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Calculus For The People

Greg Petrics, May 16, 2019

The first season of Calculus, now streaming on Geogebra! This is an introduction to the main ideas of Calculus 1: limits, derivatives and integrals. Geogebra does the algebra for you so you can focus on understanding the concepts. Enjoy!

1.

Overview

1. Preface

1.1.

Preface

[b]Welcome:[/b][br][br]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.[br][br][b]Background:[/b][br][br]I got the idea for this book by noticing two seemingly irreconcilable facts:[br][br][list=1][*]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 .[/*][/list][br]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?[br][br]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.[br][br][b]What this book is [i]not[/i]:[/b][br][br]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.[br][br]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 [i]after[/i] you understand something, but as a tool for learning something for the first time, they are challenging. If you really need to know [i]everything[/i] 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: [url=https://www.amazon.com/gp/offer-listing/0495554669]https://www.amazon.com/gp/offer-listing/0495554669[/url] (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 [i]Calculus[/i], and Rudin's [i]Principles of Mathematical Analysis[/i].[br][br]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 [i]Calculus for the People[/i]. 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. [br][br][b]What this book [i]is[/i]:[/b][br][br]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 [i]between[/i] [i]concepts[/i] [i]and objects[/i]. I personally believe that [i]making these connections[/i] 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 challenge[sup][b]1[/b][/sup].[br][br]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 [url=https://www.geogebra.org/m/x39ys4d7#material/ftvamrcu]function[/url], a [url=https://www.geogebra.org/m/x39ys4d7#material/zruqdnrq]limit[/url], the [url=https://www.geogebra.org/m/x39ys4d7#material/rwdrnrw6]derivative[/url] and the [url=https://www.geogebra.org/m/x39ys4d7#material/ufsyvbbx]integral[/url]. 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. [br][br]That said, there still is [i]some[/i] 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 [url=https://www.geogebra.org/m/x39ys4d7#material/wrafy53s]Atomic Functions[/url].[br][br][b]Prerequisites:[/b][br][br]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 [i]does[/i] transfer out to other calculus courses.[br][br]The traditional prerequisites of trigonometry, exponential and logarithmic functions are [i]not[/i] needed, and for the little bit we do need, it will be introduced in a "just in time" fashion as we go. [url=https://www.cambridge.org/core/books/fresh-start-for-collegiate-mathematics/college-precalculus-can-be-a-barrier-to-calculus-integration-of-precalculus-with-calculus-can-achieve-success/932DB960AAF47DB4F566D17543F814B0]Current research[/url] 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. [br][br]Finally, even though we will use Geogebra to do a lot of calculations on our behalf, [i]no computer programming[/i] [i]experience is required[/i]. All the code you need is introduced as we go. The [code]code font face[/code] is used throughout to indicate code you can copy and paste and use in Geogebra.[br][br][b]A word about open-source materials:[/b][br][br]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 [url=https://www.geogebra.org/license]Geogebra License[/url], everything here is entirely open source, so please make copies, edit, change, etcetera, but you are forbidden from MONETIZING THIS. See the [url=https://www.geogebra.org/license]Geogebra License[/url] for all the details.[br][br]If you haven't done so already, you can download Geogebra (I recommend version 5) here: [url=https://wiki.geogebra.org/en/Reference:GeoGebra_Installation]https://wiki.geogebra.org/en/Reference:GeoGebra_Installation[/url]. Scroll down to find Version 5. Version 6 is OK too, but I prefer 5.[br][br][b]Navigation:[/b][br][br]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.[br][br][b]Feedback:[/b][br][br]This book is always changing, so your [url=https://docs.google.com/forms/d/e/1FAIpQLScXbXCapUfqMVoIffzQ8BomFLz06ajvZ2kanfBXSjj45J_0mQ/viewform?usp=sf_link]feedback is appreciated[/url].[br][br][b]Onwards![/b][br][br]Dive right in below, and [i][b]have fun[/b][/i]![br][br][br][br][br][br][br][br][br][br][br][b][sup]1[/sup][/b] An essay on this thesis is in the [url=https://www.geogebra.org/m/x39ys4d7#material/fxpkwpt7]Miscellany chapter at the end of the book[/url].

