Dynamic Visualizations of Calculus Ideas

"Simultaneous Proof of the First Fundamental Theorem of Calculus and Integrability of Continuous Functions." The American Mathematical Monthly, 122(1), p. 23

1. Introduction 2. The Riemann integral 3. Explorations of R 4. Nowhere dense sets and the problem with the fundamental theorem of calculus 5. The development of measure theory 6. The Lebesgue integral 7. The fundamental theorem of calculus 8. Fourier series 9. Epilogue: A. Other directions B. Hints to selected exercises.

SummaryWe provide an easy argument, using the second fundamental theorem of calculus, for computing the area of the unit circle.

The paper wants to show how it is possible to develop based on an adequate basic idea (so-called “Grundvorstellung”) of the derivative a visual understanding of the (first) Fundamental theorem of Calculus.

In the introduction to Chap. 10 the notion of the definite integral of a function f on an interval [a,b] has already been mentioned. It arises from summing up expressions of the form f(x)Δx and taking limits. Such sums appear in many applications including the calculation of areas, surface areas and volumes as well as the calculation of lengths of curves. This chapter employs the notion of Riemann integrals as the basic concept of definite integration. Riemann’s approach provides an intuitive concept in many applications, as will be elaborated in examples at the end of the chapter.The main part of Chap. 11 is dedicated to the properties of the integral. In particular, the two fundamental theorems of calculus are proven. The first theorem allows one to calculate a definite integral from the knowledge of an antiderivative. The second fundamental theorem states that the definite integral of a function f on an interval [a,x] with variable upper bound provides an antiderivative of f. Since the definite integral can be approximated, for example by Riemann sums, the second fundamental theorem offers a possibility to approximate the antiderivative numerically. This is of importance, for example, for the calculation of distribution functions in statistics.KeywordsCantilever BeamFundamental TheoremPiecewise ConstantPiecewise Continuous FunctionIntuitive ApproachThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Differential operators and integral operators are linked together by the first fundamental theorem of calculus. Based on this principle, the notion of a differential Rota-Baxter algebra was proposed by Guo and Keigher. Recently, the subject has attracted more attention since it is associated with many areas in mathematics, such as integro-differential algebras. This paper considers differential Rota-Baxter algebras in the quasi-idempotent operator context. We establish a Gröbner-Shirshov basis for free commutative quasi-idempotent differential algebras (resp. Rota-Baxter algebras, resp. differential Rota-Baxter algebras). This provides a linear basis of a free object in each of the three corresponding categories by the Composition-Diamond lemma. Communicated by P. Kolesnikov

This paper is intended to offer calculus students and teachers the opportunity to motivate and discover the (first) Fundamental Theorem of Calculus in an experimental, experiential, inductive, intuitive, vernacular-based manner. Starting from the observation that a distance travelled at a constant speed corresponds to the area inside a rectangle, a precursor of the FTC is quickly and painlessly motivated and discovered. Its validity is then investigated both experimentally and theoretically. Once its plausibility is established, pictures and words are wedded to generalize this discovery to a standard statement of the fundamental theorem.

The paper investigates conceptual relationships between the ‘Continuous’ and the ‘Discrete’. Differentiation, integration and the fundamental theorem of calculus are based on corresponding operators and a corresponding theorem with functions on finite domains (‘discrete’ functions). The final section discusses the possibilities for introducing elements of a ‘discrete’ analysis into the mathematics curriculum.

Function identity and the fundamental theorem of calculus

Inquiry into Practice Yungbokhap Education in mathematics: Based on “the Fundamental Theorem of Calculus”

Fractional differentiation and integration have witness a continuous revolution in the last decades as they have been found to capture behaviors that resemble properties of some mathematical functions for example power law, exponential decay and crossover from exponential decay to power law. We have presented different differential operators with power law, exponential decay and the generalized Mittag-Leffler kernels. Using the fundamental theorem of calculus, integral operators associated to these differential operators were presented. Different properties of these nonlocal operators have been presented in details. Using some approximation techniques, both fractional differential and integral operators were discretized.KeywordsFractional integralsFractional derivativesInequalityLipschitz conditionsNumerical approximation

Defining a quadratic function through the slopes of its secant/tangent lines leads to the fundamental theorem of calculus (FTC) and an alternative way of understanding integration.

In this chapter we introduce the two basic notions of calculus: the notion of derivative and the one of integral; we shall then discuss a few simple but relevant results such as Fermat’s and Lagrange’s theorems, the integrability of bounded continuous and bounded monotonic functions, and the fundamental theorem of calculus which states that the processes of derivation and integration are inverse to each other. These are the fundamental facts of the new calculus (as opposed to algebraic calculus).

Using only fairly simple and elementary considerations–essentially from first year undergraduate mathematics–we show how the classical Stokes' theorem for any given surface and vector field in ℝ3 follows from an application of Gauss' divergence theorem to a suitable modification of the vector field in a tubular shell around the given surface. The two stated classical theorems are (like the fundamental theorem of calculus) nothing but shadows of the general version of Stokes' theorem for differential forms on manifolds. However, the main point in the present article is first, that this latter fact usually does not get within reach for students in first year calculus courses and second, that calculus textbooks in general only just hint at the correspondence alluded to above. Our proof that Stokes' theorem follows from Gauss' divergence theorem goes via a well-known and often used exercise, which simply relates the concepts of divergence and curl on the local differential level. The rest of this article uses only integration in 1, 2 and 3 variables together with a ‘fattening’ technique for surfaces and the inverse function theorem.

Example 3.1.9 illustrates that the basic definition of Ito integrals is not very useful when we try to evaluate a given integral. This is similar to the situation for ordinary Riemann integrals, where we do not use the basic definition but rather the fundamental theorem of calculus plus the chain rule in the explicit calculations.