مثال‌های نقض در حساب دیفرانسیل و انتگرال (منابع کلاس درس)

Counterexamples in Calculus serves as a supplementary resource to enhance the learning experience in single variable calculus courses. This book features carefully constructed incorrect mathematical statements that require students to create counterexamples to disprove them. Methods of producing these incorrect statements vary. At times the converse of a well-known theorem is presented. In other instances crucial conditions are omitted or altered or incorrect definitions are employed. Incorrect statements are grouped topically with sections devoted to: Functions, Limits, Continuity, Differential Calculus and Integral Calculus. This book aims to fill a gap in the literature and provide a resource for using counterexamples as a pedagogical tool in the study of introductory calculus. There are ninety incorrect statements in the book. Three examples are:“The tangent to a curve at a point is the line which touches the curve at that point but does not cross it there.” “If a function is twice differentiable at a local minimum point, then its second derivative is positive at that point.” “If a function is continuous on R and the tangent line exists at any point on its graph then the function is differentiable at any point on R.” The book then constructs a function that disproves each statement. This pedagogical technique of asking for counter-examples moves students beyond over-reliance on learned strategies, to an improvement in their appreciation of the properties of limits, continuity, functions, and calculus.The incorrect statements will sharpen an awareness of definitions and notation, and prompt the reader to re-examine ‘certainties’ with greater scrutiny! Most of the problems will be within the reach of first year Calculus students.The solutions are provided with large, clear graphs of each function discussed. Counterexamples in Calculus serves as a supplementary resource to enhance the learning experience in single variable calculus courses. This book features carefully constructed incorrect mathematical statements that require students to create counterexamples to disprove them. Methods of producing these incorrect statements vary. At times the converse of a well-known theorem is presented. In other instances crucial conditions are omitted or altered or incorrect definitions are employed. Incorrect statements are grouped topically with sections devoted functions, limits, continuity, differential calculus and integral calculus. This book aims to fill a gap in the literature and provide a resource for using counterexamples as a pedagogical tool in the study of introductory calculus. In that light it may well be useful for high school teachers and university faculty as a teaching resource, for high school and college students as a learning resource, and as a professional development resource for calculus instructors. Cover......Page 1 Copyright page......Page 3 Title page......Page 4 Foreword......Page 8 Contents......Page 10 Introduction......Page 12 Counterexamples in Calculus......Page 18 I Statements......Page 22 1. Functions......Page 24 2. Limits......Page 26 3. Continuity......Page 28 4. Differential Calculus......Page 30 5. Integral Calculus......Page 34 II Suggested Solutions......Page 38 1. Functions......Page 40 2. Limits......Page 50 3. Continuity......Page 56 4. Differential Calculus......Page 68 5. Integral Calculus......Page 96 References......Page 110 About the Author......Page 112 Counterexamples In Calculus Serves As A Supplementary Resource To Enhance The Learning Experience In Single Variable Calculus Courses. This Book Features Carefully Constructed Incorrect Mathematical Statements That Require Students To Create Counterexamples To Disprove Them. --from Publisher Description Statements. Functions -- Limits -- Continuity -- Differential Calculus -- Integral Calculus -- Suggested Solutions. Sergiy Klymchuk. Originally Published: New Zealand : Maths Press, 2004. Includes Bibliographical References (p. 99-100). "Counterexamples in Calculus serves as a supplementary resource to enhance the learning experience in single variable calculus courses. This book features carefully constructed incorrect mathematical statements that require students to create counterexamples to disprove them. Methods of producing these incorrect statements vary. At times the converse of a well-known theorem is presented. In other instances crucial conditions are omitted or altered or incorrect definitions are employed."--Back cover.