Mathematica

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Resource

Pages

This is where my students can come to download the labs we will be doing in class. I have also added some more general labs that others might find interesting. This is a work "in progress", so I am open to suggestions for other topics, improvements on these labs, and corrections.

**Before you begin** - Check out my Lab
Submission Guidelines.

**Lab 1: Functions, Graphs, and Integrals**The composition of functions of more than one variable, functions as arguments, ParametricPlot3D, restrictions on functions. Using Mathematica to help you visualize and compute double and triple integrals. (Mathematica notebook - 3.4 megabyte file)

**Lab 2a: Non-Cartesian Coordinate Systems - Part a**Using a change of coordinates to evaluate integrals over difficult regions of integration. Mathematica's built-in non-Cartesian coordinate systems and how to visualize them. (Mathematica notebook)

**Lab 1: Curves and Surfaces**Curves and curvature. Functions of 2 variables and limits. (Mathematica notebook. - 6 megabyte file.)

**Lab 2a: Surfaces and Optimization**Surfaces in spherical and cylindrical coordinates. Optimization. (Mathematica notebook)

**Lab 1: Curves, Surfaces, and Optimization**Curves and curvature. Functions of 2 variables, cylindrical and spherical coordinates, and limits. Optimization and constrained optimization. (Mathematica notebook. - 6 megabyte file.)

**Lab 1a: Functions, Graphs, and Integrals - Part a**The composition of functions of more than one variable, functions as arguments, ParametricPlot3D, restrictions on functions. (Mathematica notebook - 1.6 megabyte file) If you downloaded the original version of this, I have deleted one of the last graphing problems (the one in parametric cylindrical coordinates), so be sure you double-check.

**Lab 1b: Functions, Graphs, and Integrals - Part b**Using Mathematica to help you visualize and compute double and triple integrals. (Mathematica notebook - 1.6 megabyte file)

**Lab 2a: Non-Cartesian Coordinate Systems - Part a**Using a change of coordinates to evaluate integrals over difficult regions of integration. Mathematica's built-in non-Cartesian coordinate systems and how to visualize them. (Mathematica notebook - 5.6 megabyte file. A much smaller file with the output deleted can be downloaded here, if you have a slowish connection. You will need to re-execute everything.)

**Lab 2b: Non-Cartesian Coordinate Systems - Part b**Using a change of coordinates to evaluate integrals over difficult regions of integration. Mathematica's built-in non-Cartesian coordinate systems and how to work with them. (Mathematica notebook - 2 megabyte file.)

**Lab 3: Integration**Nothing very deep here; just some different types of integrals (path, line, surface, etc.) to work. You must also identify which can be worked using the different integral theorems we have studied and verify the theorems are indeed true on these. (Mathematica notebook)

**Lab 1a: Functions, Graphs, and Integrals - Part a**The composition of functions of more than one variable, functions as arguments, ParametricPlot3D, restrictions on functions. (Mathematica notebook - 1.6 megabyte file) If you downloaded the original version of this, I have deleted one of the last graphing problems (the one in parametric cylindrical coordinates), so be sure you double-check.

**Lab 1b: Functions, Graphs, and Integrals - Part b**Using Mathematica to help you visualize and compute double and triple integrals. (Mathematica notebook - 1.6 megabyte file)

**Lab 2a: Non-Cartesian Coordinate Systems - Part a**Using a change of coordinates to evaluate integrals over difficult regions of integration. Mathematica's built-in non-Cartesian coordinate systems and how to visualize them. (Mathematica notebook - 5.6 megabyte file. A much smaller file with the output deleted can be downloaded here, if you have a slowish connection. You will need to re-execute everything.)

**Lab 2b: Non-Cartesian Coordinate Systems - Part b**Using a change of coordinates to evaluate integrals over difficult regions of integration. Mathematica's built-in non-Cartesian coordinate systems and how to work with them. (Mathematica notebook - 2 megabyte file.)

