Counterexamples in Calculus

Counterexamples in Calculus

Sergiy Klymchuk
Copyright Date: 2010
Edition: REV - Revised, 1
Pages: 112
https://www.jstor.org/stable/10.4169/j.ctt5hh9t5
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  • Book Info
    Counterexamples in Calculus
    Book Description:

    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. Counterexamples 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; high school and college students as a learning resource;and as a professional development resource for calculus instructors.

    eISBN: 978-1-61444-109-0
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-vi)
  2. Foreword
    (pp. vii-viii)
    John Mason

    This book offers to students and teachers who know that there is more to learning calculus than solving problems mechanically, a welcome and refreshing antidote to rote learning. It is consistent with views I have put forward with A. Watson inMathematics as a Constructive Activity: The Role of Learner-Generated Examples(Mahwah: Erlbaum, 2005). Mathematics is a constructive activity, and a central aspect of learning mathematics is enriching the space of examples that come to mind when one encounters a technical term. The care and precision needed for using and doing mathematics require access to a wide range of “familiar”...

  3. Table of Contents
    (pp. ix-x)
  4. Introduction
    (pp. 1-6)

    Counterexamples in Calculusis a resource for single-variable calculus courses. The book challenges students to provide counterexamples to carefully constructedincorrectmathematical statements. Some of the incorrect statements are converses of well-known theorems. Others come from altering or omitting conditions in theorems, or from applying incorrect definitions. I have grouped the incorrect statements into five sections: Functions, Limits, Continuity, Differential Calculus and Integral Calculus. And I have arranged the statements in each section in order of increasing difficulty, emphasizing early in each section some standard misconceptions. The more challenging statements oftenseemcorrect to students, who may be hard pressed...

  5. Counterexamples in Calculus
    (pp. 7-10)

    This book is about counterexamples. Deciding on an assertion’s validity is important in the information age. A counterexample can quickly and easily show that a given statement is false. One counterexample is all you need to disprove a statement! Counterexamples thus offer powerful and effective tools for mathematicians, scientists, and researchers. They can indicate that a hypothesis is wrong or a research proposal, misguided. Before attempting a proof for an assertion, looking for counterexamples may save an investigator lots of time and effort.

    The search for counterexamples has been important in the history of mathematics. I mention three famous instances....

  6. I Statements

    • 1 Functions
      (pp. 13-14)

      1.1 The tangent to a curve at a point is the line that touches the curve at that point, but does not cross it there. (page 29)

      1.2 The tangent line to a curve at a point cannot touch the curve at infinitely many other points. (page 30)

      1.3 A quadratic function ofxis one in which the highest power ofxis two. (page 30)

      1.4 If both functionsf(x) andg(x) are continuous and monotone on$\mathbb{R}$, then their sumf(x) +g(x) is also monotone on$\mathbb{R}$. (page 30)

      1.5 If both functionsf(x)...

    • 2 Limits
      (pp. 15-16)

      2.1 Iff(x) <g(x) for allx> 0 and both\[\underset{x\to \infty }{{\lim }}\,f(x)\quad \text{and}\quad \underset{x\to \infty }{{\lim }}\,g(x)\]exist, then\[\underset{x\to \infty }{\lim }\,f(x) < \underset{x\to \infty }{\lim }\,g(x).\quad (\text{page}\ 39)\]

      2.2 The following definitions of a non-vertical asymptote are equivalent:

      a) The straight liney=mx+cis called a non-vertical asymptote to a curvef(x), asxtends to infinity, if\[\underset{x\to \infty }{{\lim }}\,\left( f(x)-(mx+c) \right)=0.\]

      b) A straight line is called a non-vertical asymptote to a curve, asxtends to infinity, if the curve gets closer and closer (as close as we like) to the straight line asxtends to infinity without touching or crossing it. (page 40)

      2.3 The tangent line to a...

    • 3 Continuity
      (pp. 17-18)

      3.1 If the absolute value of the functionf(x) is continuous on (a,b), then the function is also continuous on (a,b). (page 45)

      3.2 If both functionsf(x) andg(x) are discontinuous atx=a, thenf(x) +g(x) is also discontinuous atx=a. (page 46)

      3.3 If both functionsf(x) andg(x) are discontinuous atx=a, thenf(x)g(x) is also discontinuous atx=a. (page 47)

      3.4 A function always has a local maximum between any two local minima. (page 48)

      3.5 For a continuous function there is always a...

    • 4 Differential Calculus
      (pp. 19-22)

      4.1 If both functionsf(x) andg(x) are differentiable andf(x) >g(x) on the interval (a,b), thenf′(x) >g′(x) on (a,b). (page 57)

      4.2 If a nonlinear function is differentiable and monotone on (0, ∞), then its derivative is also monotone on (0, ∞). (page 57)

      4.3 If a function is continuous at a point, then it is differentiable at that point. (page 58)

      4.4 If a function is continuous on$\mathbb{R}$and the tangent line exists at any point on its graph, then the function is differentiable at any point on$\mathbb{R}$. (page 59)

      If a...

