For a well-behaved region the infimum of the areas of polygons that contain the region (the outer area) is equal to the supremum of the areas of the polygons contained in the region (the inner area). The usual textbook treatment of arc length considers only an analogue of the inner arc length, and as a result one can easily construct a sequence of polygonal approximations that approach a given smooth curve pointwise but whose lengths do not approach the arc length of the curve. For concave curves, however, an outer arc length can be defined and it is shown to coincide with the inner arc length. The discussion not only clarifies the arc length concept, but it shows what sorts of approximating curves are effective in approximating arc length.
Steven M. Hetzler, A Continuous Version of Newton's Method.
Newton's method for finding a zero of a function is equivalent to applying Euler's method with step size 1 to approximate the solution of an associated initial value problem. Thus this initial value problem may be considered a continuous version of Newton's iterative method. Under quite general conditions its solution is shown to converge to a zero of the given function. Examples show that established numerical methods for solving initial value problems lead quickly to a zero even in cases where Newton's method fails spectacularly. The discussion is accessible to calculus students and provides a novel application of differential equations.
Paulo Ribenboim, Are There Functions That Generate Prime Numbers?
Formulas are known that produce the nth prime number, for any natural number input n, but each has a flaw that makes it useless for computing primes. Even when one demands only that distinct natural number inputs produce distinct primes as output, the known formulas are unsatisfactory. A polynomial in one variable, with integer coefficients, always has composite values for an infinite number of natural number inputs. But if the coefficients are relatively prime, must some natural number input produce a prime output? This is an old conjecture of Schinzel and Sierpinski, whose proof would have important consequences. Other results about generating primes with polynomials are described and shown to be intimately related to classical problems in number theory.
Underwood Dudley, Is Mathematics Necessary?
Does the importance of mathematical skill in the world of work justify the prominent place of mathematics in the school and college curriculum? This skill-based rationale for studying mathematics is shown to be a rather recent invention, and one that is difficult to sustain. A case is made for the traditional justification: that mathematical study delights and strengthens the mind. The mathematical way of thinking is widely useful even though the specific facts and techniques studied in school may never be called upon at work.
Mario Martelli and Gerald Gannon, Weighing Coins: A Divide and Conquer Strategy
A classic puzzle problem is to find the largest number, m, of otherwise identical coins among which a single coin of different weight can be detected with a balance scale in n weighings. This problem is solved in an elegant and constructive way by solving three simpler weighing problems, and then showing how the original problem can be divided into cases each of which is equivalent to one that has already been conquered. The argument showcases mathematical induction and recursion, techniques that are used extensively in discrete mathematics.
Rick Kreminski, Using Simpson's Rule to Approximate Series Sums
By combining Simpson's Rule with the integral test a simple correction term is found that, added to the partial sums of an infinite series with positive terms, greatly accelerates the convergence. The derivation requires only a few lines and the result allows students to quickly and accurately estimate the sum of typical textbook series using a calculator or computer. The method is modified for estimating sums of alternating series and for estimating Euler's constant. Typical error bounds are derived.
Wolf von Ronik, Doughnut Slicing.
A plane through the center of a torus and tangent to the surface cuts the torus in two overlapping circles. This result is derived by rotating the torus to make the tangent plane become a coordinate plane. This provides another example, in addition to the familiar problems about intersections of a cone or cylinder with various planes, where a rotation of axes and standard analytic geometry make quick work of a three dimensional intersection problem.
Allan A. Struthers, Counterexamples to a Weakened Version of the Two-Variable
Second Derivative Test.
A hypothesis in the second derivative test for determining the nature of a critical point for a function of two variables is that the second partial derivatives be continuous in a neighborhood of the critical point. Simple rational functions are provided as examples to show that, unlike the single variable case, mere existence of the partial derivatives at the critical point is not sufficient in the two-variable case.
Kathy Liebars, Tying Up Loose Ends with Probability.
Suppose three strings are folded in half and held so that the six ends appear identical. Students then choose at random three pairs of ends and tie each pair together. What are the probabilities that when the knots have been tied the result will be three loops, two loops, or one loop? This classroom activity engages students in computing probabilities in an unusual setting, and by comparing the observed classroom frequencies with the theoretical probabilities it can be used to demonstrate the relationship between relative frequencies and probabilities.
William Barnier and Douglas Martin, Unifying a Family of Extrema Problems.
The problem of maximizing the product of several variables subject to the constraint that their sum is a given constant, or the dual problem of minimizing the sum subject to the product being fixed, is seen to underlie several calculus book chestnuts. Examples are problems on finding the minimal cost of fencing a rectanglular field of a given area, or the minimal cost box of a given volume, or even the shape of a rectangular building that minimizes heat loss from its walls, floor and ceiling. Symmetry hidden in the solutions of the special problems is exposed when the problems are seen as instances of the more general dual problems.
Paul Eenigenburg, A Note on the Ratio of Arc Length to Chordal Length
Does the ratio of the arc length to chordal length between two points on the graph of a differentiable function tend to 1 as the distance between the points approaches 0? This is shown to be the case if the derivative is continuous, and an example is provided to prove that the continuity of the derivative is necessary.
Richard A. Jacobson, Paths of Minimum Length in a Regular Tetrahedron.
What is the shortest path inside a regular tetrahedron that begins on one face, touches the other three and returns to its starting point? Perhaps the path joining the centroids of the faces? No, by vector geometry the length of a typical path can be found explicitly as a function of two parameters, and a straightforward calculus argument identifies the minimum length path. The points of contact are found, near the centroids of the faces, but no geometric characterization of them is known.