2.

Functions

An introduction to the concept of functions.

2.1.

A Quick Word about Functions

This first chapter is about [b]functions[/b], which is the only true mathematical pre-requisite for calculus. As we will see in the [url=https://www.geogebra.org/m/x39ys4d7#material/ftvamrcu]definition of a function[/url], [b]functions[/b] are mathematical computer programs. [br][br]This chapter--like every chapter--begins with an example to help you discover the main idea of the chapter on your own. Then, in the middle of the chapter, the idea is defined formally. It's not a bad idea to jump ahead to the [url=https://www.geogebra.org/m/x39ys4d7#material/ftvamrcu]definition of a function[/url] to take a look now, but don't worry if it doesn't make sense. Come back to the start of the chapter, and work your way towards the definition, and by the time you get to the definition you will be prepared to understand it. [br][br]A note to people who have already taken calculus: If you already understand the concept of a function, I still think you should review how I define a function. Everything here is done quickly, and things are a little different than you might be accustomed to.

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2.13.

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2.19.

2.20.

2.21.

3.

Limits

An introduction to the concept of limits.

3.1.

A Quick Word about Limits

This chapter is about the concept of Limits. This is the first calculus topic in this book. [br][br]Limits provide an essential method for investigating, or "studying" functions. Indeed, as we'll see in the [url=https://www.geogebra.org/m/x39ys4d7#material/zruqdnrq]definition of a limit[/url], limits [i]are[/i] a method of study. Usually we study functions with the aid of limits, but a limit can be used to investigate any process that doesn't terminate. This might sound outlandish now, but I hope you'll see what I mean before this chapter is concluded.[br][br]In this chapter we'll investigate just a few functions with limits. We will only be concerned with limits in so much as they lead us to the concept of the [b]Derivative[/b] which we'll continue studying at length in the next chapter. This is the least "applied" chapter of the book because the application of limits is derivatives, but we can't talk about derivatives until we've discussed limits. [br][br]This is the shortest chapter in this book. I really put the pedal to the metal here to get us to applications as quickly as possible.

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3.4.

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3.6.

3.7.

3.8.

3.9.

3.10.

4.

Derivatives

An introduction to the concept of derivatives.

1. A Quick Word About Derivatives
2. Jumping Right In
3. Almost a Derivative
4. Connecting the Dots
5. The Dots Connected
6. But Wait, It's A Little Off...
7. The First of Many Shortcuts
8. Definition of The Tangent Line
9. The Tangent Line and the Derivative
10. The Tangent Line Part 3
11. Definition of the Derivative
12. Another Derivative
13. A Quick Application
14. The Units of the Derivative
15. What's Geogebra Doing?
16. The Monkey Rules
17. Monkey Rule 0
18. Figuring out the rest of the Monkey Rules
19. Monkey Rules 1 and 2
20. A Bit More About Monkey Rule 2
21. Monkey Rule 3
22. Monkey Rule 4
23. Monkey Rule 5
24. Monkey Rule 6
25. Monkey Rule 7
26. Monkey Rule 8
27. Monkey Rule 8 Part 2
28. Monkey Rule Practice
29. Another Application
30. Application: Maximums and Minimums -- Critical Points
31. Application: Maximums and Minimums -- 2nd Derivative Test
32. Application: Modeling Traffic In Johnson, Vermont Pt. 1
33. Application: Modeling Traffic In Johnson, Vermont Pt. 2
34. Application: Modeling User Growth Pt. 1
35. Application: Modeling User Growth Pt. 2 -- Concavity
36. Absolute Maximums and Minimums of a Function on a Closed Interval

4.1.