**Lab 3: Integration**Nothing very deep here; just some different types of integrals (path, line, surface, etc.) to work. You must also identify which can be worked using the different integral theorems we have studied and verify the theorems are indeed true on these. (Mathematica notebook)

**Lab 0 - Using Mathematica to Solve Differential Equations**This isn't really a true lab, but it is an introduction to how to use Mathematica to solve differential equations and systems of differential equations. Use this for reference and to do the other labs. Warning: This explanation isn't complete, though it should have plenty of information in it to get you started (you won't need all of it right away). (html / Mathematica notebook - 6 Megabyte file / Mathematica notebook with all output deleted - much smaller file, but you have to execute the commands to see the output)

**Lab 1 - Equilibrium behavior, sensitivity to initial conditions, and an example of population growth**

This lab covers techniques for analyzing autonomous equations, investigating the sensitivity of a first order equation to initial conditions and how that effects approximation methods, and investigates another model of population growth. (html / Mathematica notebook)

**Lab 1: Limits and Functions**Limits of functions of two variables, the composition of functions of more than one variable, functions as arguments, ParametricPlot3D, and graphing too many dimensions. (html / Mathematica notebook - 6 megabyte file / Mathematica notebook with all output deleted - much smaller file, but you have to execute the commands to see the output)

**Lab 2: Non-Cartesian Coordinate Systems**How to work in non-Cartesian coordinate systems. This examines graphing, differential operators, integration, and Mathematica's built-in support for doing these things in different non-Cartesian coordinate systems. (html / Mathematica notebook - Long form - 11 Mb / Mathamtica notebook - Short form with all the output deleted - 60 Kb - If you download the short form, you should go to the

*Kernal*menu, choose*Evaluation*, and then*Evaluate Notebook.*)

**Lab 0 - Using Mathematica to Solve Differential Equations**This isn't really a true lab, but it is an introduction to how to use Mathematica to solve differential equations and systems of differential equations. Use this for reference and to do the other labs. Warning: This explanation isn't complete, though it should have plenty of information in it to get you started (you won't need all of it right away). (html / Mathematica notebook - 6 Megabyte file / Mathematica notebook with all output deleted - much smaller file, but you have to execute the commands to see the output)

**Lab 1 - Equilibrium behavior, sensitivity to initial conditions, and an example of population growth**

This lab covers techniques for analyzing autonomous equations, investigating the sensitivity of a first order equation to initial conditions and how that effects approximation methods, and investigates another model of population growth. (html / Mathematica notebook)

**Lab 1: Limits and Functions**Limits of functions of two variables, the composition of functions of more than one variable, functions as arguments, and ParametricPlot3D. (html / Mathematica notebook)

**Lab 2: Non-Cartesian Coordinate Systems**How to work in non-Cartesian coordinate systems. This examines graphing, differential operators, integration, and Mathematica's built-in support for doing these things in many different non-Cartesian coordinate systems. (html / Mathematica notebook - Long form - 8 Mb / Mathamtica notebook - Short form with all the output deleted - 50 Kb - If you download the short form, you should go to the

*Kernal*menu, choose*Evaluation*, and then*Evaluate Notebook.*)

**Lab 0 - Using Mathematica to Solve Differential Equations**This isn't really a true lab yet, but it is an introduction to how to use Mathematica to solve differential equations and systems of differential equations. Use this for reference and to do the other labs. Warning: This explanation isn't complete, though it should have plenty of information in it to get you started (you won't need all of it right away). (html / Mathematica notebook - 6 Megabyte file / Mathematica notebook with all output deleted - much smaller file, but you have to execute the commands to see the output)

**Lab 1 - Equilibrium behavior, sensitivity to initial conditions, and an example of population growth**

This lab covers techniques for analyzing autonomous equations, investigating the sensitivity of a first order equation to initial conditions and how that effects approximation methods, and investigates another model of population growth. (html / Mathematica notebook)

