Mathematics Manual: Calculus
Hello! Welcome to this online textbook on Calculus!
To students: This textbook was made for you! Mathematics is a subject often taught in a way that discourages exploration; that is, all of the figures are frozen or lacking, every explanation too verbose, the equations difficult to understand, and the concepts needlessly abstract. My goal in creating this book was to demonstrate to my readers that math is a subject that is teeming with life. It is a playground of thought, freedom made rigorous: I have done my best to give you a glimpse of these perspectives in this book. Mathematics (and particularly Calculus) will add clarity, color, and structure to your perspective of the world, but it requires struggle and intense effort. Do the exercises, as they will help reinforce what you learned while reading the text and playing with its many illustrations.
To teachers: Don’t think I forgot about you! I was once in your shoes! After focusing on student understanding, my second aim was to provide a free, open-source text that you can access anytime, anywhere (as long as you have Internet access 😄). You can use this resource however you wish: you can just use the diagrams, or you can have students read the text, too. You might even discover something new yourself!
This book was produced using Rbookdown, a package within the R programming language. Yihui Xie is the brilliant software developer behind Rbookdown and many other packages in R capable of generating professional documents. The plots were produced using the wonderful online Desmos Graphing Calculator. Eli Luberoff and his excellent team at Desmos Studio PBC have produced a tool to unleash student creativity and facilitate transmission of mathematical ideas.
This book will require a solid understanding of Geometry, Algebra, and the study of functions typically completed in a Precalculus course. While the author will devote time to laying some foundations, others will not receive such treatment.
Don’t panic! Often, the viewing window is not aligned with the graph. You can zoom in and out with a mouse or two fingers on the trackpad. The sidebar contains the equations used to make the plots, but it can be totally minimized by clicking the two arrows in the upper right hand corner of the sidebar.
Relax! You can simply reload the page or click the undo arrow at the top of the sidebar. You can play with the following picture of Sir Isaac Newton to get a sense of some of the controls:
There are many illustrations in this text used to aid the reader’s understanding. Play with the illustrations as much as you wish! Be sure to read the text too, so that you understand what the graphic is trying to convey.
Much of this life is having fun and exploring creation. Math is the grid of and provides structure to life, and Calculus is at its center. So in the end, I hope that, whoever you are, you sincerely enjoy using this book and learning what the great physicist Richard Feynman referred to as “the language God talks”: Calculus.