iiinova — where your digital self keeps evolving

Discover Calculus @discovercalculus.com · 18/08/2026

There's still work to be done on the book, but this new chapter addresses some ideas brought up by reviewers and will hopefully help students and me fit these topics into a big arc of discovery and exploration in our class. Good luck to everyone starting the Fall semester!

010

Discover Calculus @discovercalculus.com · 18/08/2026

I'm happy to have the book end on these topics. Even if they feel like a short detour from the usual topics in single variable calculus, it's nice to get at the big idea of accumulation right at the end of the book. www.discovercalculus.com/web/sec-Pola...

Instructions: Move the slider to change theta_k^* and move the point along the polar curve.

Slider labeled theta_k^*=1.18
Checkbox labeled Show the kth slice.

Polar plot of a curve that looks like a figure 8. It has a point on the edge, and a line connecting the center/pole and the point. It is labeled r(theta_k^*). There is a small circular sector, looking like a thin pizza slice, where the curved edge/crust passes through the point on the curve centered at the pole. The angle is labeled delta theta.

100

Discover Calculus @discovercalculus.com · 18/08/2026

Then, for the calculus topics, we can call back to the results that we already know: chain rule for differentiating things with different variables, substitution for antidifferentiating, playing with the formulation of Riemann sums for arc lengths and polar areas...

Instructions: Use the sliders and options to build the cycloid curve.

Slider labeled slide to roll the wheel.

Graph of a cycloid with a picture of a circle and a point on the edge of it. The graph is created by tracing the point on the edge of the circle as it rolls along the horizontal axis.

Checkbox labeled Show the cycloid.

100

Discover Calculus @discovercalculus.com · 18/08/2026

For both parametric and polar curves, I implemented paginated @doenet.bsky.social activities. Students can focus on individual aspects first, building to understand how we think of these curves. Parametric: www.discovercalculus.com/web/sec-Para... Polar: www.discovercalculus.com/web/sec-Pola...

nstructions: Navigate through each step one at a time to slowly reveal the parametric curve. For our example, we'll be looking at:
x=t^2-3t-4
y=1-t^2
for 0 <= t<= 4.

Slider with t=0

First we'll consider just the x-values of our parametric curve: x=t^2-3t-4.

Picture of a graph of a parabola with a point on it, as well as a number line with a point on it.

Fill in the table below using the function for x:

t 0 1 2 3 4
x

Step 1 of 3. Next Step.

100

Discover Calculus @discovercalculus.com · 18/08/2026

The new chapter on parametric and polar curves is finished and live! Practice problems are coming, but I'm happy to have translated the lectures I used to give into something that fits this book's theme: discovery and active learning. www.discovercalculus.com/web/ch-Param... #MathSky 🧮

discovercalculus.com

Parametric and Polar Curves

273

Discover Calculus @discovercalculus.com · 21/05/2026

Since the end of the Spring semester, I've made some updates with more to come. Here's a list of some tasks I'm working on this summer: - solutions for practice problems - add images in PreFigure where appropriate - update Doenet interactives - new chapter on polar and parametric curves #MathSky 🧮

021

Discover Calculus @discovercalculus.com · 16/04/2026

Here's a link to the actual activity! www.discovercalculus.com/web/sec-Riem... Pretty easy low-tech solution to computing these approximations, and it's nice to talk about how we can use the general process to get lots of different approximations based on each group's choice of x-value.

discovercalculus.com

Riemann Sums and Area Approximations

000

Discover Calculus @discovercalculus.com · 16/04/2026

Yesterday we crowd sourced some calculations in class. Each group was responsible for 2 rows in a column. Fun to do a couple of times, but I'm excited to move on to the big event. Easy to tell what we're up to? #MathSky 🧮

A whiteboard with many numbers on it. For instance, there is, in black, written [0,1/6], then in red, 1/10, and 101/100 in blue, and 101/600 on green. There are twelve of these rows of numbers.

111

Discover Calculus @discovercalculus.com · 10/03/2026

It's here! @acidlich.bsky.social created this beautiful cover art, perfect as a representation of the book. While students will mostly be interacting with the book digitally, I'm very excited for the printed copies that we'll be making to give to students as well! #MathSky 🧮

Cover art for Discover Calculus. The background shows a plane flying over water in an arc. Reflected in the water is the plane and its arc but also diverging curves, rippling in the water. In the background is a bridge and some lights. The sky is dark with shining stars, a crescent moon, and beautiful clouds. The whole picture is a warm grey/brown with black hatch shading and white highlights.