**Lab 1: Limits and Functions**Limits of functions of two variables, the composition of functions of more than one variable, functions as arguments, and ParametricPlot3D. (html / Mathematica notebook)

**Lab 2: Non-Cartesian Coordinate Systems**How to work in non-Cartesian coordinate systems. This examines graphing, differential operators, integration, and Mathematica's built-in support for doing these things in many different non-Cartesian coordinate systems. (html / Mathematica notebook - Long form - 8 Mb / Mathamtica notebook - Short form with all the output deleted - 50 Kb - If you download the short form, you should go to the

*Kernal*menu, choose*Evaluation*, and then*Evaluate Notebook.*)

**Lab 1: Limits and Functions**Limits of functions of two variables, the composition of functions of more than one variable, functions as arguments, and ParametricPlot3D. (html / Mathematica notebook)

**Lab 2: Non-Cartesian Coordinate Systems**How to work in non-Cartesian coordinate systems. This examines graphing, differential operators, integration, and Mathematica's built-in support for doing these things in many different non-Cartesian coordinate systems. (html / Mathematica notebook - Long form - 8 Mb / Mathamtica notebook - Short form with all the output deleted - 50 Kb - If you download the short form, you should go to the

*Kernal*menu, choose*Evaluation*, and then*Evaluate Notebook.*)

**Lab 3: Vector Integration**Nothing very deep here; just some different types of integrals (path, line, surface, etc.) to work. You must also identify which can be worked using the different integral theorems we have studied and verify the theorems are indeed true on these. (html / Mathematica notebook)

**Lab 0 - Using Mathematica to Solve Differential Equations**This isn't really a true lab yet, but it is an introduction to how to use Mathematica to solve differential equations and systems of differential equations. Use this for reference and to do the other labs. Warning: This explanation isn't complete, though it should have plenty of information in it to get you started (you won't need all of it right away). (html / Mathematica notebook - 6 Megabyte file / Mathematica notebook with all output deleted - much smaller file, but you have to execute the commands to see the output)

**Lab 1 - Equilibrium behavior, sensitivity to initial conditions, and an example of population growth**

This lab covers techniques for analyzing autonomous equations, investigating the sensitivity of a first order equation to initial conditions and how that effects approximation methods, and investigates another model of population growth. (html / Mathematica notebook)

**Extra credit lab**Finding Taylor series expansions of a function and the error introduced by using a finite Taylor polynomial. (html / Mathematica notebook)

**Lab 1: Limits and Functions**Limits of functions of two variables, the composition of functions of more than one variable, and ParametricPlot3D. (html / Mathematica notebook)

**Lab 2: Non-Cartesian Coordinate Systems**How to work in non-Cartesian coordinate systems. This examines graphing, differential operators, integration, and Mathematica's built-in support for doing these things in many different non-Cartesian coordinate systems. (html / Mathematica notebook - Long form - 8 Mb / Mathamtica notebook - Short form with all the output deleted - 50 Kb - If you download the short form, you should go to the

*Kernal*menu, choose*Evaluation*, and then*Evaluate Notebook.*)**Lab 3: Vector Integration**Nothing very deep here; just some different types of integrals (path, line, surface, etc.) to work. You must also identify which can be worked using the different integral theorems we have studied and verify the theorems are indeed true on these. (html / Mathematica notebook)

**Lab 0 - Using Mathematica to Solve Differential Equations**This isn't really a true lab yet, but it is an introduction to how to use Mathematica to solve differential equations and systems of differential equations. Use this for reference and to do the assignement I announced in class. I hope to have a real lab or three for you soon, but we'll see how time goes. Warning: This explanation isn't complete, though it should have plenty of information in it to get you started. (html / Mathematica notebook)

**Lab 1 - Damped Harmonic Oscillation with a Driving Force**

This is just the assignment I gave in class on Wednesday, but fleshed out a little. It asks you to give graphical examples of resonance, beating, and the 3 different types of damping for a driven, damped harmonic oscillator. (html / Mathematica notebook)

