Classroom Resource Materials : Paradoxes and Sophisms in Calculus.

cover -- copyright page -- title page -- Contents -- Introduction -- Acknowledgments -- I Paradoxes -- Functions and Limits -- Laying bricks -- Spiral curves -- A paradoxical fractal curve: the Koch snowflake -- A tricky fractal area: the Sierpinski carpet -- A mysterious fractal set: the Cantor ternary set -- A misleading sequence -- Remarkable symmetry -- Rolling a barrel -- A cat on a ladder -- Sailing -- Encircling the Earth -- A tricky equation -- A snail on a rubber rope -- Derivatives and Integrals -- An alternative product rule -- Missing information? -- A paint shortage -- Racing marbles -- A paradoxical pair of functions -- An unruly function -- Jagged peaks galore -- Another paradoxical pair of functions -- II Sophisms -- Functions and Limits -- Evaluation of lim_n _k=1n1n2+k proves that 1=0. -- Evaluation of lim_x0 (xsin1x ) proves that 1 = 0. -- Evaluation of lim_x0+ (xx) shows that 1 = 0. -- Evaluation of lim_n [n]n demonstrates that 1=. -- Trigonometric limits prove that sinkx = k sinx. -- Evaluation of a limit of a sum proves that 1=0. -- Analysis of the function x+yx-y proves that 1 = -1. -- Analysis of the function ax+yx+ay proves that a = 1a, for any value a 0. -- One-to-one correspondences imply that 1 = 2. -- Aristotle's wheel implies that R = r. -- Logarithmic inequalities show 2 &gt -- 3. -- Analysis of the logarithm function implies 2 &gt -- 3. -- Analysis of the logarithm function proves 14 &gt -- 12. -- Limit of perimeter curves shows that 2 = 1. -- Limit of perimeter curves shows = 2. -- Serret's surface area definition proves that = . -- Achilles and the tortoise -- Reasonable estimations lead to 1,000,000 2,000,000. -- Properties of square roots prove 1 = -1. -- Analysis of square roots shows that 2=-2. -- Properties of exponents show that 3 = -3. -- A slant asymptote proves that 2 = 1.

Euler's interpretation of series shows 12 = 1-1+1-1+@let@token . -- Euler's manipulation of series proves -1&gt -- &gt -- 1. -- A continuous function with a jump discontinuity -- Evaluation of Taylor series proves ln2=0. -- Derivatives and Integrals -- Trigonometric integration shows 1 = C, for any real number C. -- Integration by parts demonstrates 1 = 0. -- Division by zero is possible. -- Integration proves sin2 x = 1 for any value of x. -- The u-substitution method shows that 2 &lt -- 0 &lt -- . -- ln2 is not defined. -- is not defined. -- Properties of indefinite integrals show 0=C, for any real number C. -- Volumes of solids of revolution demonstrate that 1 = 2. -- An infinitely fast fall -- A positive number equals a negative number. -- The power rule for differentiation proves that 2=1. -- III Solutions to Paradoxes -- Functions and Limits -- Laying bricks -- Spiral curves -- A paradoxical fractal curve: the Koch snowflake. -- A tricky fractal area: the Sierpinski carpet -- A mysterious fractal set: the Cantor ternary set -- A misleading sequence -- Remarkable symmetry: Reuleaux polygons -- Rolling a barrel -- A cat on a ladder -- Sailing -- Encircling the Earth -- A tricky equation -- A snail on a rubber rope -- Derivatives and Integrals -- An alternative product rule -- Missing information? -- A paint shortage -- Racing marbles -- A paradoxical pair of functions -- An unruly function -- Jagged peaks galore -- Another paradoxical pair of functions -- IV Solutions to Sophisms -- Functions and Limits -- Evaluation of lim_n n_k=1 1n2+k proves that 1 = 0. -- Evaluation of lim_x 0 ( xsin1x ) proves that 1 = 0. -- Evaluation of lim_x 0+ (xx) shows that 1 = 0. -- Evaluation of lim_n [n]n demonstrates that 1 = . -- Trigonometric limits prove that sinkx = ksinx. -- Evaluation of a limit of a sum proves that 1 = 0.

Analysis of the function x + yx - y proves that 1 = -1. -- Analysis of the function ax + yx + ay proves that a = 1a, for any value a 0. -- One-to-one correspondences imply that 1 = 2. -- Aristotle's wheel implies that R = r. -- Logarithmic inequalities show 2 &gt -- 3. -- Analysis of the logarithm function implies 2 &gt -- 3. -- Analysis of the logarithm function proves 14 &gt -- 12. -- Limit of perimeter curves shows that 2 = 1. -- Limit of perimeter curves shows = 2. -- Serret's surface area definition proves that = . -- Achilles and the tortoise -- Reasonable estimations lead to 1,000,000 2,000,000. -- Properties of square roots prove 1 = -1. -- Analysis of square roots shows that 2 = -2. -- Properties of exponents show that 3=-3. -- A slant asymptote proves that 2=1. -- Euler's interpretation of series shows 12 = 1-1+1-1+@let@token . -- Euler's manipulation of series proves -1&gt -- &gt -- 1. -- A continuous function with a jump discontinuity. -- Evaluation of Taylor series proves ln2=0. -- Derivatives and Integrals -- Trigonometric Integration shows 1 = C, for any real number C. -- Integration by parts demonstrates 1 = 0. -- Division by zero is possible. -- Integration proves sin2 x = 1 for any value of x. -- The u-substitution method shows that 2 &lt -- 0 &lt -- . -- ln2 is not defined. -- is not defined. -- Properties of indefinite integrals show 0 = C, for any real number C. -- Volumes of solids of revolution demonstrate that 1 = 2. -- An infinitely fast fall -- A positive number equals a negative number. -- The power rule for differentiation proves that 2=1. -- Bibliography -- About the Authors.

Paradoxes and Sophisms in Calculus offers a delightful supplementary resource to enhance the study of single variable calculus. By the word paradox the authors mean a surprising, unexpected, counter-intuitive statement that looks invalid, but in fact is true. The word sophism describes intentionally invalid reasoning that looks formally correct, but in fact contains a subtle mistake or flaw. In other words, a sophism is a false proof of an incorrect statement. A collection of over fifty paradoxes and sophisms showcases the subtleties of this subject and leads students to contemplate the underlying concepts. A number of the examples treat historically significant issues that arose in the development of calculus, while others more naturally challenge readers to understand common misconceptions. Sophisms and paradoxes from the areas of functions, limits, derivatives, integrals, sequences, and series are explored.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.