In calculus, the second derivative, or the second order derivative, of a function f is the derivative of the derivative of f.Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, the second derivative of the position of an object with respect to time is the instantaneous acceleration of the object, or the rate at which …

For a function of more than one variable, the second-derivative test generalizes to a test based on the eigenvalues of the function's Hessian matrix at the critical point. In particular, assuming that all second-order partial derivatives of f are continuous on a neighbourhood of a critical point x, then if the eigenvalues of the Hessian at x are all positive, then x is a local minimum.

The first fundamental theorem may be interpreted as follows. For a continuous function y = f(x) whose graph is plotted as a curve, each value of x has a corresponding area function A(x), representing the area beneath the curve between 0 and x.The area A(x) may not be easily computable, but it is assumed to be well-defined.. The area under the curve between x and x + …

The n th derivative is also called the derivative of order n and denoted f (n). If x(t) represents the position of an object at time t, then the higher-order derivatives of x have specific interpretations in physics. The first derivative of x is the object's velocity. The second derivative of x is the acceleration. The third derivative of x is ...

In continuum mechanics, the material derivative describes the time rate of change of some physical quantity (like heat or momentum) of a material element that is subjected to a space-and-time-dependent macroscopic velocity field.The material derivative can serve as a link between Eulerian and Lagrangian descriptions of continuum deformation.. For example, in fluid …

The second (symbol: s) is the unit of time in the International System of Units (SI), historically defined as 1 ⁄ 86400 of a day – this factor derived from the division of the day first into 24 hours, then to 60 minutes and finally to 60 seconds each (24 × 60 × 60 = 86400).. The current and formal definition in the International System of Units is more precise:

For b > 1, the function is increasing (as depicted for b = e and b = 2), because ⁡ > makes the derivative always positive; while for b < 1, the function is decreasing (as depicted for b = 1 / 2); and for b = 1 the function is constant.. Euler's number e = 2.71828... is the unique base for which the constant of proportionality is 1, since ⁡ =, so that the function is its own derivative:

Thus, the second partial derivative test indicates that f(x, y) has saddle points at (0, −1) and (1, −1) and has a local maximum at (,) since = <. At the remaining critical point (0, 0) the second derivative test is insufficient, and one must use higher order tests or other tools to determine the behavior of the function at this point.

The natural logarithm of a number is its logarithm to the base of the mathematical constant e, which is an irrational and transcendental number approximately equal to 2.718 281 828 459.The natural logarithm of x is generally written as ln x, log e x, or sometimes, if the base e is implicit, simply log x. Parentheses are sometimes added for clarity, giving ln(x), log e (x), or log(x).

A Bézier curve (/ ˈ b ɛ z. i. eɪ / BEH-zee-ay) is a parametric curve used in computer graphics and related fields. A set of discrete "control points" defines a smooth, continuous curve by means of a formula. Usually the curve is intended to approximate a real-world shape that otherwise has no mathematical representation or whose representation is unknown or too complicated.