Preface

Author:Greg Petrics

Welcome: The goal of this book is to quickly get you using the core ideas of calculus to answer real world questions, and to leverage the power of the computer to help you learn. It was written for people who think they can't understand calculus because of traditional algebraic hurdles. Most (but not all) of the algebra has been shunted to the computer. If you want to get to work, AWESOME, don't bother reading the rest of this page, and instead just jump right into "Functions" on the left sidebar. Background: I got the idea for this book by noticing two seemingly irreconcilable facts:

Most people think calculus is absolutely impossible no matter how hard they think.
Proficient users of calculus generally do so without even having to think .

How could these two facts simultaneously be true? Is it that proficient users of calculus have such advanced brains, that they can process vast amounts of material more quickly than the rest of us? Or is it that the core ideas of calculus are surprisingly straightforward, and by understanding just them, this enables proficient users to utilize the tools of calculus with ease? Or is it that standardized calculus tests are so focused on sorting people for college admissions, that they have ramped up the algebraic difficulty to spread students out, but the outcome is that the student population is left with the impression that the only way to understand calculus is to be an algebra wizard? I can tell you that it's definitely not option one. I tend to think it's mostly option two, with a little bit of option three sprinkled in. What this book is not: This book is not intended to prepare you for an AP or other standardized test. Those tests do a great job of assessing skill with calculus calculations and computations, but many students get turned back by the algebra in AP Calculus, and fail to become proficient users of calculus. Therefore, we won't focus much on the algebra in this book, and will instead be focusing on big ideas. If you want to get into the nitty gritty algebra of calculus after you have learned what calculus is all about, that's always an option down the road. So one bit of bad news: if you came here hoping for a quick pathway to a 4 or a 5 on the AP test, I haven't written this for you. Additionally, this book is not intended to provide exhaustive coverage of all the topics of calculus. The thinking on this is pretty simple: It's very hard to learn from an encyclopedia. Encyclopedias are great for reference after you understand something, but as a tool for learning something for the first time, they are challenging. If you really need to know everything about calculus, one of the best encyclopedic treatments is by Stewart; buy a used early edition for under 10 bucks on Amazon. At last check the 5th edition was under $10: https://www.amazon.com/gp/offer-listing/0495554669 (and in case you were wondering, there is no commission cookie or tracking code in that link). Other notable texts on the subject that go into all the nitty gritty include Spivak's Calculus, and Rudin's Principles of Mathematical Analysis. Finally, this book is not titled anything along the lines of "Calculus is easy." Calculus is not easy, and no one can change that. Instead, this book is titled Calculus for the People. This title reflects my belief that for whatever reason, calculus has been cordoned off from the populace, and put behind a "wall of algebra." I don't see any reason why this should be so. Of course high level professional users of calculus need to know a good deal of algebra, but they need that knowledge in the same way a master carpenter must be familiar with a wide variety of tools. But to build a shed you don't need to know how to use every tool in Home Depot; similarly I believe there's no reason everyone needs to know every aspect of algebra in order to understand and use the core ideas of calculus. What this book is: This book makes some serious changes to the "standard" approach to calculus, and only covers the big ideas that you need to become a proficient user of calculus. I've done as much as I possibly can to remove sophisticated algebra as a barrier to understanding, and have replaced that algebra with over 75 Geogebra "applets". These applets were designed to help you see calculus concepts that are traditionally visualized as algebraic phenomena. The thinking for doing so is to let the computer do the algebraic computing so that us humans can do what we're good at: understanding and making connections between concepts and objects. I personally believe that making these connections is the biggest challenge in calculus, not the algebra, so in essence, this book is about freeing you up from the algebra so you can focus on the real challenge¹. The key activities the book is built around are the four core definitions of calculus, one per chapter. If you want to skim them right now, it's not a bad idea, but they are meant to be read in the context of examples and real-world applications. They are: the definition of a function, a limit, the derivative and the integral. As you work your way towards them, you'll see that these four definitions are introduced and motivated as answers to tangible puzzles and real-world applications. I've worked hard to not rely on algebraic curios to get your interest. This approach is definitely not the most efficient way to regurgitate technical material, but I do think it is an effective way to learn it for the first time. That said, there still is some algebra. However, I've also written this book in a way that you can skip the "algebra parts" of the book, and still be able to understand the big definitions. I try to make it clear in the book when this occurs, and what the pros and cons are of doing so. The first time this happens is in the activity on the Atomic Functions. Prerequisites: This book assumes you are competent, if not a Jedi, at basic algebra and arithmetic. Specifically, an understanding of lines, their equations, slope, y-intercepts, x-intercepts, and so on is more or less assumed. I don't have any intent of "building calculus from scratch" in this book. I recognize the importance of the core ideas of algebra and linear functions, and lean on them. Furthermore, I stick with standard notation used in other texts on calculus, and don't "invent" anything. So the notation you learn in this book does transfer out to other calculus courses. The traditional prerequisites of trigonometry, exponential and logarithmic functions are not needed, and for the little bit we do need, it will be introduced in a "just in time" fashion as we go. Current research suggests that a "just in time" approach to pre-requisites is the best approach for improving student success in calculus for the entire student population. Finally, even though we will use Geogebra to do a lot of calculations on our behalf, no computer programming experience is required. All the code you need is introduced as we go. The code font face is used throughout to indicate code you can copy and paste and use in Geogebra. A word about open-source materials: You can get the source materials for every lesson by clicking on the three dots in the TOP RIGHT of every activity, and selecting "DETAILS". On the next page, select "DOWNLOAD" and choose the ".GGB" version of the file. This is the native file format of the desktop version of Geogebra. As per the Geogebra License, everything here is entirely open source, so please make copies, edit, change, etcetera, but you are forbidden from MONETIZING THIS. See the Geogebra License for all the details. If you haven't done so already, you can download Geogebra (I recommend version 5) here: https://wiki.geogebra.org/en/Reference:GeoGebra_Installation. Scroll down to find Version 5. Version 6 is OK too, but I prefer 5. Navigation: There's a few ways to get around the book. At the bottom of each page is an option to go back on the left, or forward on the right. Also, on the left sidebar is a table of contents for the entire book enabling you to jump around as needed. Finally, I have hyper-linked the book so that important material from earlier in the book can be easily found when it is needed in a later activity. Feedback: This book is always changing, so your feedback is appreciated. Onwards! Dive right in below, and have fun! ¹ An essay on this thesis is in the Miscellany chapter at the end of the book.

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