Calculus

To compare Merleau-Ponty’s and Deleuze’s phenomenal bodies, I first examine how for Merleau-Ponty phenomena appear on the basis of three levels of integration: 1) between the parts of the world, 2) between the parts of the body, and 3) between the body and its world. I contest that Deleuze’s attacks on phenomenology can be seen as constructive critiques rather than as being expressions of an anti-phenomenological position. By building from Deleuze’s definition of the phenomenon and from his more phenomenologically relevant writings, we find that phenomena for him are given to the body under exactly the opposite conditions as for Merleau-Ponty, namely that 1) the world’s differences 2) appear to a disordered body that 3) comes into shocking affective contact with its surroundings. I argue that a Deleuzian theory of bodily-given phenomena is better suited than Merleau-Ponty’s model in the task of accounting for the intensity of phenomenal appearings.

Since the first volume of this work came out in Germany in 1924, this book, together with its second volume, has remained standard in the field. Courant and Hilbert's treatment restores the historically deep connections between physical intuition and mathematical development, providing the reader with a unified approach to mathematical physics. The present volume represents Richard Courant's second and final revision of 1953.

Since the first volume of this work came out in Germany in 1924, this book has remained a classic in its field. Courant and Hilbert's treatment restores the historically deep connections between physical intuition and mathematical development, providing the reader with a unified approach to mathematical physics. The present volumes represent Richard Courant's second and final revision of 1953. It will be of interest to mathematicians and physicists.

Matematik sadece işlemlerden ibaret olan bir disiplin değildir. Matematiğin içerisinde kavramlar, bu kavramlara ait tanımlar ve tanımlarla ilişkili işlemler de yer alır. Bahsi geçen bu unsurlar matematik öğrenenler için oldukça önemlidir. Aynı şekilde matematik müfredatı içerisinde yer alan Analiz derslerine ait tanımlar ve tanımlarla ilgili işlemler de ilköğretim matematik öğretmenliği öğrencilerinin yeterli alan bilgisine sahip olmaları açısından oldukça önemlidir. Bu doğrultuda yapılan bu araştırmanın amacı ilköğretim matematik öğretmenliği öğrencilerinin Analiz-III dersine ait tanımlar ve yakınsaklık kriterlerini ilgili problemlerin çözümünde kullanabilme becerilerini incelemektir. Araştırmada model olarak durum çalışması kullanılmıştır. Araştırmanın katılımcıları ise Kırıkkale Üniversitesi Eğitim Fakültesi İlköğretim matematik öğretmenliği anabilim dalında öğrenim görmekte olan 32 üçüncü sınıf öğrencisinden oluşmaktadır. Veri toplama aracı olarak açık uçlu test maddeleri ve yarı yapılandırılmış mülakat formu kullanılmıştır. Araştırmada açık uçlu test maddelerinden elde edilen veriler betimsel içerik analizine tabi tutulmuştur. Yarı yapılandırılmış mülakattan elde edilen veriler ise içerik analizi tekniğiyle transkript edilmiştir. Araştırmadan elde edilen sonuçlara göre öğrencilerin bazı tanımlar ve tanımlarla ilgili işlemsel problemler arasında ilişki kurmakta problem yaşadıkları tespit edilmiştir.

Two general methods for establishing the logarithmic behavior of recursively defined sequences of real numbers are presented. One is the interlacing method, and the other one is based on calculus. Both methods are used to prove logarithmic behavior of some combinatorially relevant sequences, such as Motzkin and Schr\"oder numbers, sequences of values of some classic orthogonal polynomials, and many others. The calculus method extends also to two- (or more-) indexed sequences.

This Monthly problem investigates the frequency of values of an arbitrary real-valued continuous function on the closed unit interval.  Two solutions were selected for publication.  The first was by James Propp and the second was by myself.  We were both undergraduates when we submitted our solutions to the Monthly.

While the notion of formal contract regulating the behavior of software agents is accepted, the concept of contract regulating the activities of software developers is quite vague. In general there is not documented contract establishing obligations and benefits of members of the development team. However, a disciplined software development methodology should encourage the use of formal contracts between developers. We propose to apply the notion of formal contract to the object-oriented software development process itself. That is to say, the software development process can be seen as involving a number of agents (the development team and the software artifacts) carrying out actions with the goal of building a software system that meets the user requirements. In this way, contracts can be used to reason about correctness of the development process, and comparing the capabilities of various groupings of agents (coalitions) in order to accomplish a particular contract.

