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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.

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.

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.

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.

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 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 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.

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 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.