A Quick Word About Derivatives

In this chapter we'll study [b]derivatives [/b]which offer us a powerful way to study the rate of change of a function. For instance, think about the [url=https://www.geogebra.org/m/x39ys4d7#material/aayx7rmz]model of the height of an incoming missile[/url], [code]g(x)[/code] from earlier in the book. The derivative of this model will give us insight into the rate of change of the missile.[br][br]This chapter has four parts. [br][br][list=1][*]In the first part, we start out by seeing how the concept of a limit from the previous chapter can help us calculate the "tangent lines" of a function. This will give us very detailed information about how a function [i]changes[/i].[/*][*]Next, we'll spend a bit of time [url=https://www.geogebra.org/m/x39ys4d7#material/rwdrnrw6]defining[/url] exactly what the derivative of a function [i]is[/i]. This is the first big conceptual speed bump in calculus. I strongly encourage you to go back and forth between the [url=https://www.geogebra.org/m/x39ys4d7#material/rwdrnrw6]definition[/url] and the [url=https://www.geogebra.org/m/x39ys4d7#material/ybp9bfdt]first application[/url] to help yourself solidify your understanding.[/*][*]After learning about the concept, we'll proceed to acquiring some algebraic procedural knowledge, and study some shortcuts for calculating derivatives, called the [url=https://www.geogebra.org/m/x39ys4d7#material/p8jdmayj]Monkey Rules[/url]. However, because Geogebra already knows all the Monkey Rules, we won't bother spending much time with them. In fact, if you only care about understanding calculus, you can skip the algebraic procedural Monkey Rules. [/*][*]Finally, we'll wrap up our study by seeing how to use derivatives to accomplish some [url=https://www.geogebra.org/m/x39ys4d7#material/mznjt3yx]important quantitative tasks[/url].[/*][/list]

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4.34.

4.35.

4.36.

5.

Integrals

An introduction to the concept of integrals.

1. A Quick Word about Integrals
2. Accumulated Effect of a Rate Part 1
3. Accumulated Effect of a Rate Part 2
4. Accumulated Effect of a Rate Part 3
5. Definition of the Integral
6. Nuances of the Integral
7. Calculating Integrals
8. The Fundamental Theorem of Calculus
9. Visualizing the Fundamental Theorem of Calculus
10. The Intuitive Proof of The Fundamental Theorem of Calculus
11. Constants of Integration
12. Falling Stuff on Earth
13. Finding Antiderivatives -- Lucifer's Rules
14. Lucifer's 1st Rule
15. Lucifer's 2nd Rule
16. Lucifer's 3rd Rule
17. Lucifer's 4th and 5th Rules
18. Lucifer's Rules Practice

5.1.

A Quick Word about Integrals

In the [url=https://www.geogebra.org/m/x39ys4d7#chapter/398515]last chapter[/url] we studied [b]derivatives[/b]. As you know, the derivative of a function allows us to study the rate of change of the function. For instance, when we modeled the height of an incoming missile with the function [code]g(x)[/code], [url=https://www.geogebra.org/m/x39ys4d7#material/ybp9bfdt]the derivative of the model, [code]g'(x)[/code], was a model of the rate of change of height of the incoming missile[/url]. [br][br]In this chapter, we study the opposite process, called the [b]integral[/b]. The very short story goes like this: if we construct a functional model of the rate of change of some physical quantity, the [b]integral[/b] of the function will tell us the [i]accumulated effect[/i] of that rate of change. [br][br]For instance, think about [url=https://www.geogebra.org/m/x39ys4d7#material/zdrhsxcx]the model of the rate of cars (in cars per minute) traveling along Route 15 in Johnson, Vermont[/url]. The [b]integral[/b] of this model will give us insight into the total number of cars that travel along Route 15 during a period of time. In the next few lessons, we'll refer to the total number of cars that travel along a road during a period of time as a "car count".[br][br]One unfortunate fact is that the [url=https://www.geogebra.org/m/x39ys4d7#material/ufsyvbbx]mathematical definition[/url] however is quite different than the interpretation that is described above. One of the main goals of this chapter is to help you connect the interpretation of the integral and its mathematical definition. [br][br]The integral and the derivative together comprise the two core ideas of calculus. You already know a lot about the derivative. Click ahead to get started learning about the [b]integral[/b]!

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6.

Miscellany

A few closing comments on Season 1, and what to expect in Season 2.

