Discover Calculus

@discovercalculus.com

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An open copyright, online textbook for single variable calculus with a discovery-based perspective from @mathprofpeter.bsky.social. We build the big ideas together through guided and interactive activities. www.discovercalculus.com

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.
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
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.
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
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.
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."
A screenshot of an interactive plot of f(x)=sin(x)/x when x is not equal to 0 and 1 when x=0. The instructions say to list the zeros of the function, and an infinite product of the function is displayed below it. This is a step in Euler's proof of the sum of reciprocal squares being pi^2/6.
Pixel book logo with links underneath: Home Read Online Other Downloads Instructor Resources On the right is the following text: Instructor Resources Python Demonstrations These python notebooks are hosted on Google Colab, with no python installations necessary. Each notebook demonstrates some concept from the text, and are suitable for in-class demonstrations or out-of-class explorations. These python notebooks have a mixture of pre-written code and text instructions and explanations, making them useful even without much (or any!) python background. Images and Animations There are many images and animations included in the text already, but several interesting relevant examples have not been included. These can still be used in classes for additional explanation or small deviations from the specific content in the text. Doenet Interactives Interactive Doenet elements are included throughout this text, whether in the activities or in the exposition between student explorations. Source code for all of these interactive elements can be found on the Doenet website. The source code for all of these can also be found in the Github repository for this text, but the Doenet website includes a free editor, making it easy to edit or copy any of these interactive elements.
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.
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."
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?
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).
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!
The graph of a parabola is displayed, labeled "Graph of f'(x)." There are two hints in drop down boxes encouraging students to think about how we might interpret the height of the point on f'(x), and how we might learn about f(x). As the mouse drags the point above and below the x-axis, we can see information displayed about how this impacts the direction that the function f(x) will travel. When a button labelled "Show graph of f(x)" is clicked, a second graph displays with the graph of a cubic function. A point is labeled with a tangent line.
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.