    • 5 Integral Calculus
      (pp. 23-26)

      5.1 If the functionF(x) is an antiderivative of a functionf(x), then\[\int_{a}^{b}{f(x)dx=F(b)-F(a).\quad (\text{page}\ 85)}\]

      5.2 If a functionf(x) is continuous on [a,b], then the area enclosed by the graph ofy=f(x),y= 0,x=aandx=bnumerically equals\[\int_{a}^{b}{f(x)dx.\quad (\text{page}\ 85)}\]

      5.3 If\[\int_{a}^{b}{f(x)dx\ge 0,}\]thenf(x) ≥ 0 for allx∈ [a,b]. (page 86)

      5.4 Iff(x) is a continuous function andkis any constant, then:\[\int{kf(x)dx=k\int{f(x)dx.\quad \quad (\text{page}\ 86)}}\]

      5.5 A plane figure of an infinite area rotated about an axis always produces a solid of revolution of infinite volume. (page 87)...

  7. II Suggested Solutions

    • 1 Functions
      (pp. 29-38)

      1.1 The tangent to a curve at a point is the line that touches the curve at that point, but does not cross it there.

      Counterexample

      a) Thex-axis is the tangent line to the curvey=x³, but it crosses the curve at the origin.

      b) The three straight lines just touch and do not cross the curve below at the point, but none of them is the tangent line to the curve at that point.

      1.2 The tangent line to a curve at a point cannot touch the curve at infinitely many other points.

      Counterexample The tangent...

    • 2 Limits
      (pp. 39-44)

      2.1 Iff(x) <g(x) for allx> 0 and both$\underset{x\to \infty }{\mathop{\lim }}\,f(x)$and$\underset{x\to \infty }{\mathop{\lim }}\,g(x)$exist, then$\underset{x\to \infty }{\mathop{\lim }}\,f(x) > \underset{x\to \infty }{\mathop{\lim }}\,g(x)$.

      Counterexample For the functions\[f(x)=-\frac{1}{x}\quad \text{and}\quad g(x)=\frac{1}{x},\]f(x) <g(x) for allx> 0, but\[\underset{x\to \infty }{\mathop{\lim }}\,f(x)=\underset{x\to \infty }{\mathop{\lim }}\,g(x)=0.\]

      2.2 The following definitions of a non-vertical asymptote are equivalent:

      a) The straight liney=mx+cis called a non-vertical asymptote to a curvef(x) asxtends to infinity if$\underset{x\to \infty }{\lim }\,\left( f(x)-(mx+c) \right)=0$.

      b) A straight line is called a non-vertical asymptote to a curve asxtends to infinity if the curve gets closer and closer to the straight line (as close as we like) as...

    • 3 Continuity
      (pp. 45-56)

      3.1 If the absolute value of the functionf(x) is continuous on (a,b), then the function is also continuous on (a,b).

      Counterexample The absolute value of the function\[y(x)=\left\{\begin{array}{rl} -1,\ & \text{if }x\le 0 \\ 1,\ & \text{if }x > 0 \\\end{array} \right.\]is$|y(x)|=1$for all realxand it is continuous, but the functiony(x) is discontinuous.

      3.2 If both functionsf(x) andg(x) are discontinuous atx=a, thenf(x) +g(x) is also discontinuous atx=a.

      Counterexample\[\begin{array}{ll} & f(x)=-\frac{1}{x-a},\quad \text{if}\ x\ne a \\ & g(x)=x+\frac{1}{x-a},\quad \text{if}\ x\ne a \\ & f(x)=g(x)=\frac{a}{2},\quad \text{if}\ x=a \\ \end{array}\]

      Both functionsf(x) andg(x) are discontinuous atx=a, but the function\[f(x)+g(x)=\left\{ \begin{array}{rl} x,\ & \text{if }x\ne a \\ a,\ & \text{if }x=a \\\end{array} \right.\]is continuous atx=a. For example, ifa= 2:...

    • 4 Differential Calculus
      (pp. 57-84)

      4.1 If both functionsf(x) andg(x) are differentiable andf(x) >g(x) on the interval (a,b), thenf′(x) >g′(x) on (a,b).

      Counterexample Both functionsf(x) andg(x) below are differentiable andf(x) >g(x) on the interval (a,b), butf′(x) <g′(x) on (a,b).

      4.2 If a nonlinear function is differentiable and monotone on (0, ∞), then its derivative is also monotone on (0, ∞).

      Counterexample The nonlinear functiony=x+ sinxis differentiable and monotone on (0, ∞), but its derivativey′= 1 + cosxis not monotone on (0, ∞).

      4.3...

    • 5 Integral Calculus
      (pp. 85-98)

      5.1 If the functionF(x) is an antiderivative of a functionf(x), then\[\int_{a}^{b}{f(x)dx=F(b)-F(a).}\]

      Counterexample The function\[F(x)=\ln |x|\]is an antiderivative of the function\[f(x)=\frac{1}{x},\]but the (improper) integral\[\int_{-1}^{1}{\frac{1}{x}dx}\]does not exist.

      CommentsTo make the statement truewe need to add that the functionf(x) must be continuous on [a,b].

      5.2 If a functionf(x) is continuous on [a,b], then the area enclosed by the graph ofy=f(x),y= 0,x=aandx=bnumerically equals\[\int_{a}^{b}{f(x)dx.}\]

      Counterexample For any continuous functionf(x) that takes only negative values on [a,b]...

  8. References
    (pp. 99-100)
  9. About the Author
    (pp. 101-101)