Counter Examples in Calculus (PDF)
(Sprache: Englisch)
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...
“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...
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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.
“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.
Autoren-Porträt von Dr. Sergiy Klymchuk
Dr Sergiy KlymchukAssociate Professor Auckland University of Technology, New Zealand
Qualifications: BSc Odessa, PhD Odessa
Professional Membership
Member of the Royal Society of New Zealand (RSNZ) – since 2006
Member of the Editorial Board of the International Journal “Mathematics and Informatics Quarterly” published by SCT Publishing, Singapore – since 2005
Member of the International Group on Psychology in Mathematics Education (PME) – since 2001
Member of the International Group on Improvements in Mathematics Education (CIEAEM) – since 1999
Member of the New Zealand Association of Mathematics Teachers (NZAMT) – since 1999
Member of the International Group on Teaching Mathematical Modelling and Applications (ICTMA) – since 1997
Teaching Areas: Calculus, operations research, statistics
Research Areas: Mathematics education
Current Research, Mathematics education: Enhancing students’ conceptual understanding and critical thinking in mathematics rigour, precision and students’ misconceptions in mathematics analysing the transition from secondary to university education in mathematics
effective teaching of mathematical modelling skills
Previous Research
Differential equations: Asymptotic methods for differential equations and inclusions optimal control theory. For a complete bio of university papers and research go to the AUT staff website: http://www.aut.ac.nz/schools/computing_and_mathematical_sciences/our_staff/sergiy_klymchuk.htm
Bibliographische Angaben
- Autor: Dr. Sergiy Klymchuk
- 2007, Englisch
- ISBN-10: 0473127962
- ISBN-13: 9780473127961
- Erscheinungsdatum: 03.12.2007
Abhängig von Bildschirmgröße und eingestellter Schriftgröße kann die Seitenzahl auf Ihrem Lesegerät variieren.
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- Größe: 1.50 MB
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Sprache:
Englisch
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