problem isn't always what's most important to a student: a student may also be interested in whether a specific technique they'd like to try will work. This project presents some different strategies for using Maple to do this. (The advantage of using Maple and such techniques is that Maple is a dynamic tool, not a static tool as is a text or solutions manual.)

  • Partial Fractions
    Maple has a built-in command that allows you to compute a partial fraction decomposition in one line of code. This project doesn't use this command: the goal of this project is to illustrate how Maple can be used to check work done by hand. (Computations involving partial fraction decompositions are sometimes tedious. While they are important - since they arise in future courses in Differential Equations - it may be useful to provide students with a tool that can save class time.)

  • Approximate Integration
    In this project we compare approximate integration by Riemann sums, the Trapezoidal Rule, and Simpson's Rule.


    Chapter 9: Further Applications of Integration

  • Surfaces of Revolution
  • Surfaces of Revolution
  • Surfaces of Revolution
  • Surfaces of Revolution
    These projects are concerned with producing graphics that can be used to get a better feel for surfaces of revolution: students are not expected to understand the Maple code, although they should be able to modify it to draw surfaces that might be of interest to them. In the first project the student is led through revolving the graph of y = f(x) about the x- and y-axes, and around the lines x = k and y = k; in the second the student is led through revolving the graph of y = f(x) just about the x- and y-axes; in the third the student is led through revolving the graph of y = f(x) about the x- and y-axes, and around the lines x = k and y = k, and then asked to revolve the graph of a different function about the same axes; in the fourth the student is led through revolving the graph of y = f(x) about just the x- and y-axes, and then asked to revolve the graph of a different function about the same axes. (These variations are aimed at accommodating both class schedules and student abilities.)


    Chapter 10: Differential Equations

  • Solutions of a Differential Equation
  • Solutions of a Differential Equation
    These projects are slightly different variations of each other. Both involve solving a differential equation and plotting solution curves.

  • Direction Fields
    This project asks the student to draw a direction field and several integral curves using Maple, then to draw a solution curve by hand.

  • Exponential Growth and Decay
  • Exponential Growth and Decay
    These projects discuss the general exponential growth and decay equations. The first specifies the Maple code to be used, and the second leaves the writing of Maple code to the student.

  • The Logistic Equation
  • The Logistic Equation
    These projects discuss the logistic equation. The first specifies the Maple code to be used, and the second leaves the writing of Maple code to the student.

  • Newton's Law of Cooling
  • Newton's Law of Cooling
    These projects discuss Newton's Law of Cooling. The first specifies the Maple code to be used, and the second leaves the writing of Maple code to the student.

  • Another Population Model, Equilibria and Stability
  • Another Population Model, Equilibria and Stability
    These projects discuss a population model that requires a certain critical mass of inhabitants for survival of the population, as well as equilibria and stability. The first project specifies the Maple code to be used, and the second leaves the writing of Maple code to the student.

  • Ballistics with Air Resistance
    This project discusses the ballistics problem assuming air resistance is an effective force. This project builds on previous work in which air resistance was ignored.

  • Euler's Method
    This project discusses Euler's Method, both numerically and graphically.

  • Differential vs. Difference Equations
    This project presents essentially the same analysis that is the basis for Euler's method, but from a slightly more general point of view.


    Chapter 11: Parametric Equations and Polar Coordinates

    In various projects - such as the first project in this chapter on Parameterized Curves - students are encouraged to use their creativity to draw something interesting. To make things a little more interesting for the class as a whole, you might want to consider offering a small prize - such as a candy bar - for the "best" offering.

  • Parametrized Curves
  • Parametrized Curves
    In these projects the student is led through the graphing of several parameterized curves in the plane. In the first project, the code is provided; in the second project, the code is provided for one curve, and the student is then expected to write the code for the others.

  • Ballistics
  • Ballistics
  • Ballistics
  • Ballistics
    In these projects the student investigates the path of a projectile in xy-coordinates. (Earlier they were investigated in ts-coordinates.) The first and second of these projects uses English units, and the later two use metric units. The first and third specifies the Maple code to be used, and the second and fourth leave writing the Maple code to the student.

  • Polar Curves in Cartesian Coordinates
    This project illustrates how to draw polar curves in Cartesian coordinates using Maple.

