Calculus and Trigonometry

We discuss how derivatives can be considered as a game of cubes and beams, and of geometric means. The same principles underlie wide classes of polynomials. This results in an unconventional view on the history of the dierentiation and dierentials.

The hyperbolic and circular trigonometric functions exhibit periodicity under repeated differentiation. The hyperbolic sine and cosine functions, whilst distinct from their first derivatives, are each equal to their own second derivative. Similarly, the circular sine and cosine functions are each equal to their own fourth derivative. Very little appears in the literature about functions of a real variable that are equal to their own third derivative. The purpose of this note is to define three such functions and to investigate their properties.

Recorrido crítico por la figura y la obra de Menelao de Alejandría (ca. 45/50 – ca. 110/120 d.C.), matemático (geómetra) y astrónomo griego (inmediatamente anterior a Tolomeo), cuya producción sólo nos es conocida por citas o traducciones (al árabe y al latín). Debe su celebridad sobre todo a sus contribuciones en el campo de la trigonometría esférica, ciencia de la que puede ser considerado como el padre. Esquema del capítulo: Bibliografía. Biografía y cronología. Obra: Geometría. Astrología. Mecánica. Influencia.

In 2016, the Mathematical Society of the Philippines (MSP) testified that among Filipino learners of ages 14-18, 85% experienced the dilemma on dealing with the evaluation of special angles in Trigonometry wherein 45% was from public high schools. This study aimed to determine the potential of a visual mathematical hand mnemonic tactic (VMHMT) as fast and accurate classroom mathematical mnemonic strategy in evaluating special angles in Trigonometry. A 20-item multiple choice type test with 50-minute time allotment. Reliability of the test items was also measured via Pearson's. The test was administered three times with 7-day interval for each strategy with uniform starting time. ANOVA results showed no significant difference among the three manual strategies in terms of accuracy of answers (p=0.373) which means that the accuracy of the VMHMT is comparable with that of the other existing manual strategies. Meanwhile, significant difference was obtained in terms of speed of calculations (p=0.000) wherein the use of unit circle rendered longest time for calculations (M=21.234) while the VMHMT (M=14.70) and table of trigonometric ratios (M=14.23) appeared to be very fast. Learners find the strategy more relevant and experiential which could make a more lifelong learning.

This academic paper is a collection of notes I created while taking an online Calculus course. To assist much of my self-taught knowledge, I wrote these notes in a handheld notebook. This notebook would serve as the foundation for what turned out to be a platform for many independent discoveries years later. I began to notice repetitive underlying patterns, sequences, and constants. This paper further investigates these discoveries as a way to discover a simpler execution. Disclaimer I am an architecture alumnus by way of educational training and a project manager by way of profession. I am not a mathematician so my findings may be rudimentary to some scholars with a higher degree of acumen in the related subjects. This was my way of trying to make sense of calculus. If anyone can point out any erroneous nature to my findings, please do so in a respectable and patient manner. I have attached images of my handwritten notes so one can see the thought process behind the execution. I am a visual learner so I have adopted tendencies in showcasing my work in a similar fashion, making it aesthetically pleasing to the eyes.

This academic paper is a collection of notes I created while taking an online Calculus course, focusing on Finding Limits. To assist much of my self-taught knowledge, I wrote these notes in a handheld notebook. This notebook would serve as the foundation for what turned out to be a platform for many independent discoveries years later. I began to notice repetitive underlying patterns, sequences, and constants. This paper further investigates these discoveries as a way to discover a simpler execution. Disclaimer I am an architecture alumnus by way of educational training and a project manager by way of profession. I am not a mathematician so my findings may be rudimentary to some scholars with a higher degree of acumen in the related subjects. This was my way of trying to make sense of calculus. If anyone can point out any erroneous nature to my findings, please do so in a respectable and patient manner. I have attached images of my handwritten notes so one can see the thought process behind the execution. I am a visual learner so I have adopted tendencies in showcasing my work in a similar fashion, making it aesthetically pleasing to the eyes.

