This document discusses differentiation and integration (antidifferentiation). It provides:
1) An example of differentiating and then integrating (antidifferentiating) a function to reverse the process.
2) The definition that integration is the reversing of differentiation, known as antidifferentiation or indefinite integration.
3) Several formulas for integrating common functions like polynomials, trigonometric, exponential, and logarithmic functions.
sifat - sifat logaritma yang sering kita pelajari terkadang hanya sekedar kita hafalkan saja tanpa mengetahui dari mana sifat tersebut berasal berikut saya sajikan slide dalam pembuktian masing2 sifat logaritma,.. untuk penjelasannya kalian dapat menyaksikan video di youtube...
untuk penjelasan dari slide share ini dapat kalian simak videonya pada link berikut :
https://youtu.be/JSU5gWgnrDU
Basic concepts of integration, definite and indefinite integrals,properties of definite integral, problem based on properties,method of integration, substitution, partial fraction, rational , irrational function integration, integration by parts, reduction formula, improper integral, convergent and divergent of integration
sifat - sifat logaritma yang sering kita pelajari terkadang hanya sekedar kita hafalkan saja tanpa mengetahui dari mana sifat tersebut berasal berikut saya sajikan slide dalam pembuktian masing2 sifat logaritma,.. untuk penjelasannya kalian dapat menyaksikan video di youtube...
untuk penjelasan dari slide share ini dapat kalian simak videonya pada link berikut :
https://youtu.be/JSU5gWgnrDU
Basic concepts of integration, definite and indefinite integrals,properties of definite integral, problem based on properties,method of integration, substitution, partial fraction, rational , irrational function integration, integration by parts, reduction formula, improper integral, convergent and divergent of integration
FERMAT’S LAST THEOREM PROVED BY INDUCTION (accompanied by a philosophical com...Vasil Penchev
A proof of Fermat’s last theorem is demonstrated. It is very brief, simple, elementary, and absolutely arithmetical. The necessary premises for the proof are only: the three definitive properties of the relation of equality (identity, symmetry, and transitivity), modus tollens, axiom of induction, the proof of Fermat’s last theorem in the case of n=3 as well as the premises necessary for the formulation of the theorem itself. It involves a modification of Fermat’s approach of infinite descent. The infinite descent is linked to induction starting from n=3 by modus tollens. An inductive series of modus tollens is constructed. The proof of the series by induction is equivalent to Fermat’s last theorem. As far as Fermat had been proved the theorem for n=4, one can suggest that the proof for n≥4 was accessible to him.
An idea for an elementary arithmetical proof of Fermat’s last theorem (FLT) by induction is suggested. It would be accessible to Fermat unlike Wiles’s proof (1995), and would justify Fermat’s claim (1637) for its proof. The inspiration for a simple proof would contradict to Descartes’s dualism for appealing to merge “mind” and “body”, “words” and “things”, “terms” and “propositions”, all orders of logic. A counterfactual course of history of mathematics and philosophy may be admitted. The bifurcation happened in Descartes and Fermat’s age. FLT is exceptionally difficult to be proved in our real branch rather than in the counterfactual one.
FERMAT’S LAST THEOREM PROVED BY INDUCTION (accompanied by a philosophical com...Vasil Penchev
A proof of Fermat’s last theorem is demonstrated. It is very brief, simple, elementary, and absolutely arithmetical. The necessary premises for the proof are only: the three definitive properties of the relation of equality (identity, symmetry, and transitivity), modus tollens, axiom of induction, the proof of Fermat’s last theorem in the case of n=3 as well as the premises necessary for the formulation of the theorem itself. It involves a modification of Fermat’s approach of infinite descent. The infinite descent is linked to induction starting from n=3 by modus tollens. An inductive series of modus tollens is constructed. The proof of the series by induction is equivalent to Fermat’s last theorem. As far as Fermat had been proved the theorem for n=4, one can suggest that the proof for n≥4 was accessible to him.