  • Cardioids and Limacons
    This project asks the student to draw several cardioids and limacons in the same graphic. The objective is some insight into the geometric significance of the constants a and b that specify curves whose polar equations are of the form r = a + b sin theta and r = a + b cos theta.


    Chapter 12: Infinite Sequences and Series

  • Sequences and Series
    This project leads the student through a numerical and graphical analysis of infinite series.

  • Sequences and Series: Part 1
  • Sequences and Series: Part 2
    The first of these projects has the student define two procedures - one for generating some values for, and another for generating some graphics for, a user-specified series - and then apply them to an analysis of the series. The second project asks the student to repeat the first project using some different series.

  • Maclaurin Series
    In this project the student creates and graphs some truncated Maclaurin series expansions.

  • The Second Derivative Test Again
    This views the Second Derivative Test from the standpoint of series expansions, and extends the Second Derivative Test.


    Chapter 13: Vectors and the Geometry of Space

    In most Calculus sequences, the remaining material deals with multivatiate Calculus. Since visualization and being able to draw objects in space is important in learning this material, we emphasize graphics in many of the projects.

  • 3-Dimensional Coordinate Systems
    This project has the student draw several basic geometric figures in space.

  • Vector Algebra, Lines and Planes
    This project illustrates how to find the vector equations of lines and planes in space with Maple.

  • Visualizing and Drawing Lines and Planes
    In this project we illustrate how to visualize and draw lines and planes in space.

  • Cylinders and Quadric Surfaces
  • Cylinders and Quadric Surfaces
    These projects have the student drawing and identifying cylinders and quadric surfaces in space. The first specifies the Maple code to be used, and the second leaves the writing of part of the Maple code to the student. Some parts of these projects can be omitted if there seem to be too many.

  • Quadric Surfaces
    In this project the student is asked to identify the graphs of several quadric surfaces from implicit graphs of their second order quadratic equations in x, y, and z.

  • Rectangular, Cylindrical and Spherical Coordinates
  • Rectangular, Cylindrical and Spherical Coordinates
    In this project we graph the basic coordinate surfaces in rectangular, cylindrical and spherical coordinates. The first project specifies the Maple code to be used, and the second leaves writing the Maple code to the student.

  • Surfaces of Revolution
    In this project the student uses implicit equations to graph some surfaces of revolution.


    Chapter 14: Vector Functions

  • Lines and Curves in Space
    This project has the student draw a line a two curves in space, as well as the projection planes of the line and two projection cylinders of one of the curves.

  • Curves in Space
  • Curves in Space
  • Curves in Space
    This project has the student drawing several curves in space.

  • Curvature of a Curve
  • Curvature of a Curve
    These projects are aimed at verifying that the definition of curvature is intuitively correct. The first project specifies the Maple code to be used, and the second leaves writing the Maple code to the student.

  • Motion Around a Circular Path
    This project presents an analysis of motion along a circular path. Several important formulas are derived. This project does not use Maple: all computations can be done by hand.


    Chapter 15: Partial Derivatives

  • Graphs of Functions of Two Variables
  • Graphs of Functions of Two Variables
  • Graphs of Functions of Two Variables
    These projects are slightly different variations of each other. In all three the student graphs functions of two variables.

  • Partial Derivatives
    In this project we emphasize the geometric significance of the partial derivatives of a function of two variables.

  • Tangent Planes
    In this project we discuss the similarities and differences between computing the equation of the tangent plane to the graph of a function and the equation of the tangent plane to the graph of a level surface.

  • Function Optimization
  • Function Optimization
    These projects are aimed at clearly identifying the process of finding the relative extrema for a function of two variables over an open set. The first project specifies the Maple code to be used, and the second leaves writing the Maple code to the student.

  • Function Optimization over a Closed Set
    This project illustrates the process of finding the absolute extrema of a function of two variables over a closed set.

  • Solving Constrained Optimization Equations
    This project illustrates how to use Maple to solve the systems of equations that arise in constrained optimization problems.

  • Constrained Optimization
    This project illustrates the geometry behind Lagrange Multipliers.

  • Lagrange Multipliers
    This project illustrates how to solve Lagrange Multiplier problems analytically, and the geometry behind Lagrange Multipliers.

