Understanding the Course Calculus II: Problems, Solutions, and Tips. Calculus II is the result of mastering Calculus I. This second course in the Calculus sequence introduces you to exciting new techniques and applications of one of the most powerful mathematical tools ever invented. Equipped with Calculus II skills, you can solve a wide variety of problems in the physical, biological, and social sciences, engineering, economics, and other fields. Success in Calculus II also gives you a solid foundation for further study in mathematics and fulfills the mathematics requirements of many undergraduate majors. But beyond these benefits, you’ll find that the methods you learn in Calculus II are practical, interesting, and elegant, and involve ideas that are beautifully simple. Because calculus can model real-world situations, it has amazing applications, and these applications are fully demonstrated in Calculus II.
Understanding Calculus 2: Problems, Solutions, and Tips takes you on this exciting journey in 36 half-hour intensively illustrated lectures covering all the core topics of the second complete calculus course in high school at the Advanced Placement BC or College Board level. Covering the second semester course in college of Professor Bruce H. Edwards of the University of Florida, based on decades of teaching experience, he infuses his lectures with clear explanations, consistent study tips, pitfalls to avoid, and, best of all, hundreds of specially designed examples and practical problems to enrich and reinforce key concepts. Few calculus teachers are as qualified, accessible, or entertaining as Professor Edwards, who has won numerous teaching awards and written a series of best-selling calculus textbooks. Many calculus students give up trying to understand why a particular method works and resort to memorizing the steps of a solution. With Professor Edwards, the underlying concepts are always clear and constantly reinforced, making the path to learning the material much easier. Professor Edwards begins with a three-lecture review of the basic ideas of calculus. He also includes brief reviews of key concepts throughout the course, making Understanding Second Calculus a stand-alone lecture series for anyone who is already familiar with the two core calculus operations, differentiation and integration. Professor Edwards takes these ideas beyond the definitions, rules, and formulas that are at the heart of first-semester calculus and applies them in engaging ways. For example:
- Differential equations: This broad area uses derivatives – modeling population growth, nuclear decay, collapsing objects, and countless other processes involving change. Professor Edwards recalls that as a young mathematician, he spent a summer working for NASA, solving the equations for airplanes in flight.
- Infinite Series: Does adding an infinite sequence of numbers result in an infinite result? It doesn’t have to. The series may converge to a fixed value or diverge to infinity. Calculus can provide answers to many different types of infinite series and show familiar functions from algebra or trigonometry in surprising ways.
- Vectors: One of the geometric applications of differential and integral calculus is the analysis of vectors. These are quantities, such as velocity, that have both magnitude and direction. In Calculus II, you will learn techniques for evaluating vectors in the plane, which will allow you to solve problems involving moving and accelerating objects, whether they are on straight or curved paths.
Understanding Calculus II covers the above topics in great depth, especially infinite series, which you explore in 11 lectures. You also study other standard topics, including second semester calculus
- Integration Formulas and Techniques,
- merging areas and volumes,
- Taylor and Maclaren polynomials,
- L’Hôpital’s rule for evaluating limits,
- evaluation of improper integrals,
- calculations used in parametric equations and
- Calculations used in polar coordinates
Each of these very different applications of calculus involves the basic idea of limits. Professor Edwards says that calculus can be thought of as a “limiting machine” – a set of procedures for getting infinitely close to some value. One of the interesting features of calculus is its logical exactness combined with creative use of the mysterious entity of infinity. This unusual marriage leads to surprising and precise solutions to inaccessible problems. Calculus is full of fascinating features, puzzling paradoxes, and fun problems. Among the many things you will examine in understanding the second account are:
- Gabriel’s Horn: Rotate a simple curve around its axis and a three-dimensional figure appears, resembling an infinitely long trumpet. This geometric figure, called Gabriel’s horn, has an unusual property: its surface area is unlimited but volume is finite. See, calculus proves that this must be the case.
- Exciting Baseball: A baseball is hit at a speed of 100 feet per second at a 45-degree angle from 3 feet above home plate. Use Newton’s second law of motion and the derivative of the position function to determine whether or not the ball will be a home run, crossing a 10-foot-high fence 300 feet away.
- Cantor set: Remove the middle third of a line segment. Repeat with the remaining two pieces. Repeat ad infinitum. The endpoints of all parts form an infinite set. But what about the total length of all line segments? The sum of this infinite series reveals a surprising answer.
Understanding Calculus II is a valuable experience that you can study at your own pace. Professor Edwards often encourages you to pause the video and test yourself by solving a problem before revealing the answer. Those who can benefit from this attractive and flexible offer include:
- High school or college students currently in, or about to enroll in, Calculus II who want personalized coaching from an excellent teacher.
- High school students preparing for the College Board Advanced Placement Test in Calculus at the BC level.
- Students taking higher level math courses or professionals who want to review calculus. and
- Anyone interested in following one of life’s greatest intellectual adventures, one that has been solving tough problems for over 300 years.
A three-time Teacher of the Year at the University of Florida, Professor Edwards knows how to help students overcome obstacles on the way to mastering calculus. In this course, he uses a steady stream of on-screen equations, graphs, and other visual aids to document the key steps in solving sample problems. The accompanying workbook is designed to enhance each lecture with additional practice problems and worked solutions, as well as lecture summaries, tips, and problems, and formulas for derivatives, integration, and power series. Professor Edwards’ lectures also include a feature he calls “You Be the Teacher,” in which he reverses roles and challenges you to answer a specific question asked in class, come up with a suitable problem to explain, design a theory, or otherwise make yourself available to the instructor. Shoes – A valuable exercise in learning to think for yourself in the language of calculus. Calculus’s place at the end of the high school math curriculum makes it seem like the final destination. But that’s just the beginning. Calculus is a world unto itself, an ever-expanding set of tools that can solve the most complex problems in simple and often surprising ways. The deeper you go into calculus, the richer and better prepared you will be for the more advanced mathematics courses that will open their doors. In his final lecture, Professor Edwards focuses on where your mathematical studies can take you after this course. This country is exciting. Imagine entering a foreign country with the ability to speak the country’s language. Your opportunities to explore, interact, and learn more are nearly limitless. Understanding Calculus II helps you master one of the greatest accomplishments of the human mind.