In order to effectively integrate technology into mathematics teaching and learning, teachers could create a technology-based learning environment which provides students with opportunities to experience the process of mathematical investigation from exploring with mathematical ideas to making and testing conjectures, as well as extending their conjectures to a general form if possible. Additionally, the learning environment should support students in ways that encourage them to articulate not only what they know about the mathematical ideas in their exploration, but also how they arrive at their conjectures and how they generalise the ideas. This article offers a framework that encompasses the processes of exploring, conjecturing, verifying and generalising to help mathematics teachers plan and design effective technology-based lessons to create an environment which engages students in meaningful learning in the mathematics classroom. An interactive spreadsheet template based on a ...

The main purpose of Calculus courses is to introduce students to the scientific method of analysis, providing a suitable language and useful skills in order to effectively face other disciplinary courses. Unfortunately, this does not happen frequently. Many students perceive the Calculus exam as a stumbling block and try to postpone it as much as possible. This negative mindset can quickly turn into a cycle of low confidence, less motivation, and poor performance. Math anxiety is always just around the corner. Added to this emotional stress is the impact of the COVID-19 pandemic on education, with the distinctive rise of e-learning and digital platforms, forcing teachers to quickly develop new teaching strategies. In this framework, designing preliminary actions aimed at fostering motivation and increasing engagement of students attending university basic calculus courses is an essential challenge. It is not by chance that students with higher learning motivation “achieve significantly higher test scores, enjoy learning more, have more positive self-concepts, make greater use of deep learning strategies and engage to a greater extent in autonomous self-regulated learning”. The purpose of this research is to explore the possible advantages of mixing digital tools and pedagogical techniques in order to keep students engaged and help them to approach the study of mathematics effectively, improving both their satisfaction level and their results. In particular, this paper describes a learning activity called Advent Calendar. Proposed within a calculus course, it seeks to involve students in their learning process throughout a non conventional, gradual experience which recalls the logic of the special calendar used to count the days until Christmas. Every day, students carry out a Christmas-themed mathematical exercise. Santa Claus, the elves and the rein- deer are facing problems that can be solved by exploiting suitable mathematics tools introduced during the course. This storytelling process gives concrete form and a familiar connotation to abstract mathematical concepts, involving emotions and imagination at the same time: “a story tends to have more depth than a simple example”.

In this paper is discussed an approach for the use of augmented reality in mathematics education that takes advantage of affordances in this technology to offer a tangible experience when interacting with solids of revolution. A multidisciplinary team that must get involved in the field of interaction design and human computer interaction supports the integration of this technology. After giving some elements to understand the perspective created for innovation in the pedagogy of mathematics, the perspective used for the design of the augmented reality application will also be explained. The decisions to perform a vision-based use that involves dynamical visualizations are supported with the idea of the creation of an immersive experience. The solids of revolution application will then be exposed focusing in the affordances that are considered in the learning design. Conclusions about the changes that the use of this technology could bring for spatial visualization development will be suggested.

Abstract. We develop an imperative calculus that provides a formal model for both single and mixin inheritance. By introducing classes and mixins as the basic object-oriented constructs in a λ-calculus with records and references, we obtain a system with an intuitive operational ...