An idea for an elementary arithmetical proof of Fermat’s last theorem (FLT) by induction is suggested. It would be accessible to Fermat unlike Wiles’s proof (1995), and would justify Fermat’s claim (1637) for its proof. The inspiration for a simple proof would contradict to Descartes’s dualism for appealing to merge “mind” and “body”, “words” and “things”, “terms” and “propositions”, all orders of logic. A counterfactual course of history of mathematics and philosophy may be admitted. The bifurcation happened in Descartes and Fermat’s age. FLT is exceptionally difficult to be proved in our real branch rather than in the counterfactual one.
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Errata:
Slide 13: "The way 𝑓𝑜𝑙𝑑𝑙 does it is by associating to the right" - should, of course ,end in "to the left".
Theory of Relativity
Maybe travelling in time is an interesting topic. Also the idea of the flow of time at high speeds is a difficult idea to understand. But did you know that in 1905, someone dared to think differently. He is Albert Einstein. Questions such as, what will you see if you are moving at the speed of light? Well, it is argued that light speed is the maximum speed that is available in the entire universe. The speed of light was calculated by Maxwell using the equations of Electromagnetic wave.
c=√(1/(ε_o μ_o ))
We were able to understand that anything that has speed travels a certain distance in space in amount of time.
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Another instance is when we are on board a plane with some velocity Vplane and we fire a bullet the relative velocity of the bullet on an stationary observer will be;
V=Vbullet+Vplane
Which is correct in Galilean transformation. Now what if we turn on the headlight of a plane? Would it mean that the speed of light will be the velocity of the plane + the speed of light? (v=c) ?. Absolutely not, because this will violate the premise that, “the speed of light is constant in a vacuum”.
Clearly from the two instances there must be a different formula that will unify measurements made on different reference frame. This method is called transformation.
So let us create two equation that will unify measurements in these two instances. The first instance is at the plane, the observer at the plane will have (x,y,z,t). and the observer from the earth will us the coordinates (x^',y^',z^',t^'). So which is it the spaceship is moving away from the earth or the earth is moving away from the spaceship. To fix this, we assume that the origin O and O^'coincide and are parallel to one another at all times. Further more we let t and t^' be equal that is t= t^'.
and more....
IOSR Journal of Mathematics(IOSR-JM) is an open access international journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Periodic Function, Dirichlet's Condition, Fourier series, Even & Odd functions, Euler's Formula for Fourier Coefficients, Change of Interval, Fourier series in the intervals (0,2l), (-l,l) , (-pi, pi), (0, 2pi), Half Range Cosine & Sine series Root mean square, Complex Form of Fourier series, Parseval's Identity
On ranges and null spaces of a special type of operator named 𝝀 − 𝒋𝒆𝒄𝒕𝒊𝒐𝒏. – ...IJMER
In this article, 𝜆 − 𝑗𝑒𝑐𝑡𝑖𝑜𝑛 has been introduced which is a generalization of trijection
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This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
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The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
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Module 7 the antiderivative
1. Just like any other mathematical operation, the process of differentiation can be reversed. For example,
when we perform the differentiation of 𝑓(𝑥) = 𝑥3
.
𝒇(𝒙) = 𝒙𝟑
𝒇′(𝒙) = 𝟑𝒙𝟑−𝟏
𝒇′(𝒙) = 𝟑𝒙𝟐
Now if you begin with the function 𝒇(𝒙) = 𝟑𝒙𝟐
, reversing the process should yield the possible functions
below:
𝒇(𝒙) = 𝒙𝟑
𝒇(𝒙) = 𝒙𝟑
+ 𝟏
𝒇(𝒙) = 𝒙𝟑
− 𝟏
The reversing of the operation of differentiation is known as ANTIDIFFERENETIATION or INDEFINITE
INTEGRATION. If the derivative of 𝑓(𝑥) = 𝑥3
is 𝑓′(𝑥) = 3𝑥2
, then we say that an antiderivative of 𝑓(𝑥) =
3𝑥2
is 𝑓(𝑥) = 𝑥3
.