1. Calculus On Your Own
2. A Word About Mathematical Models
3. Domain of a Function
4. Why do People say "Calculus is Impossible"?
5. An Application of Limits: Archimedes' Infinite Sum

6.1.

Calculus On Your Own

In order for you to step out of this book and start doing calculus on your own, the first step is to construct a functional mathematical model of your data. Geogebra has a powerful toolkit for constructing single variable functional models. We used it multiple times in this book to construct the models of the [url=https://www.geogebra.org/m/x39ys4d7#material/emxhjg78]height of the incoming missile[/url], the [url=https://www.geogebra.org/m/x39ys4d7#material/we84syqz]length of the day in Johnson Vermont[/url], the [url=https://www.geogebra.org/m/x39ys4d7#material/zdrhsxcx]rate of traffic on Route 15[/url] in Johnson Vermont, and the [url=https://www.geogebra.org/m/x39ys4d7#material/jqdpbend]user headcount of Instagram[/url]. As you'll see if you go back and review those four activities, I progressively made you more autonomous. In the first activity where you modeled the height of the incoming missile, I only asked you to select the regression model (polynomial of order 2, AKA a quadratic), and then in later exercises I pushed you a little harder and had you do more of the task by yourself. For instance when you built the model of the length of the day in Johnson Vermont I asked you to highlight the data and open the Two Variable Regression Analysis Tool on your own.[br][br]The only skill you need now to "close the loop" and be fully autonomous is for you to understand how to get bivariate data into Geogebra. This is a straightforward procedural task that is no harder than any other office computer task you are likely to encounter at work.[br][br][list=1][*](Hardest step) Your bi-variate data must be formatted so that it is stored in two columns. The term "bi-variate" means that each row of your two-column data set [i]must[/i] represent a pair of values of an independent variable and an associated dependent variable. For instance, the first column could be time, and the second column could be height as in the [url=https://www.geogebra.org/m/x39ys4d7#material/emxhjg78]incoming missile activity[/url]. If this doesn't make sense to you, go back to the examples listed above and see how each model started from a bi-variate data set that was in just such a two column format. The best software to use to store and format your data is something like Microsoft Excel, Google Spreadsheets, or Open Office. Any spreadsheet software is fine.[/*][*]Open the Spreadsheet View in Geogebra. You can read more about the view [url=https://wiki.geogebra.org/en/Spreadsheet_View]here[/url]. In general, the Spreadsheet View of Geogebra can be thought of just like any other spreadsheet, except each cell can be thought of as its own little input bar. In this way the Spreadsheet View deeply integrates with Geogebra. So if you store a number in a cell, the number won't show up in the Graphics View since number objects aren't plotted, but if you store a point such as [code](1,2)[/code], that will show up in the Graphics View.[/*][*]Copy and paste your bi-variate data columns from Excel or comparable into the spreadsheet view of Geogebra. Tidy it as you would in any other spreadsheet software. If you run into any troubles with the copy and paste, I recommend saving your data in ".CSV" in your spreadsheet software first, and closing the program, re-opening it, and then trying again. The format ".CSV" stands for [b]c[/b]omma [b]s[/b]eparated [b]v[/b]alues, and is kind of like the least-common-denominator data format of all spreadsheet and database software.[/*][*]Proceed as you did in any of the modeling applications we did in this course (see the links above in the header paragraph for examples) to construct your model. [/*][/list]This is neither time nor the place to discuss model type selection or testing. That is an entire other course. That said, there's nothing to stop you from using the Two Variable Regression Analysis tool now to begin exploring. In the near future I will be writing a few example activities to illustrate this process. They will appear below.[br][br]Once you have a mathematical model from Geogebra, you can then use what you've learned to do a derivative-analysis of it to find the model's predictions of maximums and minimums of the dependent variable. If the model is of a rate of some physical quantity, you can use the integral to see the accumulated effect of the rate. [br][br]As noted, additional examples of this process will be added to this chapter in the future and listed here as they become available. Here are those that are currently available:[br][br]**NONE YET AS OF AUGUST 8, 2019**

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