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The article solves the ladder problem-using envelope, which is a direct approach wherein maximum line that will fit around the corner is calculated. The traditional approach was to find minimal line that cannot go around the corner.

Concepts of calculus are used to calculate the floor space required to accommodate the opening and closing of a bifold door mounted on a closet. A parameterization of the floor space reveals that the same astroidal and elliptical curves occur, irrespective of the length of the door panels.

A planimeter is a device used to measure the area enclosed by a curve and the most familiar is the polar planimeter. Proofs using Green's theorem for the rolling and polar planimeters are illustrated, which can be used in a vector calculus course. Both these planimeters are available in mechanical and electronic versions for commercial use.

Bernhard Bolzano and Augustin Louis Cauchy defined the concepts of continuity and limit to expand the uses of calculus in the early 19th century. They initiated the use of delta and epsilon to prove statements that involved limits and continuity in the context of infinite processes.

Trochoids are a family of curves created by a tack in a wheel that is traveling around a circle. In this article, an alternate model is presented that uses a tack in a bungee cord that is fixed to a circular track to create series of trochoids.

This article introduces students to the study of fractional calculus and derivatives. Problems involving fractional derivatives of exponential functions, trigonometric functions, and iterated integrals are presented.

The author argues that there is a topology that can examine partial derivatives of a function. Proof of this concept and implications for the study of multivariable calculus are discussed.

A basic theorem of calculus for gauge integrals was proven by describing gauge integrals and incorporating the concept of the parametric derivative. One of the conditions for which F can be considered as a parametric derivative of f on {a,b} is that a differentiable function phi should exist and map an arbitrary interval {alpha, beta} onto {a,b}. The theorem stipulates that a function f(x) is gauge integrable over an interval {a,b} and that the integral from a to b of f(x)dx is equal to F(b) F(a), provided that f(x) is a parametric derivative of F(x) on the same interval.

The calculus formula F'(j) = {F(j+h)-F(j-h)}/2h was used to approximate the value of slowly convergent series. Generalizations were then made regarding more precise calculations of derivatives and the alternating series summation from one to infinity of (-1)(super j+1)a(sub j). The method was then compared with Euler-Maclaurin summation formula. The method generates certain error bounds of the form f(super m-2)(k), which are similar to error bounds that result in the summation estimates using the Euler-Maclaurin summation formula.

Calculus is characterized by concepts of geometric series. Concerns associated with finite geometric series can be generalized by direct multiplication and a standard definition of derivatives. Integrals directly correlate with Riemann sums during the computation of simple integrals with uniform partitions. Moreover, geometric dissections of intervals can be solved by utilizing the Riemann sum definitions characterized by positive integral powers.

An alternative to the Mean Value Theorem called the Increasing Function Theorem (IFT) is proposed. IFT states that if f' is greater than or equal to 0 on an interval, then f is increasing on that interval. IFT is claimed to be easier to state, more intuitive and more effective. The IFT shows how the theoretical structure of calculus can be made more thoughtful and sensible.

The Increasing Function Theorem (IFT) has been proposed as an easier, more intuitive and more effective replacement to the Mean Value Theorem (MVT). Although the IFT does have its strengths, it cannot be used to demean the value of MVT. MVT provides an effective test of the effectiveness of Weierstrass' definition of limit and continuity and can be used more often than IFT in establishing various kinds of results.

Infinitesimals, the infinitesimally small numbers, cannot be seen or observed through practical means, but can be visualized using mathematical tools. All the elementary properties of addition and multiplication of real numbers are also valid for infinitesimals. These elementary properties of infinitesimals help in proving the continuity of quadratic functions. Calculus can be better understood by taking into account the concept of infinitesimals. A brief description of the real number system in conjunction with Tennenbaum's theorem for polynomials leads to better understanding of infinitesimals.

Gerardus Mercator's map-projection technique of the 16th century can be used to solve the calculus problem of multivariable arc length. Mercator's map enabled rhumb lines, which are constant bearing paths connecting two points. These rhumb lines can be parametrized using rectangular coordinates and trigonometric identities. An improper integral then gives arc length in real three-dimensional space.

The importance of tangents in calculus is emphasized by the relation between tangents and derivatives. The graph of any function F possesses a tangent at the location which has the coordinates (c, F(c)), if the derivative F(c) is present. The reverse of this does not exist. Tangents are also related to the derivatives through tangent vectors. The tangent of a simple arc has a line at P, when the simple arc possesses a non-zero tangent vector at the location P. The reverse generally does not exist, but under certain conditions, when P is a cusp, it may exist.

Concavity and convexity of functions in calculus can be defined without resort to derivatives. This approach uses the concept of lower bounding line. Convexity is first defined and then the definition is expanded to include concavity and differentiability. Five conditions, one lemma and one theorem are used.