This paper will investigate the general class theory of polycalculii, and theory originally developed and introduced on the Quora platform as a kind of philosophy of logic and mathematics. The ostensible goal at minima will be to demonstrate the exclusion of an array of class types which edicate the theorem of the polycalculus. The polycalculus is a general method for deriving equivalent calculus’s other than the traditional intelligent calculus. The domain of this discussion is not only mathematics, but also logic, philosophy, and philosophy of science, as well as philosophy of the foundations of mathematics.

It is an interesting trigonometrical equation related to the radii of infinite circles and here it is prooved geometrically by initially taking a circle of unit radius. This method of taking unit radius circle to discover new equations had been being adopted by various Indian mathematicians like Aryabhata ( 476 - 550 AD ) and Madhava of Kerala for centuries. It can also be prooved by general mathematical-algebraical methods by applying Taylor expansion formula, sum of infinite GP series and limit concepts.

This work briefly describes, in introductory manner, some important concepts and methods in multivariate calculus, including differential operators, multivariate Taylor series, extrema, parametric curves and surfaces, and surface integration.

We refine Oppenheim's inequality as well as generalized Cusa-Huygens type inequalities established recently by some researchers. One of the results where the bounds of sin x/x are tractable will be used to obtain a sharp version of Yang's inequality.

This academic paper is a collection of notes I created while taking an online Calculus course. To assist much of my self-taught knowledge, I wrote these notes in a handheld notebook. This notebook would serve as the foundation for what turned out to be a platform for many independent discoveries years later. I began to notice repetitive underlying patterns, sequences, and constants. This paper further investigates these discoveries as a way to discover a simpler execution. Disclaimer I am an architecture alumnus by way of educational training and a project manager by way of profession. I am not a mathematician so my findings may be rudimentary to some scholars with a higher degree of acumen in the related subjects. This was my way of trying to make sense of calculus. If anyone can point out any erroneous nature to my findings, please do so in a respectable and patient manner. I have attached images of my handwritten notes so one can see the thought process behind the execution. I am a visual learner so I have adopted tendencies in showcasing my work in a similar fashion, making it aesthetically pleasing to the eyes.

This academic paper is a collection of notes I created while taking an online Calculus course. To assist much of my self-taught knowledge, I wrote these notes in a handheld notebook. This notebook would serve as the foundation for what turned out to be a platform for many independent discoveries years later. I began to notice repetitive underlying patterns, sequences, and constants. This paper further investigates these discoveries as a way to discover a simpler execution. Disclaimer I am an architecture alumnus by way of educational training and a project manager by way of profession. I am not a mathematician so my findings may be rudimentary to some scholars with a higher degree of acumen in the related subjects. This was my way of trying to make sense of calculus. If anyone can point out any erroneous nature to my findings, please do so in a respectable and patient manner. I have attached images of my handwritten notes so one can see the thought process behind the execution. I am a visual learner so I have adopted tendencies in showcasing my work in a similar fashion, making it aesthetically pleasing to the eyes.

We discuss how derivatives can be considered as a game of cubes and beams, and of geometric means. The same principles underlie wide classes of poly-nomials. This results in an unconventional view on the history of the differ-entiation and differentials. In On Proof and Progress in Mathematics William Thurston describes how people develop an "understanding" of mathematics [1]. He uses the exam-1

Standard calculus is developed using standard real numbers R, usually plotted on the number line. In Leibniz's vision of calculus, there are infinitesimally small numbers dx, that is |dx| < for all n = 1, 2, 3, . . . , yet dx ≠ 0. This requires an extension of the standard reals R called the hyperreal numbers, denoted often by R*. Then the standard real numbers R form a proper subset of the hyperreal numbers R*, that is, R ⊂ R* with R ≠ R*. Leibniz himself could not formally establish the theory of hyperreal numbers, which was done much later in the 1960s by Abraham Robinson, using a more general theory of mathematical logic called Nonstandard Analysis, an advanced subject area.

We discuss how derivatives can be considered as a game of cubes and beams, and of geometric means. The same principles underlie wide classes of poly-nomials. This results in an unconventional view on the history of the differ-entiation and differentials. In On Proof and Progress in Mathematics William Thurston describes how people develop an &quot;understanding&quot; of mathematics [1]. He uses the exam-1