Discover Calculus
Single-Variable Calculus Topics with Motivating Activities

Peter Keep

25511

Discover Calculus @discovercalculus.com · 20/02/2026

If you want to see more of what we're up to, here's how we'll move from thinking about the Power Rule for derivatives into thinking about derivatives of trig and exponential functions. www.discovercalculus.com/web/sec-Deri...

discovercalculus.com

Some Early Derivative Rules

010

Discover Calculus @discovercalculus.com · 20/02/2026

Can you guess what we're talking about in class recently? 🧮

Triangle with the binomial coefficients, ending with the row that begins 1, 8, 28, ... In each row, the first term (1) is red. The second number (1, 2, 3, 4, ..., 8) is red. The remaining numbers are blue.

221

Discover Calculus @discovercalculus.com · 16/01/2026

I got this very fun message the other day! I wrote this book for my own classes, but I'm happy to learn that it seems to be working for other people's students, too. If you want to see the u-sub interactive, it's here: www.discovercalculus.com/web/sec-uSub... #MathSky 🧮 #ITeachMath

Can I link your book in canvas or no?

Yah, as like an extra resource for your class?

Yeah. I showed a student the u-sub interactive and they asked if they could access it on their own. Thought I'd ask first

030

Discover Calculus @discovercalculus.com · 14/01/2026

I've added links to every activity on Canvas, for those following along with a laptop or other device, and #PreTeXt automatically generates a QR code for any embedded interactive element.

A page of one of the booklets with an activity titled "Activity 4.2.3 First Derivative Test Graphically." There is text and a picture of an interactive graph with a QR code beside it, linking to the interactive graph.

000

Discover Calculus @discovercalculus.com · 14/01/2026

It's "print shop pickup" day! Because this text is written in #PreTeXt, there are some pretty easy ways to automatically export and format activity booklets for Calculus I and Calculus II. These are what students will be working with in class all semester, and the full textbook will be online.

A box full of spiral bound booklets. On top are two booklets, with covers that read "Discover Calculus I - Activity Book" and "Discover Calculus II - Activity Book."

120

Reposted by Discover Calculus

Discover Calculus @discovercalculus.com · 12/12/2025

This was a great excuse to gather up some of the interesting problems for end-of-chapter "Explorations." Here's the Basel Problem Exploration, in case you'd like to check it out: www.discovercalculus.com/web/explore-... I like this one because it's secretly about more than the reciprocal squares. 🧮

111

Discover Calculus @discovercalculus.com · 12/12/2025

Anyways, there are some more problems in here that are fun, and I'd like to keep adding more. This probably needs to take a backseat for now, though, since this project is coming up on some "end dates" (even though, of course, I'll keep adding to it).

000

Discover Calculus @discovercalculus.com · 12/12/2025

It's a shame to leave the most fun mathematical thinking only to the few students that might pursue math further. Maybe including these Explorations at the end of most chapters in this book will remind me (and hopefully others) to give students the exciting problems so that they can love them too.

100

Discover Calculus @discovercalculus.com · 12/12/2025

I think all of these Explorations should just be some way of thinking a bit more about something included in a standard calculus curriculum, but maybe from a slightly different perspective. And fun! The kinds of problems that got me hooked on mathematics because they're *fun* to think about.

100

Discover Calculus @discovercalculus.com · 12/12/2025

111

Discover Calculus @discovercalculus.com · 24/11/2025

I probably *should* leave this until next semester, but I think I'll likely leave this as an end-of-chapter exploration, once we finish going over Taylor series. I'll try to remember to send it to you when it's ready!

000

Discover Calculus @discovercalculus.com · 24/11/2025

Ok quick, is writing a "practice problem" that guides students through re-creating Euler's proof that the sum of reciprocal squares converges to π²/6 unhinged, yes or no? #MathSky #iTeachMath

450

Discover Calculus @discovercalculus.com · 22/11/2025

Even though the text itself is (mostly) finished, there are still lots of things to add! Textbooks *should* be good teaching and learning resources, so this one will have lots of extras to share with people to use (or change) as they want! #MathSky 🧮 #iTeachMath

020

Discover Calculus @discovercalculus.com · 21/11/2025

Anyways, that was my small win for the semester, and I'm glad that this text was helpful to students. If you want to see what they were up to, they were told to do these two activities: 1. www.discovercalculus.com/web/sec-FTOC... 2. www.discovercalculus.com/web/sec-FTOC...