í µí°· í µí±¥ (í µí±¢) í µí±› = í µí±›(í µí±¢) í µí±›−1 í µí±‘í µí±¢ í µí°· í µí±¥ [í µí±¢ * í µí±£] = í µí±¢í µí°· í µí±¥ í µí±£ + í µí±£í µí°· í µí±¥ í µí±¢ í µí°· í µí±¥ [ í µí±¢ í µí±£ ] = í µí±£í µí°· í µí±¥ í µí±¢−í µí±¢í µí°· í µí±¥ í µí±£ í µí±£ 2 í µí°· í µí±¥ [í µí±™í µí±›í µí±¢] = 1 í µí±¢ í µí°· í µí±¥ í µí±¢ í µí°· í µí±¥ [í µí°¿í µí±œí µí±” í µí±Ž í µí±¢] = 1 í µí±¢í µí±™í µí±›í µí±Ž í µí°· í µí±¥ í µí±¢ í µí°· í µí±¥ [í µí±’ í µí±¢ ] = í µí±’ í µí±¢ í µí°· í µí±¥ í µí±¢ í µí°· í µí±¥ [í µí±Ž í µí±¢ ] = í µí±Ž í µí±¢ lní µí±Ží µí°· í µí±¥ í µí±¢ í µí°· í µí±¥ [í µí±†í µí±’í µí±›í µí±¢] = í µí° ¶í µí±œí µí± í µí±¢í µí°· í µí±¥ í µí±¢ í µí°· í µí±¥ [í µí° ¶í µí±œí µí± í µí±¢] = −í µí±†í µí±’í µí±›í µí±¢í µí°· í µí±¥ í µí±¢ í µí°· í µí±¥ [í µí±‡í µí±Ží µí±›í µí±¢] = í µí±†í µí±’í µí± 2 í µí±¢í µí°· í µí±¥ í µí±¢ í µí°· í µí±¥ [í µí° ¶í µí±œí µí±¡í µí±¢] = −í µí° ¶í µí± í µí± 2 í µí±¢í µí°· í µí±¥ í µí±¢ í µí°· í µí±¥ [í µí±†í µí±’í µí±í µí±¢] = í µí±†í µí±’í µí±í µí±¢í µí±‡í µí±Ží µí±›í µí±¢í µí°· í µí±¥ í µí±¢ í µí°· í µí±¥ [í µí° ¶í µí± í µí±í µí±¢] = −í µí° ¶í µí± í µí±í µí±¢í µí° ¶í µí±œí µí±¡í µí±¢í µí°· í µí±¥ í µí±¢ í µí°· í µí±¥ [í µí±†í µí±’í µí±›ℎí µí±¢] = í µí° ¶í µí±œí µí± ℎ(í µí±¢)í µí°·í µí±¢ í µí°· í µí±¥ [í µí° ¶í µí±œí µí± ℎí µí±¢] = í µí±†í µí±’í µí±›ℎ(í µí±¢)í µí°·í µí±¢ í µí°· í µí±¥ [í µí±‡í µí±Ží µí±›ℎí µí±¢] = í µí±†í µí±’í µí±ℎ 2 (í µí±¢)í µí°·í µí±¢ í µí°· í µí±¥ [í µí° ¶í µí±œí µí±¡ℎí µí±¢] = −í µí° ¶í µí± í µí±ℎ 2 (í µí±¢)í µí°·í µí±¢ í µí°· í µí±¥ [í µí±†í µí±’í µí±ℎí µí±¢] = −í µí±†í µí±’í µí±ℎ(í µí±¢)í µí±‡í µí±Ží µí±›ℎ(í µí±¢)í µí°·í µí±¢ í µí°· í µí±¥ [í µí° ¶í µí± í µí±ℎí µí±¢] = −í µí° ¶í µí± í µí±ℎ(í µí±¢)í µí° ¶í µí±œí µí±¡ℎ(í µí±¢)í µí°·í µí±¢ í µí°· í µí±¥ [ í µí°´í µí±Ÿí µí±í µí±†í µí±’í µí±›í µí±¢] = í µí°· í µí±¥ í µí±¢ √1−í µí±¢ 2 í µí°· í µí±¥ [ í µí°´í µí±Ÿí µí±í µí° ¶í µí±œí µí± í µí±¢] = −í µí°· í µí±¥ í µí±¢ √1−í µí±¢ 2 í µí°· í µí±¥ [ í µí°´í µí±Ÿí µí±í µí±‡í µí±Ží µí±›í µí±¢] = í µí°· í µí±¥ í µí±¢ 1+ í µí±¢ 2 í µí°· í µí±¥ [ í µí°´í µí±Ÿí µí±í µí° ¶í µí±œí µí±¡í µí±¢] = −í µí°· í µí±¥ í µí±¢ 1+ í µí±¢ 2 í µí°· í µí±¥ [ í µí°´í µí±Ÿí µí±í µí±†í µí±’í µí±í µí±¢] = í µí°· í µí±¥ í µí±¢ |í µí±¢|√í µí±¢ 2 −1 í µí°· í µí±¥ [ í µí°´í µí±Ÿí µí±í µí° ¶í µí± í µí±í µí±¢] = −í µí°· í µí±¥ í µí±¢ |í µí±¢|√í µí±¢ 2 −1 í µí°· í µí±¥ [ í µí±†í µí±’í µí±›ℎ −1 í µí±¢] = í µí°· í µí±¥ í µí±¢ √í µí±¢ 2 + 1 í µí°· í µí±¥ [ í µí° ¶í µí±œí µí± ℎ −1 í µí±¢] = í µí°· í µí±¥ í µí±¢ √í µí±¢ 