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Oct 9, 2012 · A classic textbook by Silvanus Thompson that introduces calculus in a simple and intuitive way. It covers differentiation, integration, maxima and minima, curvature, compound interest, and more, with examples and exercises.
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- CONTENTS
- INTRODUCTION
- Learning Strategy: Blurry To Sharp
- Email Updates
- Book Webpage
- 1 MINUTE CALCULUS: X-RAY AND TIME-LAPSE VISION
- 1.1 Calculus In 10 Minutes: See Patterns Step-By-Step
- PRACTICING X-RAY AND TIME-LAPSE VISION
- 2.3 Board-by-board Analysis
- 2.4 Getting Organized
- 2.5 Questions
- 3.1 Exploring The 3d Perspective
- LEARNING THE OFFICIAL TERMS
- 4.2 The Integral, Arrows, and Slices
- MUSIC FROM THE MACHINE
- IMPROVING ARITHMETIC AND ALGEBRA
- 6.2 Better Formulas
- 7.5 Creating The Abstract Rules
- 8.1 Bring On The Calculus
- THE THEORY OF DERIVATIVES
- 10.3 Throwing Away Artificial Results
- THE FUNDAMENTAL THEOREM OF CALCULUS (FTOC)
- 11.2 Part 2: Finding The Indefinite Integral
- Accumulation(x) Æ steps(x)dx
- 11.3 Next Steps
- 12.1 Addition
- 15.1 Changing Circumference To Area
- 15.4 2000 Years Of Math In A Day
- AFTERWORD
- Keep In Touch
- About the Author
- APPENDIX: LEARNING CHECKLIST
- Technical Description (Chapters 4-5)
- Theory I (Chapters 6-8)
- Performance (Chapter 15)
Introduction 1 Minute Calculus: X-Ray and Time-Lapse Vision Practicing X-Ray and Time-Lapse Vision Expanding Our Intuition Learning The Official Terms Music From The Machine Improving Arithmetic And Algebra Seeing How Lines Work Playing With Squares Working With Infinity The Theory Of Derivatives The Fundamental Theorem Of Calculus (FTOC) The Basic...
Hi! It looks like you’re interested in learning Calculus. I like you already. This book isn’t a collection of practice problems or formal theories. Hundreds of textbooks handle that quite well; this is the guide I wish they tucked into their front cover. The goal is to help you: Grasp the essence of Calculus in hours, not months Develop lasting, pr...
What’s a better learning strategy: covering a subject in full detail from top-to-bottom, or progressively sharpening a quick overview? After a single class, which strategy gives you a better understanding of the material? Which helps you predict how later parts fit together? Which is more fun? The linear, official, approach doesn’t work for me. Sta...
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We usually take shapes, formulas, and situations at face value. Calculus gives us two superpowers to dig deeper: X-Ray Vision: You see the hidden pieces inside a pattern. You don’t just see the tree, you know it’s made of rings, with another growing as we speak. Time-Lapse Vision: You see the future path of an object laid out before you (cool, righ...
What do X-Ray and Time-Lapse vision have in common? They examine patterns step-by-step. An X-Ray shows the individual slices inside, and a Time-Lapse puts each future state next to the other. This seems pretty abstract. Let’s look at the equations for circumference, area, surface area, and volume: We have a vague feeling these formulas are connecte...
Calculus trains us to use X-Ray and Time-Lapse vision, such as re-arranging a circle into the “ring triangle” we saw in the previous chapter. This makes finding the area. . . well, if not exactly easy, much more manageable. But we were a little presumptuous. Must every circle in the universe be made from rings? Heck no! We’re more creative than tha...
Getting the hang of X-Rays and Time-lapses? Great. Look at the progression above, and spend a few seconds thinking of the pros and cons. Don’t worry, I’ll wait. Ready? Ok. Here’s a few of my observations: This is a very robotic pattern, with boards neatly arranged left-to-right. The contribution from each step starts small, gradually gets larger, m...
So far, we’ve been using natural descriptions to explain our thoughts: “Take a bunch of rings” or “Cut the circle into pizza slices”. This conveys a general notion, but it’s a bit like describing a song with “Dum-de-dum-dum” – you’re probably the only one who knows what you mean. But a little organization can make your intent perfectly clear. To st...