040

Discover Calculus @discovercalculus.com · 21/11/2025

Having these interactive activities has mainly been geared towards what to do in class, but I also hoped that they could be helpful for students who miss class or want to recreate the same kind of environment. It seems like it worked really well in this specific instance!

120

Discover Calculus @discovercalculus.com · 21/11/2025

In class when we returned, about a third of the students told me about how they just kept reading and kept doing more of the "in-class" activities. They were excited about finishing the Fundamental Theorem of Calculus and learning what how to compute definite integrals.

110

Discover Calculus @discovercalculus.com · 21/11/2025

Students were supposed to do two activities to build the first part of the Fundamental Theorem of Calculus. Then, I figured we'd construct the second part together, and not lose too much time. Not only did most students actually do it, but many of them worked ahead!

100

Discover Calculus @discovercalculus.com · 21/11/2025

Here's a cool proof-of-concept that happened recently. I missed some class time recently with conference travel, and instead of falling behind I sent some instructions to students on what to read about while we cancelled class. I figured I'd have to make it up in class still, like normal. 🧮 #MathSky

130

Discover Calculus @discovercalculus.com · 31/10/2025

Still lots to do compiling everything and updating the website, but we can celebrate for now!

010

Discover Calculus @discovercalculus.com · 07/09/2025

These triangles—used to demonstrate how to think about d/dθ sin(θ)=cos(θ) and d/dθ cos(θ)=-sin(θ)—are better than the other triangles—used to show the limit as θ→0 of sin(θ)/θ = 1, which you would then use to show d/dθ sin(θ)=cos(θ) and d/dθ cos(θ)=-sin(θ). It's all about which triangles to show.

A triangle on a unit circle, with standard lengths 1, sin(theta), and cos(theta). Another point is labeled on the unit circle. There is a triangle formed by the line connecting the two points, and then the vertical and horizontal components of the distance between the two points. The vertical distance is labeled sin(theta+Delta theta)-sin(theta) and the horizontal distance is labeled cos(theta)-cos(theta+Delta theta). The hypotenuse is labeled h, but it is very close to the same as the arclength between the two points, labeled Delta theta.

150

Discover Calculus @discovercalculus.com · 24/07/2025

I'm going to be giving a talk about accessibility, OER, and general updates on this project soon, and couldn't help but include this slide. I'm going to also need to work in this statement in the quoted post about resisting LLMs and generative AI. Should be fun!

A slide with a dark gradient background from a dark grey color to a green color. In white, the text on the top says "Why not just use MS Word?"

In gold below it, the text says "Because it's terrible."

051

Discover Calculus @discovercalculus.com · 24/07/2025

I think it's missing in a lot of early college courses, too! Students get the feeling (in calculus courses, at least) that math is more about computing things instead of exploring objects and their properties!

010

Discover Calculus @discovercalculus.com · 24/07/2025

We should explicit about what mathematics *is*, even (especially?) in courses that are largely students not majoring in math. So we can (1) remind students why we impose conditions or restrictions for certain results and (2) explore removing those conditions or restrictions. #MathSky 🧮 #iTeachMath

8.5 Alternating Series and Conditional Convergence

Before we move too far forward, let’s circle back to a point made in Subsection 8.4.3 Why Do We Need These Conditions?. In the Integral Test, we required the terms of our series (and the continuous function we connected it with) to be positive. This was really just a mechanism that allowed us to say, in our proof, that the sequence of partial sums was monotonic. When we accumulate more of a positive thing, the total gets bigger. This is half of what we needed for us to employ the Monotone Convergence Theorem. Because this is such a useful tool, we’ll see more of this "positive term series" condition showing up in the tools we use to see if a series converges.

But that makes this a perfect time to stop and ask a hallowed mathematical question: What happens if that property isn’t there? What happens when our series does not only have positive terms?

130

Discover Calculus @discovercalculus.com · 04/07/2025

The third one is! The first two interactive ones are built using Doenet and the last one is just plain tikz.