𝒇(𝒙) = 𝒙𝟑
𝒇′(𝒙) = 𝟑𝒙𝟐
∫ 𝒇′(𝒙) 𝒅𝒙 = 𝒇(𝒙) + 𝑪
FINDING THE ANTIDERIVATIVE OF A FUNCTION
BASIC INTEGRATION FORMULAS
Differentiation Formulas Integration Formulas
𝒅
𝒅𝒙
(𝑪) = 𝟎 ∫ 𝟎 𝒅𝒙 = 𝑪
𝒅
𝒅𝒙
(𝒌𝒙) = 𝒌 ∫ 𝒌 𝒅𝒙 = 𝒌𝒙 + 𝑪
𝒅
𝒅𝒙
(𝒌𝑭(𝒙)) = 𝒌𝑭′(𝒙) ∫ 𝒌 𝒇(𝒙) 𝒅𝒙 = 𝒌 ∫ 𝒇( 𝒙) 𝒅𝒙
𝒅
𝒅𝒙
(𝑭(𝒙) + 𝑮(𝒙)) = 𝑭′(𝒙) + 𝑮′(𝒙) ∫[𝒇(𝒙) + 𝒈(𝒙)]𝒅𝒙 = ∫ 𝒇(𝒙) 𝒅𝒙 + ∫ 𝒈(𝒙) 𝒅𝒙
DERIVATIVE
INTEGRAL
![Just like any other mathematical operation, the process of differentiation can be reversed. For example,
when we perform the differentiation of 𝑓(𝑥) = 𝑥3
.
𝒇(𝒙) = 𝒙𝟑
𝒇′(𝒙) = 𝟑𝒙𝟑−𝟏
𝒇′(𝒙) = 𝟑𝒙𝟐
Now if you begin with the function 𝒇(𝒙) = 𝟑𝒙𝟐
, reversing the process should yield the possible functions
below:
𝒇(𝒙) = 𝒙𝟑
𝒇(𝒙) = 𝒙𝟑
+ 𝟏
𝒇(𝒙) = 𝒙𝟑
− 𝟏
The reversing of the operation of differentiation is known as ANTIDIFFERENETIATION or INDEFINITE
INTEGRATION. If the derivative of 𝑓(𝑥) = 𝑥3
is 𝑓′(𝑥) = 3𝑥2
, then we say that an antiderivative of 𝑓(𝑥) =
3𝑥2
is 𝑓(𝑥) = 𝑥3
.
𝒇(𝒙) = 𝒙𝟑
𝒇′(𝒙) = 𝟑𝒙𝟐
∫ 𝒇′(𝒙) 𝒅𝒙 = 𝒇(𝒙) + 𝑪
FINDING THE ANTIDERIVATIVE OF A FUNCTION
BASIC INTEGRATION FORMULAS
Differentiation Formulas Integration Formulas
𝒅
𝒅𝒙
(𝑪) = 𝟎 ∫ 𝟎 𝒅𝒙 = 𝑪
𝒅
𝒅𝒙
(𝒌𝒙) = 𝒌 ∫ 𝒌 𝒅𝒙 = 𝒌𝒙 + 𝑪
𝒅
𝒅𝒙
(𝒌𝑭(𝒙)) = 𝒌𝑭′(𝒙) ∫ 𝒌 𝒇(𝒙) 𝒅𝒙 = 𝒌 ∫ 𝒇( 𝒙) 𝒅𝒙
𝒅
𝒅𝒙
(𝑭(𝒙) + 𝑮(𝒙)) = 𝑭′(𝒙) + 𝑮′(𝒙) ∫[𝒇(𝒙) + 𝒈(𝒙)]𝒅𝒙 = ∫ 𝒇(𝒙) 𝒅𝒙 + ∫ 𝒈(𝒙) 𝒅𝒙
DERIVATIVE
INTEGRAL](https://image.slidesharecdn.com/module7theantiderivative-210225034500/85/Module-7-the-antiderivative-1-320.jpg)