2 − 1 í µí°· í µí±¥ [ í µí±‡í µí±Ží µí±›ℎ −1 í µí±¢] = í µí°· í µí±¥ í µí±¢ 1 − í µí±¢ 2 í µí°· í µí±¥ [ í µí° ¶í µí±œí µí±¡ℎ −1 í µí±¢] = í µí°· í µí±¥ í µí±¢ 1 − í µí±¢ 2 í µí°· í µí±¥ [ í µí±†í µí±’í µí±ℎ −1 í µí±¢] = −í µí°· í µí±¥ í µí±¢ í µí±¢ √1−í µí±¢ 2 í µí°· í µí±¥ [ í µí° ¶í µí± í µí±ℎ −1 í µí±¢] = −í µí°· í µí±¥ í µí±¢ |í µí±¢|√1−í µí±¢ 2 REGLAS BASICAS DE LA INTEGRACION ∫[í µí±“(í µí±¥) ± í µí±”(í µí±¥)]í µí±‘í µí±¥ = ∫ í µí±“(í µí±¥) ± ∫ í µí±”(í µí±¥)í µí±‘í µí±¥ ∫ í µí±‘í µí±¥ = í µí±¥ + í µí° ¶ ∫ í µí±¥ í µí±› í µí±‘í µí±¥ = í µí±¥ í µí±›+1 í µí±›+1 + í µí° ¶ ∫ í µí°¾í µí±“(í µí±¥)í µí±‘í µí±¥ = í µí°¾ ∫ í µí±“(í µí±¥)í µí±‘í µí±¥ í µí°Š = í µí°œí µí°­í µí°ž CAMBIO DE VARIABLE ∫ í µí±’ í µí±¢ í µí±‘í µí±¢ = í µí±’ í µí±¢ + í µí° ¶ ∫ í µí±Ž í µí±¢ í µí±‘í µí±¢ = í µí±Ž í µí±¢ ln í µí±Ž + í µí° ¶ ∫ í µí±¢ í µí±› í µí±‘í µí±¢ = í µí±¢ í µí±›+1 í µí±›+1 + í µí° ¶ í µí° § ≠ −í µí¿ En donde u es una función polinomial o trascendental í µí²† = í µí±ªí µí²•í µí²†. í µí²…í µí²† í µí±¬í µí²–í µí²í µí²†í µí²“ = í µí¿. í µí¿•í µí¿í µí¿– í µí±ƒí µí±Ÿí µí±œí µí±í µí±–í µí±’í µí±‘í µí±Ží µí±‘: í µí±’ í µí±™í µí±›í µí±¥ = í µí±¥ FUNCION LOGARITMICA í µí°¿í µí±› 1 = 0 ∫ í µí±‘í µí±¢ í µí±¢ = í µí±™í µí±›|í µí±¢| + í µí° ¶ Propiedades: Ln (pq) = Ln p + Ln q Ln e=1 Ln(í µí± í µí±ž) = í µí°¿í µí±›(í µí±) − í µí°¿í µí±›(í µí±ž) Ln í µí± í µí±Ÿ = í µí±Ÿ í µí°¿í µí±› í µí± FUNCIONES EXPONENCIALES FUNCIONES TRIGONOMÉTRICAS ∫ í µí±†í µí±’í µí±›(í µí±¢)í µí±‘í µí±¢ = −í µí° ¶í µí±œí µí± (í µí±¢) + í µí° ¶ ∫ í µí° ¶í µí±œí µí± (í µí±¢)í µí±‘í µí±¢ = í µí±†í µí±’í µí±›(í µí±¢) + í µí° ¶ ∫ í µí±‡í µí±Ží µí±›(í µí±¢)í µí±‘í µí±¢ = ln|í µí±†í µí±’í µí±(í µí±¢)| + í µí° ¶ = −ln|í µí° ¶í µí±œí µí± (í µí±¢)| + í µí° ¶ ∫ í µí° ¶í µí±œí µí±¡(í µí±¢)í µí±‘í µí±¢ = −ln|í µí° ¶í µí± í µí±(í µí±¢)| + í µí° ¶ = ln|í µí±†í µí±’í µí±›(í µí±¢)| + í µí° ¶ ∫ í µí±†í µí±’í µí±(í µí±¢)í µí±‘í µí±¢ = ln|í µí±†í µí±’í µí±(í µí±¢) + í µí±‡í µí±Ží µí±›(í µí±¢)| + í µí° ¶ ∫ í µí° ¶í µí± í µí±(í µí±¢)í µí±‘í µí±¢ = ln|í µí° ¶í µí± í µí±(í µí±¢) − í µí° ¶í µí±œí µí±¡ (í µí±¢)| + í µí° ¶ ∫ í µí±†í µí±’í µí± 2 (í µí±¢)í µí±‘í µí±¢ = í µí±‡í µí±Ží µí±›(í µí±¢) + í µí° ¶ ∫ í µí° ¶í µí± í µí± 2 (í µí±¢)í µí±‘í µí±¢ = − í µí° ¶í µí±œí µí±¡(í µí±¢) + í µí° ¶ ∫ í µí±†í µí±’í µí±(í µí±¢)í µí±‡í µí±Ží µí±›(í µí±¢)í µí±‘í µí±¢ = í µí±†í µí±’í µí±(í µí±¢) + í µí° ¶ ∫ í µí° ¶í µí± í µí±(í µí±¢)í µí° ¶í µí±œí µí±¡(í µí±¢)í µí±‘í µí±¢ = −í µí° ¶í µí± í µí±(í µí±¢) + í µí° ¶