  • Lagrange Multipliers
    This project illustrates how to solve Lagrange Multiplier problems analytically, and includes several more for the student to do.


    Chapter 16: Multiple Integrals

  • Double Integrals over Type I and Type II Regions
  • Double Integrals over Type I Regions
  • Double Integrals over Type II Regions
    These projects illustrate how to set up and evaluate double integrals over Type I and Type II regions in the plane.

  • Visualizing Regions in Space
    This project illustrates one method for doing clipping. The focus here is on clipping as an aid to setting up iterated integrals.

  • Double Integrals
    This project is based on a general discussion of double integrals.

  • Double Integrals in Polar Coordinates
    This project examines the connection between polar and rectangular (r,theta)-coordinates.

  • Transformations of the Plane
  • Transformations of the Plane
    In these projects a test pattern in the plane is defined and transformed by various transformations. The first project specifies the Maple code to be used, and the second leaves writing the Maple code to the student.

  • Rotation of Coordinate Axes
    This project presents a brief discussion of rotation of coordinate axes.

  • Transformation from Rectangular to Polar Coordinates
    This project examines the connection between polar and rectangular (r,theta)-coordinates as a transformation.

  • Transformations of Space
    This project examines some simple transformations of space.

  • Polar Rectangles
    In this project the student uses a procedure to draw several polar rectangles. The focus here is on setting up the limits on double integrals in polar coordinates.

  • Cylindrical Boxes
    In this project the student uses a procedure to draw some cylindrical boxes. The focus here is on setting up the limits on triple integrals in cylindrical coordinates.

  • Spherical Boxes
    In this project the student uses a procedure to draw some spherical boxes. The focus here is on setting up the limits on triple integrals in spherical coordinates.


    Chapter 17: Vector Calculus

  • Vector Fields in the Plane and in Space
    This project illustrates how Maple can be used to define vector fields and draw them in the plane and in space.

  • The Gradient
  • The Gradient
    In these projects the student is asked to draw the graph of a function f = f(x,y), a contour plot of f, and the gradient field of f, and then to relate the graphics. The difference between these projects is the function f.

  • Integral Curves
    This project asks the student to draw some integral curves of a vector field. It also relates the integral curves of a conservative field to the level curves of the potential function.

  • Divergence
  • Divergence
    In these projects the student is asked to work with the divergence of some simple vector fields. The emphasis is on understanding what the divergence represents.

  • Line Integrals in the Plane
  • Line Integrals in the Plane
    These projects discuss the computation of line integrals in the plane. The difference between them is that the first makes reference to Green's Theorem and the Fundamental Theorem of Line Integrals and the second does not.

  • Line Integrals in the Plane, and Work
    This project discusses the computation of line integrals in the plane, and emphasizes their interpretation in terms of work.

  • Drawing Surfaces
    This project illustrates how to draw parametric surfaces in Maple, and the strengths and weaknesses of drawing surfaces implicitly, as graphs of functions, and parametrically.

  • Parameterized Surfaces
    This project illustrates how to draw parametric surfaces in Maple.

  • A Comparison of Approaches to Drawing Surfaces
    In this project we compare the graphs of a cone obtained through implicit, function, and parametric plotting.

  • Different Strokes ...
    This project requires student to write code in Maple. In this project we illustrate that different types of information can come from viewing surfaces that have been graphed using different parameterizations.

  • Surface Area and Surface Integrals
  • Surface Area and Surface Integrals
    The goal of these projects is to clarify the steps to be taken in the computation of surface area and surface integrals. The case considered here is for parameterized surfaces.

  • Surface Area and Surface Integrals
    The goal of these projects is to clarify the steps to be taken in the computation of surface area and surface integrals. The case considered here is for surfaces that are graphs of functions (Monge patches).

  • The Flux of a Vector Field through a Surface
  • The Flux of a Vector Field through a Surface
    The student is led through the computation of the flux of a vector field through a surface. The primary goal of this project is to clarify the steps involved in computing flux. The first project deals with parameterized surfaces, and the second deals with graphs of functions (Monge patches).


    Chapter 18: Second-Order Differential Equations



    COPYRIGHT ©2005 Brooks/Cole, a division of Thomson Learning, Inc.