In teaching a second-semester calculus course for college students who have studied some calculus in high school, I think that using Calculus in Context is the way to go. Covering Chapters 1 and 2 reviews Calc I concepts in the context of modelling diseases, Chapters 4 and 8 apply these concepts to differential equations and dynamical systems, and Chapters 10 and 11 cover sequences/series and techniques of integration.
While teaching such a class at Smith College, I developed Mathematica notebooks to accompany the course, and here they are for others to use. These notebooks were last revised in Fall 2007; they all work with Mathematica 7.0. As far as I know, Mathematica commands are backwards-compatible with Mathematica 7.0, so these notebooks still work. There are no notebooks for Taylor series or techniques of integration, because Mathematica has built-in commands for use with these ideas.
While teaching a similar class at Sarah Lawrence College in Spring 2013, I created a matching set of Sage notebooks to accompany the course. In 2017, I created Jupyter notebook versions of them
(1) Introduction to
Mathematica: I use this on the first day of class to introduce
students to Mathematica command structure/syntax, show them how to get help,
and start them on the most basic S-i-R code.
Introduction to Sage (SageNB) (Jupyter) and Basic S-I-R code (SageNB) (Jupyter)
(2) Flexible S-i-R Code: For use with
Chapter 1. This contains more-customizable S-i-R code, two ways to implement
there-and-back, and basic data generation and plotting commands. If you look
ahead to other notebooks, you may wonder why the code is so rudimentary in this
notebook; the reason is to help students understand how the code works before
having them use it more like a black box later.
There-and-Back and Plotting (SageNB) (Jupyter)
(3) Interesting and Fancy Data
Plotting: Still using the S-i-R example, this notebook helps students plot
multiple sets of data on the same axes, animate the display of data sets, and
generate and plot successive approximations to the solutions of a system of
differential equations. There's also a section with instructions for adding
mouseovers, color, labels, etc.
Fancy Data Plotting (SageNB) (Jupyter)
(4) Chapter 2 Exercises:
Unsurprisingly, this is for use with Chapter 2. It contains code for
somewhat-tabular display of S-i-R data, and several exercises designed by Jim
Callahan to help students analyze issues including the stabilization (or lack
thereof) of successive approximations, and interpret the numerical results of
running various scenarios through S-i-R code.
Chapter 2 Exercises (SageNB) (Jupyter)
Investigations: This reviews the measles epidemic analysis and then sets up
an analysis of seasonal influenza spread customized to your college or
Epidemic Investigations (SageNB) (Jupyter)
(5) Play with the May Model: For use
with Chapter 4. The code in this notebook is designed to let students vary
parameters in systems of differential equations and see how these changes
affect the corresponding solutions. The example function is the May Model, but
Lotka-Volterra and other models can easily be implemented by altering the code
appropriately. The Manipulate function makes its first
appearance here, and is particularly helpful in finding Hopf bifurcations.
Play with the May Model (SageNB) (Jupyter)
(6) Fun with Vector Fields: For
use with Chapter 8. Here there are customizable chunks of code to generate and
display data trajectories atop vector fields (and of course to generate/display
trajectories separately from vector fields), and more
Fun with Vector Fields (SageNB) (Jupyter)
(7) Lorenz Attractor Craziness!
What would a study of dynamical systems be without a sample Lorenz Attractor to
play with? This notebook generates a 3D trajectory for the Lorenz Attractor and
an accompanying vector field, and the display is live-rotatable. Yes. There are
also projections of the trajectory into the three 2D coordinate planes. This
code can be adapted to show S-i-R data in 3D.
Lorenz Attractor (SageNB) (Jupyter)