Derivative

The graph of a function, drawn in black, and a tangent line to that function, drawn in red. The slope of the tangent line is equal to the derivative of the function at the marked point. The derivative of a function at a chosen input value describes the best linear approximation of the function near that input value. The process of finding a derivative is called differentiation. Differentiation and the derivative[edit] The simplest case, apart from the trivial case of a constant function, is when y is a linear function of x, meaning that the graph of y divided by x is a line. y + Δy = f(x + Δx) = m (x + Δx) + b = m x + m Δx + b = y + m Δx. It follows that Δy = m Δx. This gives an exact value for the slope of a line. Rate of change as a limit value Figure 1. Figure 2. Figure 3. Figure 4. The idea, illustrated by Figures 1 to 3, is to compute the rate of change as the limit value of the ratio of the differences Δy / Δx as Δx becomes infinitely small. Notation[edit] Rigorous definition[edit]

WebMaths | L'aide en mathématiques en ligne depuis 1995 Integral A definite integral of a function can be represented as the signed area of the region bounded by its graph. The term integral may also refer to the related notion of the antiderivative, a function F whose derivative is the given function f. In this case, it is called an indefinite integral and is written: However, the integrals discussed in this article are termed definite integrals. The principles of integration were formulated independently by Isaac Newton and Gottfried Leibniz in the late 17th century. Integrals and derivatives became the basic tools of calculus, with numerous applications in science and engineering. History[edit] Pre-calculus integration[edit] The next significant advances in integral calculus did not begin to appear until the 16th century. Newton and Leibniz[edit] The major advance in integration came in the 17th century with the independent discovery of the fundamental theorem of calculus by Newton and Leibniz. Formalizing integrals[edit] Historical notation[edit] or

Cours et Formations Gratuits en Vidéo | netprof.fr Vector-valued function Example[edit] A graph of the vector-valued function r(t) = <2 cos t, 4 sin t, t> indicating a range of solutions and the vector when evaluated near t = 19.5 A common example of a vector valued function is one that depends on a single real number parameter t, often representing time, producing a vector v(t) as the result. In terms of the standard unit vectors i, j, k of Cartesian 3-space, these specific type of vector-valued functions are given by expressions such as or where f(t), g(t) and h(t) are the coordinate functions of the parameter t. The vector shown in the graph to the right is the evaluation of the function near t=19.5 (between 6π and 6.5π; i.e., somewhat more than 3 rotations). Vector functions can also be referred to in a different notation: Properties[edit] Derivative of a three-dimensional vector function[edit] Many vector-valued functions, like scalar-valued functions, can be differentiated by simply differentiating the components in the Cartesian coordinate system. . or even

Physics Various examples of physical phenomena Physics is one of the oldest academic disciplines, perhaps the oldest through its inclusion of astronomy.[8] Over the last two millennia, physics was a part of natural philosophy along with chemistry, certain branches of mathematics, and biology, but during the Scientific Revolution in the 17th century, the natural sciences emerged as unique research programs in their own right.[b] Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the fundamental mechanisms of other sciences[6] while opening new avenues of research in areas such as mathematics and philosophy. Physics also makes significant contributions through advances in new technologies that arise from theoretical breakthroughs. History Ancient astronomy Astronomy is the oldest of the natural sciences. Natural philosophy Classical physics Modern physics

Newton's law of universal gravitation Newton's law of universal gravitation states that any two bodies in the universe attract each other with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. (Separately it was shown that large spherically symmetrical masses attract and are attracted as if all their mass were concentrated at their centers.) This is a general physical law derived from empirical observations by what Isaac Newton called induction.[2] It is a part of classical mechanics and was formulated in Newton's work Philosophiæ Naturalis Principia Mathematica ("the Principia"), first published on 5 July 1687. (When Newton's book was presented in 1686 to the Royal Society, Robert Hooke made a claim that Newton had obtained the inverse square law from him – see History section below.) History[edit] Early History[edit] Plagiarism dispute[edit] In this way arose the question as to what, if anything, Newton owed to Hooke. Vector form[edit]