020

Discover Calculus @discovercalculus.com · 03/07/2025

Here are some of the kinds of visuals that I've been working on! You can see a mix of interactive plots (in the first 2 images) as well as images made to demonstrate some typical calculus tasks (the 3rd image) and some more generic figures describing processes (the last one). #MathSky 🧮 #iTeachMath

A graph of a function with a two points, and a line connecting them. From each point is a faint line to the y-axis and a faint line to the x-axis. The distance between these lines on the x-axis is shaded orange and labeled Delta x. The vertical distance on the y-axis is also orange and is labeled f(x+Delta x)-f(x).

Two graphs on different axes. The first is a red graph with a point somewhere between x=1 and x=2. The point is labeled (x,g(x)), and the area between the curve and the x-axis from x=0 up to that point is shaded in. Part of this area is above the x-axis, and the part closest to the point is below.

The second graph is a blue graph with a point between x=1 and x=2. The point has a red tangent line, pointing downwards.

A 3-dimensional solid of revolution formed by a curve above the x-axis being revolved around the x-axis. The solid is approximated by a bunch of thin disks, stacked next to each other.

Two rectangles, one labeled "x Context" and the other, beside it, labeled "theta Context." Inside the "x Context" rectangle is an integral of f(x) dx with an arrow pointing towards F(x)+C. Inside the "theta Context" rectangle is an integral of g(theta) d theta with an arrow pointing towards G(theta)+C. There is an arrow connecting the integral of f(x) dx in the first rectangle to the integral of g(theta) d theta labeled x=T(theta) and dx = T'(theta) d theta. Then there is an arrow connecting G(theta)+C to F(x)+C labeled x = T(theta)

130

Discover Calculus @discovercalculus.com · 24/05/2025

We get to see that continuity sure does help to make the limit of the Riemann sum exists, and so it's an important property in terms of integrability, but not a necessary one!

000

Discover Calculus @discovercalculus.com · 23/05/2025

Students should know that definitions are choices, and these choices should come with both reasons for making them and an exploration of the consequences of those choices.

We’re going to now deal with the consequences of our decisions. A truth about mathematics, sometimes not an obvious truth, is that every time we state a definition what we are actually doing is making a decision. We are deciding on some common way of classifying and describing an object. These classifications and descriptions are choices that we are making: choices to prioritize some property or aspect over a different one, choices to include or exclude a type of object into the group of things we’re interested in, choices that come with downstream effects.

We chose to define the area bounded between a curve defined by the function f(x) and the x-axis between x=a and x=b as:

integral from x=a to x=b of f(x) dx = limit as n to infinity sum from k=1 to n of f(x_k^*) delta x.

We are going to stand by this definition. It’s a good one, for the reasons we described at the beginning of Section 5.2 Riemann Sums and Area Approximations.

But there are some weird things to notice. Let’s notice them!

2213

Discover Calculus @discovercalculus.com · 12/04/2025

I'm trying to build careful interactive elements for this book that will help students discover most of the ideas of calculus for themselves. Can you see how we'll bring this back around later on to discover the Fundamental Theorem of Calculus?

030

Discover Calculus @discovercalculus.com · 14/03/2025

This book is free from any content generated from, built by, or influenced by Large Language Models and/or generative AI products.

Disclosure about the Use of AI
This book has been lovingly written by a human.

Me.

Peter Keep.

I have used a lot of different tools, both for inspiration and for actually creating resources for this book. None of those tools has involved any form of generative AI.

I could list all of the ways that I think using generative AI in education is, at minimum, problematic. More pointedly, I believe that it is unethical. More broadly, I believe that the use of generative AI for any use-case that I have encountered to be unethical.

In my classes, I try to help students realize the joy and value of working at something and creating something and struggling with something and knowing something. Giving worth to something, even an imperfect thing. Celebrating our accomplishments, even when (especially when?) there is room to grow in those accomplishments. And so I have taken that advice in the creation of this book. I have created a book that is definitely not perfect. I have struggled to write it. There are parts of it that could be (need to be) improved.

But I was the one that created it. I struggled with it. I know it.

I hope that this book can also be a useful tool for others to use, and I have left the copyright to be about as open as possible. Others can take this, use it, can change it, add to it, subtract from it, etc.

In leaving this copyright open for others to change this book, I cannot guarantee that every version of this book is free from the mindless and joyless output from some Large Language Model. But I want to leave this note up in hopes that anyone who does inject some output from some generative AI product into this book will take it down. If this note, or some statement similar to it, is not present in the version of the book you are accessing, please be cautious. Find a different calculus textbook to read!

Find something written by a human. Find the words of some other mathematician who tries, maybe imperfectly, to share the ideas of calculus.

0121