**Lab 1: The**This lab introduces the*Rule of Three*and Other Oddities (some College Algebra content, some Calculus level content)

*Rule of Three*which has been a popular educational idea over the last few years. The idea is that you approach problems from three different perspectives: numerically, graphically, and symbolically. In seeing the same problem from three different "angles", you sometimes develop a better feel for how the problem works. While I think this idea can be abused, it does offer a useful approach to some types of problems. This lab investigates some simple examples that utilize this approach. Some of the later labs make use of these ideas in a less formal way, but with more sophistication. WARNING: THIS LAB IS NOT COMPLETE YET. USE AT YOUR OWN RISK... (html / Mathematica notebook)**Lab 2: Curve Fitting Population Growth Data (College Algebra and above level)**This lab investigates fitting several different curves to some data on the population of the U.S. over time. It compares the different "fits" and considers what happens when more data is available. (html / Mathematica notebook)

**Lab 3: Numerical Integration and Error Approximation (Calculus II and above level)**This lab investigates the simpler methods of approximating a definite integral numerically (left-hand sum, right-hand sum, Trapezoid Rule, Midpoint Rule, and Simpson's Rule) and the errors in these methods as a function of the number of subdivisions. (html / Mathematica notebook)

**Lab 4: Error Approximation in Taylor Series (Calculus II and above level)**

This lab investigates using a finite Taylor polynomial to approximate a function. In particular, it exames how to find the domain for a Taylor polynomial of given degree such that its error will be within a desired range. It also investigates the case where you need to find how many terms of a Taylor series to use to approximate a function to within a desired error tolerance over a given interval. (html / Mathematica notebook)

**Lab 1: Limits and Functions**Limits of functions of two variables, the composition of functions of more than one variable, and ParametricPlot3D. (html / Mathematica notebook)

**Lab 2: Non-Cartesian Coordinate Systems**This lab discusses how to visualize non-Cartesian coordinate systems in 3 dimensions. It also discusses the functions Mathematica has built-in to handle this. There is some discussion of integration in these coordinate systems as well. (html / Mathematica notebook - I have deleted all the output in the notebook to save space. After you download this file, you should go to the

*Kernal*menu, choose*Evaluation*, and then*Evaluate Notebook.*)**Lab 3: Vector Integration**This is actually a pretty short lab (since we are short on time due to the technical problems we had before we got the Mathematica updata earlier in the semester). I simply ask students to work the integrals from the last test in Mathematica. (html / Mathematica notebook)

**Lab 1: Polar and Parametric Equations**Lab 1 comes in two parts:

**1a - Polar Equations**:

Finding points of intersection of two polar graphs, finding the area between two polar graphs, and finding the arc-length of a polar graph. (html / Mathematica notebook)**1b - Parametric Equations**:

Use parametric equations to take a closer look at the cycloid, epicycloid, and the hypocycloid. (html / Mathematica notebook)

**Lab 2: 3-Dimensional Graphs**Lab 2 comes in two parts:

**2a - Curves**

Curvature and its relationship to the graph. Curves on a surface. (html / Mathematica notebook)**2b - Surfaces**:

Functions of two variables. Graphs in spherical and cylindrical coordinates. Limits of functions of two variables. (html / Mathematica notebook)

**Lab 3: Optimization**- Optimization without constraints, curve fitting using the method of least squares, and optimization with constraints. (html / Mathematica notebook)

**Lab I - Polar and Parametric Equations**(including a closer look at the cycloid, epicycloid, and hypocycloid) (Mathematica notebook)**Lab 2 - 3-Dimensional Graphs: Curves and Surfaces**(Mathematica notebook)**Lab 3 - Optimization (constrained and unconstrained, with a look at fitting a curve to experimental data)**(Mathematica notebook)

These will be added as I finish them.