A Versatile Approach to Calculus and Numerical Methods

K. Lee. Lerner. "The Elaboration of the Calculus." (Preprint) Originally published in Schlager, N. Science and Its Times: Understanding the Social Significance of Scientific Discovery. Thomson Gale, 2001

Many of the most influential advances in mathematics during the 18th century involved the elaboration of the calculus, a branch of mathematical analysis which describes properties of functions (curves) associated with a limit process. Although the evolution of the techniques included in the calculus spanned the history of mathematics, calculus was formally developed during the last decades of the 17th century by English mathematician and physicist Sir Isaac Newton (1643-1727) and, independently, by German mathematician Gottfried Wilhelm von Leibniz (1646-1716). Although the logical underpinnings of calculus were hotly debated, the techniques of calculus were immediately applied to a variety of problems in physics, astronomy, and engineering. By the end of the 18th century, calculus had proved a powerful tool that allowed mathematicians and scientists to construct accurate mathematical models of physical phenomena ranging from orbital mechanics to particle dynamics. Although it is clear that Newton made his discoveries regarding calculus years before Leibniz, most historians of mathematics assert that Leibniz independently developed the techniques, symbolism, and nomenclature reflected in his preemptory publications of the calculus in 1684 and 1686. The controversy regarding credit for the origin of calculus quickly became more than a simple dispute between mathematicians. Supporters of Newton and Leibniz often arguing along bitter and blatantly nationalistic lines and the feud itself had a profound influence on the subsequent development of calculus and other branches of mathematical analysis in England and in Continental Europe.

Journal of Macrodynamic Analysis, 2002

I will discuss some of the difficulties that I have encountered in teaching Calculus. I will follow this, in Part I, with certain examples that my students have been finding helpful in reaching a preliminary notion of derivative. The focus in Part II is the genesis of the Fundamental Theorem of Calculus.

Our point of view We believe that calculus can be for our students what it was for Euler and the Bernoullis: A language and a tool for exploring the whole fabric of science. We also believe that much of the mathematical depth and vitality of calculus lies in these connections to the other sciences. The mathematical questions that arise are compelling in part because the answers matter to other disciplines as well.

Dimension 77 The dual space 81 .Matrices 88 Trace and determinant 99 Matrix computations 102 *7 The diagonalization of a quadratic form 111 Chapter 3 The Differential Calculus Review in IR 117 Norms. 121 Continuity 126 4 Equivalent norms 5 Infinitesimals . 6 The differential 7 Directional derivatives; the mean-value theorem 8 The differential and product spaces 9 The differential and IR n • 10 Elementary applications 11 The implicit-function theorem 12 Sub manifolds and Lagrange multipliers *13 Functional dependence *14 Uniform continuity and function-valued mappings *15 The calculus of variations *16 The second differential and the classification of critical points *17 The Taylor formula . Chapter 4 Compactness and Completeness 1 Metric spaces; open and closed sets *2 Topology 3 Sequential convergence . 4 Sequential compactness. 5 Compactness and uniformity 6 Equicontinuity 7 Completeness. 216 8 A first look at Banach algebras 9 The contraction mapping fixed-point theorem 228 10 The integral of a parametrized arc 236 11 The complex number system 240 *12 Weak methods 245

Journal of emerging technologies and innovative research, 2021

From the beginning of time man has been interested in the rate at which physical and nonphysical things change. Astronomers, physicists, chemists, engineers, business enterprises and industries strive to have accurate values of these parameters that change with time. The mathematician therefore devotes his time to understudy the concepts of rate of change. Rate of change gave birth to an aspect of calculus know as differentiation. There is another subject known as integration. Integration And Differentiation in broad sense together form subject called calculus. Hence in a bid to give this research project an excellent work, which is of great utilitarian value to the students in science and social science, the research project is divided into four chapters, with each of these chapters broken up into sub units.

Abstract: Numerical analysis concerns the development of algorithms for solving various types of problems of mathematics; it is a vast-ranging field having deep interaction with computer science, mathematics, engineering, and the sciences. Numerical analysis mainly consists of Numerical Integration, Numerical Differentiation and finding Roots numerically.

