Calculus Problems And Solutions A. Ginzburg

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Calculus Problems and Solutions – A. Ginzburg This is the first of a series of problem books in analysis, analytic geometry, and higher algebra. The main purpose of this series is to provide the student with a rich collection of carefully selected material designed to increase his understanding and skill in handling problems in the above fields. To this end a large number of problems are presented and their solutions are given in full detail. To improve understanding, some prob­lems of more difficult character are included, the solution of which requires deeper insight in the topics treated. More than mil doscientos problems are presented in this book. Some of these were taken from examination papers of the Technion, Israel Institute of Technology. Others were drawn from various sources and there seems to be no point in endeavoring to trace their origins. The order of exposition adopted here seems quite natural; however, care has been taken so that the book may be used in a course in which the topics are arranged differently. Every section begins with a brief explanation of the basic notions and theorems to be used. In general, the theorems are given without proof. The main part of every section is devoted to problems, a large number of which are immediately followed by solutions; others have solutions to be found only at the end of the book. It is believed that this will encour­age the reader to solve the problem by himself and only afterwards to look for the printed solution. The comparison of solutions will often be beneficial, as it will afford a check on the work and, occasionally, encounters with new methods. Some problems, of course, have a variety of solutions and it may easily happen that the one given here is not the simplest possible. Remarks in this and other regards will be welcomed by the author. My thanks are due to doctor M. Edelstein, who read the manuscript and made many valuable suggestions, and to doctor Emory P. Starke who super­vised the production of the book. Contents: I: SEQUENCES1.1. Basic definitions and theorems1.2. Examples and exercises on general notions1.3. Representation of a number by sequences1.4. Evaluation of N(y también)1.5. Sequences given in the form n«+i «1.6. Methods for the evaluation of limits II: FUNCTIONS OF A SINGLE VARIABLE2.1. Definition and notation2.2. The elementary functions2.3. Domain of definition2.4. Even and odd functions2.5. Rational functions2.6. Logarithmic functions2.7. Trigonometric functions2.8. Hyperbolic functions2.9. Inverse functions2.10. The inverse trigonometric functions2.11. The inverse hyperbolic functions2.12. Composite functions2.13. Periodic functions III: LIMIT OF A FUNCTION3.1. Definitions and general exercises3.2. Evaluation of limits3.3. Continuity IV: DIFFERENTIAL CALCULUS FOR FUNCTIONS OF A SINGLE VARIABLE4.1. The notion of derivative and its physical and geometric interpretation4.2. Evaluating derivatives4.3. Evaluating derivatives of explicit functions4.4. Differentiation of implicit functions4.5. Parametric differentiation4.6. Special eases in calculating derivatives4.7. Higher derivatives4.8. Calculation of t/4.9. Graphical differentiation4.10. Various examples V: FUNDAMENTAL THEOREMS OF THE DIFFERENTIAL CALCULUS5.1. The theorems of Rolle, Lagrange, and Cauchy5.2. Taylor’s and Maclaurin’s formulas5.3. Indeterminate forms: L’Hdpital’s rule VI: APPLICATIONS OF DIFFERENTIAL CALCULUS6.1. Rate of change6.2. Locating intervals in which a function increases or decreases6.3. Mínima and maxima6.4. Concavity: points of inflection6.5. Asymptotes6.6. Curve tracing6.7. Graphs in polar coordinates6.8. Parametric equations6.9. Tangent and normal6.10. The order of contact6.11. Osculating circle, radius of curvature6.12. Evolute and involute6.13. Solution of equations by Newton’s approximation method VII: THE DIFFERENTIAL7.1. Definition of the differential7.2. The invariance of the form of the differential7.3. The differential as the primordial part of the increment of the function: application to approximate calculations7.4. Higher order differentials VIII: THE INDEFINITE INTEGRAL8.1. Definition and basic properties8.2. Immediate integrals8.3. The method of substitution8.4. Integration by parts8.5. Integrals of rational functions8.6. Irrational integrals8.7. Trigonometric integrals8.8. Integrals of exponential and hyperbolic functions8.9. Miscellaneous integrals IX: THE DEFINITE INTEGRAL9.1. Definition9.2. Basic properties of the definite integral9.3. Evaluation of the definite integral from its definition9.4. Estimation of definite integrals9.5. The orinan value theorem of integral calculus9.6. Integrals with variable limits9.7. Evaluation of definite integrals9.8. Changing the variable of integration9.9. Approximate integration9.10. Improper integrals9.11. Miscellaneous problems X: APPLICATIONS OF THE DEFINITE INTEGRAL10.1. Computation of plane areas10.2. Computation of arc length10.3. Computation of volumes10.4. Área of a surface of revolution10.5. Moment of mass: centroids10.6. Pappus’ theorems10.7. Moment of inertia10.8. Physics problems XI: INFINITE SERIES11.1. The general notion of a number series11.2. Convergence of series with positive terms11.3. Convergence of series with positive and negative terms11.4. Arithmetic operations on series11.5. Series of functions11.6. Power series: radius of convergence11.7. Taylor’s and Maclaurin’s scries: operations on power series11.8. Applications of Taylor’s and Maclaurin’s expansions XII: VARIOUS PROBLEMS PART TWO; SOLUTIONS, HINTS, ANSWERS INDEXLIST OF GREEK LETTERS Lenguaje: Inglés