Ap(r) Calculus I

This book follows my lectures on Advanced Placement Calculus given in the International School of Economics of Kazakh-British Technical University in Almaty, Kazakhstan. In 2011-2017 first-year students of the ISE together with AP Calculus studied AP Statistics, AP Microeconomics, AP Macroeconomics. At the end of the first year after the internal, they passed external exams running and graded by College Board, Washington. In case of successful pass, they became students of the...
More Description

This book follows my lectures on Advanced Placement Calculus given in the International School of Economics of Kazakh-British Technical University in Almaty, Kazakhstan. In 2011-2017 first-year students of the ISE together with AP Calculus studied AP Statistics, AP Microeconomics, AP Macroeconomics. At the end of the first year after the internal, they passed external exams running and graded by College Board, Washington. In case of successful pass, they became students of the International Program of University of London. The program followed that of London School of Economics and was directed those times by a team of the LSE.The key to a success in AP Calculus exams is knowledge of the theory (including all proofs) coupled with a serious practice. The system of the ISE worked as follows. Each of two semesters has 15 weeks. The first semester is devoted to Differential Calculus (Part I of this book), whereas the second to Integral Calculus (Part II). Each week students have 3 lecture hours, where not only theory is discussed but also typical problems are solved. There are also 2 hours of practice with assistants, when students regularly write quizzes. Finally, students write home works in class for one hour. Home works are announced on the site, and students may solve problems together, but they must write solutions in class individually. In the first semester students have one midterm, one mock, and the final exam. In the second semester, an extra mock exams is added. The problems are more difficult than those of College Board.This book contains in equal proportions a theoretical part, a practical part, and, finally, real exams in the AP format given in the ISE in 2015-16. The exams, as well as the solutions, are organized as addendums at the end of this book.Traditionally, every Calculus course begins with the so-called Pre-Calculus part. In this book, it is replaced with Descartes' theory of tangents, and Descartes Analytic Geometry. So, the main object of Differential Calculus appears at the very beginning.Being so successful in so many cases Descates' method, however, fails for logarithms. These are very important functions since according to Weber-Fenchler law people feel the outside world through logarithms. The reason for this is that people's abilities to react to outside signals are very limited. Therefore, logarithms eliminate not important signals in favor of fast growing signals of exponential character.It is the Weber-Fenchler law which stands behind any banking system. Since people feel logarithmically the interest rates are calculated as fixed proportions of invested sums of money rather than fixed additive parts of investments. This topic is related to the number e=2.71828... playing an important role in Calculus. The number e is the base of the natural logarithm ln x. Logarithmic and exponential growths are crucial for AP Calculus exams.Limits and continuity are present in Chapter 3. The intermediate value property of continuous functions is always present in AP Calculus exams. A monotonic function is continuous if and only if it has the intermediate value property. This fact is used to prove the continuity of elementary functions.A special attention is paid to graphs plotting. From the very beginning, we promote the method of plotting graphs by special points. The complete graph can be obtained just by connection of the plotted part with simple smooth curves. This is the result of the principle, saying that a simple formula implies a simple graph.Problems on related rates are considered on concrete examples in section 4.5. Fifteen such problems are solved. They actually exhaust the list of all possible problems which one can face on exams.Applications to Economics and Finance are considered.This book includes three full exams with solutions and over 300 solved problems.It can be also useful for Cambridge International AS and A LevelMathematics exams.