The mean value theorem IS the most important theorem in calculus and it should be rightly called the fundamental theorem of calculus because the theorem known by the same name is derived in one step from the mean value theorem.

The mvt is about an an arithmetic mean or a level term that is used to find slopes, areas, volumes, etc.  The mainstream statement of this theorem is unremarkable.

https://www.academia.edu/32016652/How_did_my_idea_of_positional_derivative_come_about

My New Calculus is based only on well-formed concepts that exclude infinity, infinitesimals and the circular rot of limit theory which is not required at all in calculus.

https://www.academia.edu/45154026/Teaching_the_fundamental_theorem_of_calculus

Read an introduction to my New Calculus here:

https://www.academia.edu/41616655/An_Introduction_to_the_Single_Variable_New_Calculus

The LinkedIn links are broken but those same articles can be found here on Academia.edu.

We review the modern approach to multivariable calculus in the context of normed vector spaces. This approach has two advantages: • It does away with the complicated notation of partial derivatives. • It puts into prominence the central notion that the derivative of a map is the best linear approximation to that map at a particular point. Let V be a real or complex vector space. Definition I.1 A norm on V is a map V → R + denoted by v → ||v|| such that 1. For all v, ||v|| ≥ 0 with equality if and only if v = 0. 2. ||λv|| = |λ|||v|| for all scalars λ and vectors v. 3. ||v + w|| ≤ ||v|| + ||w|| (triangle inequality). The triangle inequality implies that the norm gives rise to a metric via the expression d(u, v) = ||u − v||.

PROCEDURE Part 1: a) Construct the Series-Parallel Network. Insert the measured value of each resistor.  R1 Meas. = 2.15KΩ R1 Calc. = 2.2KΩ  R2 Meas. = 0.99 KΩ R2 Calc. = 1KΩ  R3 Meas. = 3.3 KΩ R3 Calc. = 3.3KΩ b) Calculate the total resistance Rt using the measured resistance values and record it.  Rt = R1 + (R2//R3) Rt = 2.2KΩ + ((1í µí°¾Ω * 3.3KΩ) (1í µí°¾Ω+3.3í µí°¾Ω)) Rt Calc. = 2.97KΩ Rt Meas. = 2.91KΩ c) Use the ohmmeter section of your multimeter to measure RT and record it. d) Determine the magnitude of the percent difference between the calculated and the measured values of parts 1(b) and 1(c) using the following equation and recorded. % í µí°·í µí±–í µí±“í µí±“í µí±’í µí±Ÿí µí±’í µí±›í µí±í µí±’ = | í µí° ¶í µí±Ží µí±™í µí±í µí±¢í µí±™í µí±Ží µí±¡í µí±’í µí±‘ − í µí±€í µí±’í µí±Ží µí± í µí±¢í µí±Ÿí µí±’í µí±‘ í µí° ¶í µí±Ží µí±™í µí±í µí±¢í µí±™í µí±Ží µí±¡í µí±’í µí±‘ | * 100%

This paper brings up some important points about logic, e.g., mathematical logic, and also an inconsistence in logic as per Gödel's incompleteness theorems which state that there are mathematical truths that are not decidable or provable. These incompleteness theorems have shaken the solid foundation of mathematics where innumerable proofs and theorems have a place of pride. The great mathematician David Hilbert had been much disturbed by them. There are much long unsolved famous conjectures in mathematics, e.g., the twin primes conjecture, the Goldbach conjecture, the Riemann hypothesis, etc. Perhaps, by Gödel's incompleteness theorems the proofs for these famous conjectures will not be possible and the numerous mathematicians attempting to find the solutions for these conjectures are simply banging their heads against the metaphorical wall. Besides mathematics, Gödel's incompleteness theorems will have ramifications in other areas involving logic. This paper looks at the ramifications of the incompleteness theorems, which pose the serious problem of inconsistency, and offers a solution to this dilemma. The paper also looks into the apparent inconsistence of the axiomatic method in mathematics. [Published in international mathematics journal. Acknowledgments: The author expresses his gratitude to the referees and the Editor-in-Chief for their valuable comments in strengthening the contents of this paper.]