Summation by parts can be applied to calculus and number theory. This technique is the discrete analog of integration by parts and can be applied to geometric figures composed of areas of rectangles. The technique is applicable to many topics in calculus including the definite integral, geometric series, arithmetic sequences, derivatives of powers and reduction formulae. From number theory, sequences of Fibonacci numbers are examined.

An inductive method simplifies the application of L'Hopital's rule to problems involving a tower of exponents, giving a limit of x to the xth power ... to the xth power as x tends to zero from the right. The expression tends to zero for odd number of x and tends to one for even number of x. The n-th hyperpower of x is evaluated using parentheses nesting and standard calculus principles. A theorem for determining the limits of x raised to a tower of exponents is presented and an inductive proof is considered.

Derivations of two infinite products of 'pi', Wallis's and Vieta's, are presented. A method to accelerate the convergence of Vieta using calculus is presented, and it is shown that the accelerated Vieta's formula can be used to approximate 'pi' to more than 300,000 digits.

A concise note on Bernoulli numbers and Bernoulli polynomials is presented. Some of the power-sum formulas are presented, and their proofs are given using elementary calculus concepts.

A detailed and expanded presentation of those parts of the Geometriae pars universalis that refer to the basic ideas of the calculus while keeping Gregory's own notation is provided. The basic features of the calculus such as tangents, areas and their relationships are discussed.

A new method to establish the monotonicity of a quotient of two functions is discussed. A new rule that applies to a wide class of quotient of functions is developed which is validated by application of the Cauchy mean value theorem, also known as the generalized mean value theorem.

Existence of divergent samples of 'harmonic-like' series is proved. An uncountably infinite family of examples of the divergence of balanced harmonic like series is constructed.

A theorem about volume found in Courant's calculus text is explained. It generalizes both Cavalieri's Principle and the Theorem of Pappus as a means for computing the volume swept out by a moving planar region.

The overall philosophy as well as some specific details of improvements that were undertaken to improve the learning of calculus sequence at a mid-sized state university starting in 1997 are described. The project and its underlying philosophy could serve as a blueprint for endeavors like it at similar institutions.

A solution to the problem in which a bug inside a vessel begins to crawl upward when a liquid is poured onto the vessel is provided. It is shown that the speed is the product of a geometric quantity and the flow rate of the liquid likewise speed can also be expressed as an explicit function of time.

The techniques of calculus using the theory of limits can be used to find the tangent to a given curve or the extremum of a function. Calculus problems that are free of the limit concept were based on the theory of equations and geometric properties of curves.

Constrained-optimization and related-rates word problems are biggest problems and methods suggested as solution are contrasted in first-year calculus. It is concluded that a different approach to constrained-optimization problems, similar to that widely used for related-rates problem would be advantageous.

It is reported that the In calculus and the In function is introduced by an integral and its derivative is then found by the means of the Fundamental Theorem of Calculus. The method of placing the natural logarithm and the exponential function on an equal footing is explained.

A solution to the problem of convergence of iterated exponentials, along with some historical notes is presented. The arguments used are elementary in the sense that they need knowledge of freshman calculus.

A theorem presents conditions for the intervals supporting a prescribed area arise in statistics to be the shortest. Further, the fundamental theorem of calculus shows that the rate of change of area at each x is f(x).

The language of predicate calculus is used to examine different possible meanings that Lincoln might have invented to convey. A question is raised on the interpretation that Lincoln had in his mind.

The applications of the operator (x d\dx)n in calculus is discussed. It can be used to obtain the full Maclaurin series of the elementary trigonometric functions other than sine and cosine.

The historical development of curvature, the subject comes under calculus, is discussed. The radius calculation of curvature and the existence of curvature in elastic force are described with some experimental processes.

The numerous tests done on a dog-named Elvis, to study whether dogs make use of calculus, are presented. Various diagrams and tables showing the results are mentioned.

The results of an elementary calculus course, which are the rearrangement of a conditionally convergent series are presented. This gives students the feeling that this convergence will tolerate perturbations only if they remain bounded.

An example of a situation where probability notions can be helpful in solving a calculus problem is reviewed. The solution is based only on probability arguments.

An evidence in support of the vibrant vitality of calculus is presented by providing a result that links together two theorems, one intuitive and other nonintuitive. The two theorems linked are the Fubini's theorem on exchanging the order of integration and the equality of mixed partial derivatives.

The concept of convergent series being good and divergent series bad is altered. An example is presented to help undergraduate students in calculus, numerical analysis, and real analysis, as well as professionals unfamiliar with the subject.

An expression is derived for the total volume of rain absorbed in terms of the speed of the traveler. It will be helpful for a calculus class to show that maximums and minimums are not always at the points where the slope of the function is zero.

The volume and the surface area of a sphere of radius R can be deduced using the cone formula. The methods as to how the volume of the cone can be deduced without the calculus are mentioned.

Galileo's investigations of the motion of projectiles in a nonresisting medium lead to the origin of modern dynamics. The path of the projectile is parabolic and its motion is a prime application in all calculus text.