The Proceedings of the 12th International Congress on Mathematical Education, 2015

Of all the areas in mathematics, calculus has received the most interest and investment in the use of Technology. Initiatives around the world have introduced a range of innovative approaches from programming numerical algorithms in various languages, to use of graphic software to explore calculus concepts, to fully featured computer algebra systems such as

Comments on the Difficulty and Validity of Various Approaches to the Calculus DAVIDTALL With the introduction of new infinitesimal methods in the last two decades, there are now available a number of dif ferent approaches to the calculus. In her perceptive review essay on "Infinitesimal Calculus" [3]. Peggi Marchi raised For the Learning of Mathematics 2, 2 (November 1981

Given a function f (x) explicitly or defined at a set of n + 1 distinct tabular points, we discuss methods to obtain the approximate value of the rth order derivative f (r) (x), r ≥ 1, at a tabular or a non-tabular point and to evaluate w x a b () z f (x) dx, where w(x) > 0 is the weight function and a and / or b may be finite or infinite. 4.2 NUMERICAL DIFFERENTIATION Numerical differentiation methods can be obtained by using any one of the following three techniques : (i) methods based on interpolation, (ii) methods based on finite differences, (iii) methods based on undetermined coefficients. Methods Based on Interpolation Given the value of f (x) at a set of n + 1 distinct tabular points x 0 , x 1 , ..., x n , we first write the interpolating polynomial P n (x) and then differentiate P n (x), r times, 1 ≤ r ≤ n, to obtain P n r () (x). The value of P n r () (x) at the point x*, which may be a tabular point or a non-tabular point gives the approximate value of f (r) (x) at the point x = x*. If we use the Lagrange interpolating polynomial P n (x) = l x f x i i i n () () = ∑ 0 (4.1) having the error term E n (x) = f (x) – P n (x) = () () ... () ()! x x x x x x n n − − − + 0 1 1 f (n+1) (ξ) (4.2) we obtain f (r) (x *) ≈ P x n r () () * , 1 ≤ r ≤ n and E x n r () () * = f (r) (x *) – P x n r () () * (4.3) is the error of differentiation. The error term (4.3) can be obtained by using the formula 212

The introduction of CAS and graphics calculators into mathematics teaching has had a strong impact on the way in which derivative and integral are introduced and developed. This paper provides a review of pen-and-paper and technology approaches identified in the research literature. They range from object-based to local, and all incorporate student investigation. We discuss their advantages and disadvantages with reference to research findings.

Bulletin of Science, Technology & Society, 1988

The Differential Calculus provides a precise, standardized language in which to learn about and discuss specific "changes" other than just in mathematics. Unfortunately, rather than to disseminate this language widely, the mathematical community has mostly considered the Differential Calculus as just a part of mathematics, if a fairly central one, and has taught it accordingly. We argue that a major rethinking of whom we want to teach it to and of the way we should then teach it is necessary and would have some far reaching consequences. One might want to learn the language of the Differential Calculus without necessarily wanting to learn its technique as do engineers and scientists or its theory as do mathematicians. To replace the usual Arithmetic and Algebra courses followed by the usual three semesters of Precalculus I-II and Calculus I, we propose, for students who passed or placed out of an integrated Arithmetic-Algebra I sequence with an A, a two four-hour semesters Integrated Precalculus I-II & Calculus I Sequence whose first semester is designed to impart Calculus Literacy and whose second semester is designed as bridge to the mainstream integral calculus. The characteristics of this sequence are that: i. it takes a Lagrangian viewpoint and ii. it proceeds through classes of functions of increasing complexity. This, together with a taskcard format gives realistic access to the Calculus to large numbers of students who never even dreamt of coming anywhere near it.

The research presented in this category can …

The basic to the concept of definite integral is area. So far in our study of Mathematics, we solved areas of plane figures like of that of triangles, rectangles and squares, circles, etc. On the previous lesson on calculus, one problem that led to the study of calculus was that of finding the area of any shape. It treated us to solve areas bounded by graphs of certain functions in the Cartesian plane. This led us to the concept of integral calculus. Now, in this section, you will learn how to reverse the process of differentiation using the properties of definite integrals. An anti-derivative of a function f(x) is a function f(x) such that F'(x) = f(x) for all x. Objectives 1. Define what anti-derivative is. 2. Determine the uses of signs and notation of integrals. 3. Compute the anti-derivative of a function. Recall The Derivative The first derivative of a function at a point is the slope of the tangent to the curve of the function at that point. The concept is defines precisely as follows. The slope m of a straight line is defined as-The ratio of the rise (change in vertical distance, Δy) to the run (change in horizontal distance, Δx). m = = =-The tangent of its angle of inclination. m = tan θ =