Are things starting to click a bit? Thinking better with X-Rays and Time-lapses? Can you describe a grandma-friendly version of what you’ve learned? Let’s expand our thinking into the 3rd dimension. Can you think of a few ways to build a sphere? (No formulas, plain-English descriptions are fine.) PS. It may bother you that our steps create a “circl...
In the first lesson we had the vague notion that the circle/sphere formulas were related: With our X-Ray and Time-Lapse skills, we have a specific idea for how: Circumference: Start with a single ring. Area: Make a filled-in disc with a ring-by-ring time lapse. Volume: Make the circle into a plate, and do a plate-by-plate time lapse to build a sphe...
We’ve been able to describe our thinking process with analogies (X-Rays, Time-Lapses) and diagrams: However, this is a very elaborate way to communicate. Here’s the Official Math® terms that describe our intuitive concepts: Let’s walk through the fancy names.
The integral is gluing together (Time-Lapsing) a bunch of slices and measuring the final result. For example, we glued together the rings (into a “ring triangle”) and saw it accumulated to r2, aka the area of a circle. 1⁄4 Here’s what we need to find the integral: Which direction are we gluing the steps together? Along the orange line (the radius, ...
In the previous lessons we’ve gradually sharpened our intuition: Appreciation: I think it’s possible to split up a circle to measure its area Natural Description: Split the circle into rings from the center outwards, like so: Formal Description: Performance: (Sigh) integrate 2 * pi * r * dr from r=0 to r=r I guess I’ll have to start measuring the a...
We’ve intuitively seen how calculus dissects problems with a step-by-step view-point. Now that we have the official symbols, let’s see how to bring arithmetic and algebra to the next level.
If calculus provides better, more-specific version of multiplication and division, shouldn’t we rewrite formulas with it? You bet. An equation like distance Æ speed¢time explains how to find total distance assuming an average speed. An equation like distance Æ R speed dt tells us how to find total distance by breaking time into instants (split alon...
Have an idea how linear functions behave? Great. We can make a few abstract rules – like working out the rules of algebra for ourselves. If we know our output is a scaled version of our input (f (x) Æ ax), the derivative (pattern of changes) is ¢x Æ a dx and the integral (pattern of accumulation) is Æ ax ÅC That is, the ratio of each output step to...
That’s it? The analysis just figures out the current perimeter and square footage? No way. By now, you should be clamoring to use X-Ray and Time-Lapse vision to see what’s happening under the hood. Why settle for a static description when we can know the step-by-step description too? We can analyze the behavior of the perimeter pretty easily:
The last lesson showed that an infinite sequence of steps could have a finite conclusion. Let’s put it into practice, and see how breaking change into infinitely small parts can point to the the true amount.
The founders of calculus intuitively recognized which components of change were “artificial” and just threw them away. They saw that the corner piece was the result of our test measurement interacting with itself, and shouldn’t be included. In modern times, we created official theories about how this is done: Limits: We let the measurement artifact...
The Fundamental Theorem of Calculus is the big aha! moment, and something you might have noticed all along: X-Ray and Time-Lapse vision let us see an existing pattern as an accumu-lated sequence of changes The two viewpoints are opposites: X-Rays break things apart, Time-Lapses put them together This might seem “obvious”, but it’s only because we’v...
Ok. Part 1 said that if we have the original function, we can skip the manual computation of the steps. But how do we find the original? FTOC Part Deux to the rescue! Let’s pretend there’s some original function (currently unknown) that tracks the accumulation: Z b
a The FTOC says the derivative of that magic function will be the steps we have:
Phew! These lessons were theory-heavy, to give an intuitive foundation for topics in an Official Calculus Class. The key insights are: Infinity: A finite result can be viewed with a sequence of infinite steps. Derivatives: We can take a knowingly-flawed measurement and find the ideal result it refers to. Fundamental Theorem Of Calculus: The origina...
Let’s start off easy: how does a system with two added components behave? In the real world, this could be sending two friends (Frank and George) to build a fence. Let’s say Frank gets the wood, and George gets the paint. What’s the total cost?
Describe how to turn the circumference of a circle into the area of a circle: Explain your plan in plain English Explain your plan using the official math notation Apply the rules of Calculus to your equation and calculate the result Verify the result using Wolfram Alpha Repeat the steps above, turning the area of a circle into the volume of a sphe...
