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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. In fact, the derivative at a point of a function of a single variable is the slope of the tangent line to the graph of the function at that point. The process of finding a derivative is called differentiation. The reverse process is called antidifferentiation. 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.

Related:  Mathematics

Tetration , for n = 1, 2, 3 ..., showing convergence to the infinite power tower between the two dots of infinite power tower converges for the bases In mathematics, tetration (or hyper-4) is the next hyper operator after exponentiation, and is defined as iterated exponentiation. The word was coined by Reuben Louis Goodstein, from tetra- (four) and iteration. Tetration is used for the notation of very large numbers.

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. Through the fundamental theorem of calculus, which they independently developed, integration is connected with differentiation: if f is a continuous real-valued function defined on a closed interval [a, b], then, once an antiderivative F of f is known, the definite integral of f over that interval is given by

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.

Imaginary unit i in the complex or cartesian plane. Real numbers lie on the horizontal axis, and imaginary numbers lie on the vertical axis There are in fact two complex square roots of −1, namely i and −i, just as there are two complex square roots of every other real number, except zero, which has one double square root. In contexts where i is ambiguous or problematic, j or the Greek ι (see alternative notations) is sometimes used. Parametric equation is a parametric equation for the unit circle, where t is the parameter. A common example occurs in kinematics, where the trajectory of a point is usually represented by a parametric equation with time as the parameter. The notion of parametric equation has been generalized to surfaces, manifolds and algebraic varieties of higher dimension, with the number of parameters being equal to the dimension of the manifold or variety, and the number of equations being equal to the dimension of the space in which the manifold or variety is considered (for curves the dimension is one and one parameter is used, for surfaces dimension two and two parameters, etc.).

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.

Quantum nonlocality Quantum nonlocality is the phenomenon by which the measurements made at a microscopic level necessarily refute one or more notions (often referred to as local realism) that are regarded as intuitively true in classical mechanics. Rigorously, quantum nonlocality refers to quantum mechanical predictions of many-system measurement correlations that cannot be simulated by any local hidden variable theory. Many entangled quantum states produce such correlations when measured, as demonstrated by Bell's theorem.

Polar coordinate system Points in the polar coordinate system with pole O and polar axis L. In green, the point with radial coordinate 3 and angular coordinate 60 degrees, or (3,60°). In blue, the point (4,210°). History[edit] Hipparchus From the 8th century AD onward, astronomers developed methods for approximating and calculating the direction to Makkah (qibla)—and its distance—from any location on the Earth.[3] From the 9th century onward they were using spherical trigonometry and map projection methods to determine these quantities accurately.