Saint Louis University

Department of Mathematics and Mathematical Computer Science

Maple Worksheets for Calculus III

(For Maple 10)

At Saint Louis University, we are using graphing calculators as the primary technology incorporated into teaching the first two semesters of our main calculus sequence. The easy arguments for graphing calculators are that they are relatively inexpensive (about the cost of a standard calculus textbook), moderately easy to learn to use, and are in the students' hands so that the students have the same set of tools available in class, for homework, and during quizzes and tests. Hopefully they will understand that the tools are tools for doing mathematics rather than tools for working in mathematics classes. Quite simply, we felt that graphing calculators made the best fit for the needs of the calculus students we have.

Nevertheless, as the students move through the calculus sequence there are topics where a computer algebra system is much more effective than a graphing calculator. This is particularly true in multi-variable calculus with the need to visualize in 3 dimensions. Our third semester calculus is being taught in a computer classroom where, among other tools, the students have access to the computer algebra system Maple.

One of the drawbacks of a program as powerful as Maple is that the difficulties of learning Maple and learning to use a computer need to be consciously factored in when planning the course. One pedagogical strategy is to make using Maple a routine part of the course and consistently teach use of the program along with the mathematics. This strategy works best if Maple is to be used heavily in a sequence of courses.

A second strategy (and the one I have used) is to introduce Maple through carefully designed worksheets. Depending on the material I was covering, the worksheets were designed to be used as one of the following:

• An in class lecture aid with the instructor running the worksheet with a projection system.
• A handout to be printed up, copied, and handed out to the students.
• A lab assignment that the class will start together as a substitute for a lecture.
• A supplemental homework assignment that the students are expected to find and do on their own.

Worksheets to be done by the student are set up so that the first time through a student can get through the worksheet by hitting enter repeatedly. These worksheets include a significant amount of exploratory text. The exercises tend to ask to student to repeat the examples from the worksheets with minor modification. I use the template model because I want them to use the power of the Maple to look at problems where I could not expect them to produce the code, but I can expect them to copy and modify a code template, focusing on the results of the problems.

With that long winded introduction, here are some worksheets I produced for topics in Calculus III. We are using the "Harvard Calculus" book for this course, so the worksheets are organized in line with that book. Most of the material is pretty standard across multivariable calculus texts.

You can download a zipped archive of the Calculus 3 Maple 10 worksheets.

• Preliminary
  • Just Enough Maple for worksheets - A worksheet designed to introduce the students to just enough Maple syntax for them to be able to run worksheets.
  • Writing Maple 10 documents - A worksheet to walk the students through some of the new features of the document interface in Maple 10.
    
• Chapter 12
  • Plot With Maple - An introduction to plotting with Maple, reviewing 2D plotting and introducing plotting in 3D.
    
  • Understanding Limits - is a worksheet on the formal definition of limit for functions of two variables. To simplify graphing, square neighborhood in the domain are used in the definition.
• Chapter 13
  • Cross and Dot Products - a fast handout explaining the syntax used for dot and cross product in Maple. The intent of this worksheet is simply to show the students how to use Maple to check their work with vector computations.
  • Visualizing Vectors - is a demonstration worksheet to show the students how to visualize vectors, vector arithmetic, and cross products with Maple. The worksheet does not include exercises.
• Chapter 14
  • Partial Derivatives - walks the students through the syntax for finding partial derivatives with Maple.
    
  • Easy Tangent Planes - is a worksheet that starts with an easy construction of tangent planes. We construct tangent planes to a surface by finding the lines tangent to the paths that fix x and y in turn.
    
  • Local Linearity - Looks at visualizing the book's definition, that a function is differentiable at a point if the graph near the point is locally approximated by the tangent plane. The definition is used to understand the corresponding delta-epsilon definition.
  • Visual Gradients - is a worksheet that visualizes directional derivatives and connects them to the gradient for differentiable functions.
  • Multivariable Chain rule - walks the students through the chain rule in several variables, starting with a visualization of the chain rule in a single variable.
    
  • Taylor Series - steps the students through the construction of Taylor polynomials for functions of 2 variables.
  • Animated Taylor Series - Shows an animation of the Taylor polynomial wrapping down to the surface. (This was broken off the other worksheet on this section due to the memory demands of animated 3D-graphics.)
  • Checking Differentiability- Steps the student through the process of checking differentiability of a function at a point.
  • Checking Differentiability, an example -Since the student have such trouble with this section, this is a worked example that looks at checking differentiability.
• Chapter 15
  • Rotations - is a worksheet to connect the use of the discriminant with rotations of axes and the elimination of the cross term in polynomials of degree 2.
    
  • Lagrange Multiplier Problem - an extra credit homework worksheet on section 14.3. It shows how to use maple to solve systems of equations.
  • Solving with a Gradient Search - Shows how to mechanize a gradient search to find a solution to a system of equations.
• Chapter 16
  • Integration Checker - is a fast note that demonstrates for the students the Maple commands needed to do integration. It seems useful in this chapter where many of the problems reduce to "and finish by evaluating the two or three integrals."
  • Plotting in Other Coordinate Systems - Is another "refresher in Maple commands." It looks at how to plot in coordinate systems other than Cartesian. It also shows how to combine objects described in different coordinate systems on a single plot.
  • Multiple Riemann sums - The worksheet looks at the Riemann sum definition of double integrals. It follows the usual pattern of the course by reviewing the definitions in the one variable case, then generalizing.
  • Double Integrals (Cartesian) - The worksheet looks at visualizing limits of integration in Cartesian coordinates in 2 dimensions and in changing the order of integration.
  • Triple Integrals (Cartesian) - This is similar to the double integral worksheet, but with triple integrals.
  • Extended Triple Integral Example - This is an extended example for setting up a triple integral.
  • Extended Triple Order of Integration Example - This gives an extended example of changing the order of integration for a triple integral.
    
  • Double integrals in Polar Coordinates - This Worksheet was done by the students in class. It looks at setting up integrals in polar coordinates and switching between rectangular and polar coordinates.
• Chapter 17
  • Parametric Curves - Looks at parameterizing curves in R2 and R3
  • Parametric Surfaces - Demo - This worksheet looks at parameterization of surfaces.
  • Planetary motion - This worksheet steps through parameterizing planetary motion under gravity. This is a demonstration worksheet. It solves for planetary motion with a flow line solution of a vector field in 4 dimensions.
  • Vector Fields and Gradient Fields - is a worksheet that has the students look at plotting vector fields and gradient fields in 2 and 3 dimensions.
  • Plotting Flow Lines - Looks at plotting flow lines for vector fields in 2 and 3 dimensions. It also shows how Euler's method works as a numerical method.
• Chapter 18
  • Line Integrals - This worksheet steps the students through the procedure of setting up and evaluating a line integral along a parameterized curve. It is intended as a homework checker.
• Chapter 19
  • Flux Integrals - This worksheet steps the students through the procedure of setting up and evaluating a flux integral through a parameterized surface. It is intended as a homework checker.