Counter Examples in Calculus

the long version

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.