Animating Calculus

Calculus and change. The two words go together. Calculus is about change, and approaches to teaching calculus are changing dramatically. Thus it is both timely and appropriate to apply techniques of animation to the varied and important graphical aspects of calculus. AB a computer algebra system, Mathematica is an excellent tool for numerical and symbolic computation. It also has the power to generate striking and colorful graphical images and to animate them dynamically. The combination of these capabilities makes Mathematica a natural resource for exploring the changing world of calculus and approaches to mastering it. In addition, Mathematica notebooks are easy to edit, allowing flexible input for commands to Mathematica and stylish text for explanation to the reader. Much has been written about the use and importance of technology in the teaching and learning of calculus. We will not repeat the arguments or feign objectivity. We are enthusiastic believers in the value of a significant laboratory experience as part oflearning calculus, and we think Mathematica notebooks are a most appropriate and exciting way to provide that experience. The notebooks that follow represent our choice of laboratory topics for a course in one-variable calculus. They offer a balance between what we think belongs in a first-year calculus course and what lends itself well to exploration in a Mathematica laboratory setting.

Animating Calculus is designed to help you explore calculus and visualize concepts through the use of computation and animation. This collection of 22 labs, together with the computer algebra system Mathematica, can be used for self-study, demonstration, or as a laboratory supplement to an existing calculus sequence. Standard calculus topics as well as new and unusual extensions and applications are presented, including derivatives and rate of change, calculus and landing airplanes, population dynamics and iteration, the fundamental theorem, The Buffon needle problem, numerical and symbolic integration, rolling wheels (round and square), subtleties of the harmonic series, and more. Animating Calculus includes exercises and demonstrations that focus on important and fundamental ideas and applications rather than the everyday mechanics of a computer algebra system. Sophisticated animations are used to clarify geometric concepts in calculus. In addition, discussions of numerical and graphical pitfalls help the student to understand the importance of verifying results. Originally published by W. H. Freeman, this new TELOS edition of Animating Calculus includes the full set of labs for DOS/Windows as well as Macintosh platforms.

This workbook contains over two dozen laboratory modules that use Mathematica to enliven the teaching and learning of calculus. The labs range from standard calculus topics to new and unusual extensions and applications.

Animation - Process, Cognition and Actuality presents a uniquely philosophical and multi-disciplinary approach to the scholarly study of animation, by using the principles of process philosophy and Deleuzian film aesthetics to discuss animation practices, from early optical devices to contemporary urban design and installations. Some of the original theories presented are a process-philosophy based theory of animation; a cognitive theory of animation; a new theoretical approach to the animated documentary; an original investigative approach to animation; and unique considerations as to the convergence of animation and actuality. Numerous animated examples (from all eras and representing a wide range of techniques and approaches – including television shows and video games) are examined, such as Fantastic Mr. Fox (2009), Madame Tutli-Putli (2007), Gertie the Dinosaur (1914), The Peanuts Movie (2015), Grand Theft Auto V (2013) and Dr. Katz: Professional Therapist (1995–2000). Divided into three sections, each to build logically upon each other, Dan Torre first considers animation in terms of process and process philosophy, which allows the reader to contemplate animation in a number of unique ways. Torre then examines animation in more conceptual terms in comparing it to the processes of human cognition. This is followed by an exploration of some of the ways in which we might interpret or 'read' particular aspects of animation, such as animated performance, stop-motion, anthropomorphism, video games, and various hybrid forms of animation. He finishes by guiding the discussion of animation back to the more tangible and concrete as it considers animation within the context of the actual world. With a genuinely distinctive approach to the study of animation, Torre offers fresh philosophical and practical insights that prompt an engagement with the definitions and dynamics of the form, and its current literature.