Describe how to turn the circumference of a circle into the area of a circle: Explain your plan in plain English Explain your plan using the official math notation Apply the rules of Calculus to your equation and calculate the result Verify the result using Wolfram Alpha Repeat the steps above, turning the area of a circle into the volume of a sphe...
Describe how to turn the circumference of a circle into the area of a circle: Explain your plan in plain English Explain your plan using the official math notation Apply the rules of Calculus to your equation and calculate the result Verify the result using Wolfram Alpha Repeat the steps above, turning the area of a circle into the volume of a sphe...
Describe how to turn the circumference of a circle into the area of a circle: Explain your plan in plain English Explain your plan using the official math notation Apply the rules of Calculus to your equation and calculate the result Verify the result using Wolfram Alpha Repeat the steps above, turning the area of a circle into the volume of a sphe...
Describe how to turn the circumference of a circle into the area of a circle: Explain your plan in plain English Explain your plan using the official math notation Apply the rules of Calculus to your equation and calculate the result Verify the result using Wolfram Alpha Repeat the steps above, turning the area of a circle into the volume of a sphe...
Describe how to turn the circumference of a circle into the area of a circle: Explain your plan in plain English Explain your plan using the official math notation Apply the rules of Calculus to your equation and calculate the result Verify the result using Wolfram Alpha Repeat the steps above, turning the area of a circle into the volume of a sphe...
Describe how to turn the circumference of a circle into the area of a circle: Explain your plan in plain English Explain your plan using the official math notation Apply the rules of Calculus to your equation and calculate the result Verify the result using Wolfram Alpha Repeat the steps above, turning the area of a circle into the volume of a sphe...
Describe how to turn the circumference of a circle into the area of a circle: Explain your plan in plain English Explain your plan using the official math notation Apply the rules of Calculus to your equation and calculate the result Verify the result using Wolfram Alpha Repeat the steps above, turning the area of a circle into the volume of a sphe...
Describe how to turn the circumference of a circle into the area of a circle: Explain your plan in plain English Explain your plan using the official math notation Apply the rules of Calculus to your equation and calculate the result Verify the result using Wolfram Alpha Repeat the steps above, turning the area of a circle into the volume of a sphe...
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- Table of Contents (PDF)
- Chapter 0: Highlights of Calculus (PDF) 0.1 Distance and Speed // Height and Slope. 0.2 The Changing Slope of \\(y=x^2\\) and \\(y=x^n\\) 0.3 The Exponential \\(y=e^x\\)
- Chapter 1: Introduction to Calculus (PDF) 1.1 Velocity and Distance. 1.2 Calculus Without Limits. 1.3 The Velocity at an Instant. 1.4 Circular Motion. 1.5 A Review of Trigonometry.
- Chapter 2: Derivatives (PDF) 2.1 The Derivative of a Function. 2.2 Powers and Polynomials. 2.3 The Slope and the Tangent Line. 2.4 Derivative of the Sine and Cosine.
This is what makes calculus different from arithmetic and algebra. IMPORTANT FUNCTIONS Let me repeat the right name for the step from .1/to .2/:When we know the distance or the height or the function f.x/;calculus can find the speed ( velocity) and the slope and the derivative. That is differential calculus, going from Function .1/
MATH 221 { 1st SEMESTER CALCULUS LECTURE NOTES VERSION 2.0 (fall 2009) This is a self contained set of lecture notes for Math 221. The notes were written by Sigurd Angenent, starting from an extensive collection of notes and problems compiled by Joel Robbin. The LATEX and Python les
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Calculus produces functions in pairs, and the best thing a book can do early is to show you more of them. " { input t + function f -, output f (t) input 2 + function u + output v(2) 1 the domain input 7 + f (t) = 2t + 6 + f (7)= 20 rangein Note about the definition of a function. The idea behind the symbol f (t) is absolutely t
already is a version of the fundamental theorem of calculus. It will lead to the in-tegral R x 0 f(x) dx , derivative d dx f(x) and the fundamental theorem of calculus R x 0 d dt f(t )dt = x(0); d dx R x 0 1.11. This is a fantastic result. The goal of this course is to understand this theorem, and to apply it. Note that if we de ne [n]0 = 1;[n ...
