In the summary that follows, links to chapters, sections, or activities in the first edition are denoted with a parenthetical (1e), while those in the new second edition are denoted with (2e).
All of the 1e links are stable at the new URL https://activecalculus.org/single1e/; at the time of this writing (July 22, 2025), the 2e preview links were from an in-development version of the text via GitHub pages. [8/11 update: all of the 2e links in the original post have been updated to their stable, permanent ones (using “/single2e/”)].
The new second edition has some significant updates to Chapters 1 and 3, and replaces the original Chapter 8 with a different chapter focused on Taylor polynomials and series. In more detail:
- In Chapter 1, the main changes are to Sections 1.5, 1.6, and 1.8.
The new Section 1.5 (2e) has been largely rewritten to include more emphasis on using precise language and proper units to interpret derivatives in meaningful contexts, including several major changes to some of the activities. Preview Activity 1.5.1 (1e) has been replaced with an entirely different scenario in the new Preview Activity 1.5.1 (2e); Activity 1.5.3 (1e) has been replaced and re-ordered with the new Activity 1.5.2 (2e) that is more focused on language and meaning in multiple contexts; and the original Activity 1.5.2 (1e) has been revised to reflect realistic data in the updated and renumbered Activity 1.5.3 (2e).
In Section 1.6, Activity 1.6.3 (1e) has been updated to have realistic data in Activity 1.6.3 (2e) that matches updated one in Section 1.5.
In Section 1.8, there’s a new activity, Activity 1.8.4 (2e), that focuses on quantifying the error in linear approximations and helps students see the role that the size of the second derivative plays in that error. (Previously there were only two activities in Section 1.8; now there are three.) - While the content of Chapter 3 remains largely unchanged, the order of the sections has changed in the 2nd edition, and the final section of Chapter 2 has been moved into the early part of Chapter 3. The following table summarizes the changes:
| Chapter 3 in 1st edition | Chapter 3 in 2nd edition |
| 3.1 (1e) – Using derivatives to identify extreme values | 3.1 (2e) – Related rates |
| 3.2 (1e) – Using derivatives to describe families of functions | 3.2 (2e) – L’Hopital’s Rule (which is Section 2.8 (1e) in the first edition) |
| 3.3 (1e) – Global optimization | 3.3 (2e) – Using derivatives to identify extreme values |
| 3.4 (1e) – Applied optimization | 3.4 (2e) – Using derivatives to describe families of functions |
| 3.5 (1e) – Related rates | 3.5 (2e) – Global optimization |
| 3.6 (2e) – Applied optimization |
It remains the case that the sections on related rates and L’Hopital’s Rule are independent of the other sections in Chapter 3 and can be studied at any time. But this new ordering is intended to place related rates immediately after implicit differentiation so that the skills developed in Section 2.7 (implicit differentiation) can be put to use right away. In addition, this provides some space in time and content between two of the most challenging topics in first semester calculus (related rates and applied optimization).
In the renumbered Section 3.3 (2e) on using derivatives to identify extreme values, the Activity 3.3.2 (2e) has been revised to focus on a different rational function, but one that doesn’t have any vertical asymptotes like the original Activity 3.1.2 (1e).
- Chapter 8 from the 1st edition offers a full treatment of series convergence. In Fall 2022, I released the “alternate ending” version of the text with a new Chapter 8 instead focused on Taylor polynomials and Taylor series, with the only formal convergence test being the Ratio Test. This different Chapter 8 has been included in a version of the text posted most recently as the “alternate” edition on activecalculus.org. In an August 2022 post, I shared my rationale for this big change: that the Taylor series representations of the sine, cosine, and exponential functions are the most important series-related ideas in calculus 2, and the details of convergence tests are not.
In the 2nd edition, the “alternate” chapter 8 becomes canonical: Chapter 8 (2e). This summer, with support from some colleagues who have used the earlier alternate Chapter 8 (big thanks to Chrissy Safranski and Spencer Bagley!), I’ve substantially revised my original work in chapter 8 plus I have added a new section that includes the Lagrange Error Bound (Section 8.6 (2e)) as part of a discussion of how we determine the accuracy of approximations.
A few other minor edits have been made to improve clarity or fix minor errors, but none of them are significant enough to merit inclusion here.
Going forward from August 2025 on, no changes will be made to the first edition (not even to fix minor typos or errors), as all future work will be focused on the second edition and additional developments.
