DIFFERENTIAL AND INTEGRAL CALCULUS, I Contents

1 DIFFERENTIAL AND INTEGRAL CALCULUS, ILECTURE NOTES (TEL AVIV UNIVERSITY, FALL 2009)ContentsPreliminariesiPreparatory readingiReadingiProblem booksiBasic notationiiBasic Greek lettersiv1. Real Infinite decimal The Application: solution of equationsn= The distance onR62. Upper and lower Maximum/minimum Some corollaries:103. Three basic lemmas:Cantor, Heine-Borel, The nested intervals The finite subcovering The accumulation Appendix: Countable and uncountable subsets ofR144. Sequences and their Fundamental properties of the limits195. Convergent Two More examples256. Cauchy s sequences.

2 Upper and lower Cauchy s Upper and lower Convergence in wide sense317. Subsequences and partial : 29 October, NOTES (TEL AVIV, 2009) Partial limits338. Infinite Cauchy s criterion for convergence. Absolute Series with positive terms. Convergence tests389. Rearrangement of the infinite Be careful! Rearrangement of the Rearrangement of conditionally convergent series4310. Limits of functions. Basic Cauchy s definition of Heine s definition of Limits and arithmetic The first remarkable limit: limx 0sinxx= Limits at infinity and infinite Limits of monotonic functions5211.

3 The exponential function and the The functiont7 The logarithmic function The second remarkable symbols osmall and limx (1 +1x)x= Infinitesimally small values and the symbolsoand .5813. Continuous functions, Points of Local properties of continuous functions6314. Continuous functions, Global properties of continuous Uniform Inverse functions7015. The Definition and some Some Derivative of the inverse function and of the composition7516. Applications of the Local linear The tangent Lagrange Derivatives of higher Definition and examples83 DIFFERENTIAL AND INTEGRAL CALCULUS, The Leibniz Basic theorems of the DIFFERENTIAL calculus:Fermat, Rolle, Theorems of Fermat and Rolle.

4 Local Mean-value theorems9219. Applications of fundamental L Hospital s Appendix: Algebraic numbers9820. x sinx x,0 x +x<log(1 +x)< x,x > 1,x6= Bernoulli s Young s H older s Minkowski s inequality10521. Convex functions. Jensen s Fundamental properties of convex Jensen s inequality11122. The Taylor Local polynomial approximation. Peano s The Taylor remainder. Theorems of Lagrange and Cauchy11423. Taylor expansions of elementary The exponential The sine and cosine The logarithmic The binomial The Taylor series for Some Application to the limits12324.

5 The complex Basic definitions and Geometric representation of complex numbers. The Convergence inC12725. The fundamental theorem of algebra and its The theorem and its Factoring the Rational functions. Partial fraction decomposition13026. Complex exponential Absolutely convergent The complex exponent134 DIFFERENTIAL AND INTEGRAL CALCULUS, IiPreliminariesPreparatory books are intended for high-school students who likemath. All three books are great, my personal favorite is the first one.(1)R. Courant, H. Robbins, I. Stewart, What is mathematics, Oxford, 1996 (orearlier editions).(2)T.

6 W. Korner, The pleasures of counting, Cambridge U. Press, 1996.(3)K. M. Ball, Strange curves, counting rabbits, and other mathematical explo-rations, Princeton University Press, are many good textbooks in analysis, though I am not going to followany of them too closely. The following list reflects my personal taste:(1)V. A. Zorich, Mathematical analysis, , Springer, 2004.(2)A. Browder, Mathematical analysis. An introduction. Undergraduate Texts inMathematics. Springer-Verlag, New York, 1996.(3)R. Courant and F. John, Introduction to calculus and analysis, , Springer,1989 (or earlier editions).(4)D. Maizler, Infinitesimal calculus (in Hebrew).

7 (5)G. M. Fihtengol tz, Course of DIFFERENTIAL and INTEGRAL Calculus, vol. I (inRussian)(6)E. Hairer, G. Wanner, Analysis by its history, Springer, last book gives a very interesting and motivated exposition of the main ideas ofthis course given in the historical may find helpful informal discussions of various ideas related to this course (aswell to the other undergraduate courses) at the web page of Timothy ~wtg10 suppose that the students attend in parallel with this course the course Introductionto the set theory , or the course Discrete Mathematics . The notes (in Hebrew) ofMoshe Jarden might be ~jarden/ those of you who are interested to try to solve more difficultand interesting problems and exercises, I strongly recommend to look at two excellentcollections of problems:(1)B.

8 M. Makarov, M. G. Goluzina, A. A. Lodkin, A. N. Podkorytov, Selectedproblems in real analysis, American Mathematical Society, 1992.(2)G. Polya, G. Szeg o, Problems and theorems in analysis (2 volumes) Springer,1972 (there are earlier editions).iiLECTURE NOTES (TEL AVIV, 2009)Basic from logic. or and negation= yields is equivalent toExample:(x2 3x+ 2 = 0) ((x= 1) (x= 2))Quantifiers: exists !exists and unique (warning: this notation isn t standard) for everySet-theoretic notation. belongs/ does not belong subset empty set intersection of sets union of sets#(X)cardinality of the setXX\Y={x X:x / Y}complement toYinXExample:(X Y) := x( (x X) = (x Y) )We shall freely operate with these notion during the course.

9 Usually, the sets we dealwith are subsets of the set of real of reals:Nnatural numbers (positive integers)ZintegersZ+=N {0}non-negative integersQrational numbersRreal numbers[a, b] :={x R:a x b}closed interval (one point sets are closedintervals as well)(a, b) :={x R:a < x < b}open interval(a, b] and [a, b) semi-open intervalsSums and j=1aj=a1+a2+..+ann j=1aj=a1 a2 .. anDIFFERENTIAL AND INTEGRAL CALCULUS, IiiiSome if and only if wlog without loss of generality RHS, LHS right-hand side , left-hand side qed end of the proof 1. Often is replaced by the box like this one:2:= according to the definition (the same asdef= )1 quod erat demonstrandum (in Latin), which was to be demonstrated ivLECTURE NOTES (TEL AVIV, 2009)Basic Greek letters.

10 Alpha beta , gamma , delta epsilon zeta eta , theta iota kappa , lambda mu nu , xi , pi rho , sigma tau , upsilon , phi chi , psi , omegaExercise:Translate from the Greek the word . DIFFERENTIAL AND INTEGRAL CALCULUS, decimal of you have an idea what are the real instance, we often think of the real numbers as strings of elements of the set{0,1,2,3,4,5,6,7,8,9}preceded by a sign (we write only a minus sign, the absence ofthe sign means that the sign is positive). A finite string of elements of this set followed bya decimal point followed by an infinite string of elements of this set.