An accessible introduction to real analysis and its connectionto elementary calculus Bridging the gap between the development and history of realanalysis, Introduction to Real Analysis: An EducationalApproach presents a comprehensive introduction to real analysiswhile also offering a survey of the field. With its balance ofhistorical background, key calculus methods, and hands-onapplications, this book provides readers with a solid foundationand fundamental understanding of real analysis. The book begins with an outline of basic calculus, including aclose examination of problems illustrating links and potentialdifficulties. Next, a fluid introduction to real analysis ispresented, guiding readers through the basic topology of realnumbers, limits, integration, and a series of functions in naturalprogression. The book moves on to analysis with more rigorousinvestigations, and the topology of the line is presented alongwith a discussion of limits and continuity that includes unusualexamples in order to direct readers' thinking beyond intuitivereasoning and on to more complex understanding. The dichotomy ofpointwise and uniform convergence is then addressed and is followedby differentiation and integration. Riemann-Stieltjes integrals andthe Lebesgue measure are also introduced to broaden the presentedperspective. The book concludes with a collection of advancedtopics that are connected to elementary calculus, such as modelingwith logistic functions, numerical quadrature, Fourier series, andspecial functions. Detailed appendices outline key definitions and theorems inelementary calculus and also present additional proofs, projects,and sets in real analysis. Each chapter references historicalsources on real analysis while also providing proof-orientedexercises and examples that facilitate the development ofcomputational skills. In addition, an extensive bibliographyprovides additional resources on the topic. Introduction to Real Analysis: An Educational Approach isan ideal book for upper- undergraduate and graduate-level realanalysis courses in the areas of mathematics and education. It isalso a valuable reference for educators in the field of appliedmathematics.

Students studying different branches of computer graphics have to be familiar with geometry, matrices, vectors, rotation transforms, quaternions, curves and surfaces and as computer graphics software becomes increasingly sophisticated, calculus is also being used to resolve its associated problems. In this 2nd edition, the author extends the scope of the original book to include applications of calculus in the areas of arc-length parameterisation of curves, geometric continuity, tangent and normal vectors, and curvature. The author draws upon his experience in teaching mathematics to undergraduates to make calculus appear no more challenging than any other branch of mathematics. He introduces the subject by examining how functions depend upon their independent variables, and then derives the appropriate mathematical underpinning and definitions. This gives rise to a function’s derivative and its antiderivative, or integral. Using the idea of limits, the reader is introduced to derivatives and integrals of many common functions. Other chapters address higher-order derivatives, partial derivatives, Jacobians, vector-based functions, single, double and triple integrals, with numerous worked examples, and over a hundred and seventy colour illustrations. This book complements the author’s other books on mathematics for computer graphics, and assumes that the reader is familiar with everyday algebra, trigonometry, vectors and determinants. After studying this book, the reader should understand calculus and its application within the world of computer graphics, games and animation.

Have you ever wanted to include believable physical behaviors in your games and projects to give them that extra edge? Physics for JavaScript Games, Animation, and Simulations teaches you how to incorporate real physics, such as gravity, friction, and buoyancy, into your HTML5 games, animations, and simulations. It also includes more advanced topics, such as particle systems, which are essential for creating effects such as sparks or smoke. The book also addresses the key issue of balancing accuracy and simplicity in your games and simulations, and the final chapters provide you with the information and the code to make the right choice for your project. Physics for JavaScript Games, Animation, and Simulations assumes you have a basic knowledge of JavaScript and HTML5. However, no previous knowledge of physics is required—only some very basic math skills. The authors present everything from basic principles to advanced concepts in an approachable way, so you’ll be able to follow the logic and easily adapt the principles to your own applications. The book is packed full of practical examples of how you can apply physics to your own games and applications. Spring behaviors can be used for anything from tweaking lowrider suspension to creating cloth simulation; flotation mechanics enable the simulation of submersibles or dirigibles; you can even create your own solar system with accurate orbits and gravity. It doesn’t matter if you’re modeling the Lorentz force in an electromagnetic field or you’re modeling the lift force in a flight simulator, Physics for JavaScript Games, Animation, and Simulations enables you to fill your games and applications with accurate and realistic effects.

Physics for Flash Games, Animation, and Simulations teaches ActionScript programmers how to incorporate real physics into their Flash animations, games, user interfaces, and simulations. Introduces Flash physics in an accurate, but approachable way, covering what is required to produce physically realistic simulations (as opposed to animations that look roughly right) Packed full of practical examples of how physics can be applied to your own games and applications Addresses the diverse needs of game developers, animators, artists, and e-learning developers The book assumes a basic knowledge of ActionScript and Flash. However, no previous knowledge of physics is required—only some very basic math skills. The authors present everything from basic principles to advanced concepts, so you'll be able to follow the logic and easily adapt the principles to your own applications. The book builds on your physics knowledge, enabling you to create not only visual effects, but also more complex models and simulations.