TGC – Understanding Calculus II: Issues, Options, and Tips

However past these benefits, you will see that that the strategies you be taught in Calculus II are sensible, attention-grabbing, and elegant, involving concepts which might be fantastically easy. As a result of it may mannequin actual-life conditions, calculus has a tremendous vary of makes use of, and these functions come into full flower in Calculus II.

Understanding Calculus II: Issues, Options, and Tips takes you on this exhilarating journey in 36 intensively illustrated half-hour lectures that cowl all the key matters of the second full-yr calculus course in highschool on the School Board Superior Placement BC degree or a second-semester course in school. Drawing on many years of instructing expertise, Professor Bruce H. Edwards of the College of Florida enriches his lectures with crystal-clear explanations, frequent research ideas, pitfalls to keep away from, and—better of all—a whole bunch of examples and observe issues which might be particularly designed to clarify and reinforce key ideas.

Few calculus academics are as certified, accessible, or entertaining as Professor Edwards, who has gained a number of instructing awards and coauthored a finest-promoting collection of calculus textbooks. Many calculus college students hand over making an attempt to know why a specific process works and resort to memorizing the steps to an answer. With Professor Edwards, the underlying ideas are at all times clear and continually bolstered, which vastly eases the trail to studying the fabric.

Get behind the Wheel of the “Limit Machine”

Professor Edwards begins with a 3-lecture evaluate of the basic concepts of calculus. He additionally contains transient evaluations of main ideas all through the course, which makes Understanding Calculus II a self-contained lecture collection for anybody who’s already aware of the 2 predominant operations of calculus, differentiation and integration. Professor Edwards takes these concepts past the definitions, guidelines, and formulation which might be the main focus of first-semester calculus and applies them in intriguing methods. For instance:

• Differential equations: This far-reaching subject places derivatives to work—modeling inhabitants development, nuclear decay, falling objects, and numerous different processes involving change. Professor Edwards remembers that as a younger mathematician, he spent summers working for NASA, fixing differential equations for plane in flight.
• Infinite collection: Does including an infinite sequence of numbers give an infinite outcome? Not essentially. The collection could converge on a selected worth, or it could diverge to infinity. Calculus can present the reply for various kinds of infinite collection and symbolize acquainted features from algebra or trigonometry in shocking methods.
• Vectors: Among the many geometric functions of calculus is the evaluation of vectors. These are portions, similar to velocity, which have each magnitude and course. In Calculus II, you be taught methods for evaluating vectors within the aircraft, permitting you to resolve issues involving transferring and accelerating objects, whether or not they’re on a straight or curved path.

Understanding Calculus II covers the above topics in appreciable depth, notably infinite collection, which you discover in 11 lectures. You additionally research different normal matters in second-semester calculus, together with

• integration formulation and methods,
• integrating areas and volumes,
• Taylor and Maclaurin polynomials,
• L’Hôpital’s rule for evaluating limits,
• evaluating improper integrals,
• calculus utilized to parametric equations, and
• calculus utilized to polar coordinates.

These very totally different functions of calculus every contain the important concept of the restrict. Professor Edwards notes that calculus may be regarded as a “limit machine”—a set of procedures for approaching infinitely near a worth. One of many attention-grabbing options of calculus is its logical rigor mixed with its artistic use of the mysterious entity of the infinite. From this uncommon marriage emerge astonishingly exact options to in any other case inaccessible issues.

Discover the Immense Riches of Calculus

Calculus is stuffed with fascinating properties, baffling paradoxes, and entertaining issues. Among the many many you examine in Understanding Calculus II are these:

• Gabriel’s Horn: Rotate a easy curve round its axis and you get a 3-dimensional form that appears like an infinitely lengthy trumpet. Referred to as Gabriel’s Horn, this geometric determine has an uncommon property: It has infinite floor space however finite quantity. See calculus show that this have to be so.
• Baseball thriller: A baseball 3 toes above house plate is hit at 100 toes per second and at an angle of 45 levels. Make use of Newton’s second legislation of movement and the spinoff of the place operate to find out if the ball can be a house run, clearing a ten-foot-excessive fence 300 toes away.
• Cantor Set: Take away the center third of a line section. Repeat with the 2 items that stay. Repeat once more advert infinitum. The tip factors of all of the items will kind an infinite set. However what concerning the complete size of all the road segments? Summing this infinite collection reveals the shocking reply.

Grasp Calculus on Your Personal Schedule

Understanding Calculus II is an immensely rewarding expertise that you may research at your individual tempo. Professor Edwards usually encourages you to pause the video and take a look at your self by fixing an issue earlier than he reveals the reply. Those that will profit from this partaking and versatile presentation embrace:

• highschool or school college students presently, or about to be, enrolled in Calculus II who need private teaching from an impressive trainer;
• highschool college students getting ready for the School Board Superior Placement take a look at in Calculus on the BC degree;
• college students in greater-degree math programs or professionals who desire a rigorous evaluate of calculus; and
• anybody thinking about pursuing one in every of life’s biggest mental adventures, which has been fixing tough issues for over 300 years.

A 3-time Trainer of the 12 months on the College of Florida, Professor Edwards is aware of easy methods to assist college students surmount the hindrances on their path to mastering calculus. On this course, he makes use of a gradual stream of on-display equations, graphs, and different visible aids to doc the important thing steps in fixing pattern issues. The accompanying workbook is designed to strengthen every lecture with extra observe issues and labored-out options, in addition to lecture summaries, ideas, and pitfalls; and formulation for derivatives, integration, and energy collection.

Professor Edwards’s lectures additionally embrace a function he calls “You Be the Teacher,” during which he reverses roles, difficult you to reply a typical query posed within the classroom, design an appropriate drawback for instance a precept, or in any other case put your self within the teacher’s footwear—a useful train in studying to suppose for your self within the language of calculus.

Open Doorways with Your New Fluency

The place of calculus on the finish of the highschool math curriculum makes it seem to be a closing vacation spot. However additionally it is solely a starting. Calculus is a world unto itself, an ever-increasing assortment of instruments that may resolve essentially the most intractable issues in ingenious and usually shocking methods. The deeper you go into calculus, the richer it will get and the higher you’re ready for much more superior math programs that open doorways of their very own.

In his final lecture, Professor Edwards appears to be like forward to the place your math research could take you after this course. It’s thrilling terrain. Think about arriving abroad outfitted with the power to talk the nation’s language. Your alternatives for exploration, interplay, and additional studying are virtually limitless. That’s what Understanding Calculus II does to your fluency in one of many biggest achievements of the human thoughts.