Sergey Khrushchev

AP (R) Calculus II Sergey Khrushchev

This book is the second part of my course ``AP Calculus'' given in the International School of Economics of KBTU, Almaty, in 2013-17. More details on AP exams of College board, Washington, can be found in the Preface to the first book ``AP(R) Calculus I.'' Similarly to it, this one contains theoretical, practical parts, and exams in the AP format in equal proportions. The exams, as well as the solutions, are organized as the addendums at the end of the book. The problems for these exams are carefully selected. There are many, so to say, conceptual problems. Therefore, even the students, who are not going to be involved in theoretical mathematics much, can pass only because they carefully study the solutions given in the book. This part presents Integral Calculus, i.e. evaluation of areas below curves and related volumes. It included the technique of evaluation indefinite integrals. The last chapter considers Ordinary Differential Equations and their applications related to the program of ``AP Calculus''. Wallis' ''Arithmetica Infinitorum'' contains a continued fraction found by Brouncker for 2/?. Since Brouncker's proof was not completely understood by Wallis, many mathematicians thought that it was lost for quite a long time. Even Euler mentioned this in one of his papers on continued fractions. My recovery of Brouncker's proof can be found at the end of Chapter 1. Main ideas of ''Arithmetica Infinitorum'' are in a good correspondence with the program of AP Calculus exams. Therefore, Chapter 1 is an extended version of Wallis' book in modern notations. It is shown how the problem of finding areas below curves leads to definite integrals. The chapter culminates with the Newton-Leibniz formula, which demonstrates an intriguing connection between the equations of tangents to a curve and areas below it.