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The following are usually easy to carry out and give important clues as to the shape of a curve: Determine the x and y intercepts of the curve. The x intercepts are found by setting y equal to 0 in the equation of the curve and solving for x. Similarly, the y intercepts are found by setting x equal to 0 in the equation of the curve and solving ...
The stationary points are the red circles. In this graph, they are all relative maxima or relative minima. The blue squares are inflection points.. In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of the function where the function's derivative is zero.
Hamilton's principle states that the true evolution q(t) of a system described by N generalized coordinates q = (q 1, q 2, ..., q N) between two specified states q 1 = q(t 1) and q 2 = q(t 2) at two specified times t 1 and t 2 is a stationary point (a point where the variation is zero) of the action functional [] = ((), ˙ (),) where (, ˙,) is the Lagrangian function for the system.
Fermat's theorem gives only a necessary condition for extreme function values, as some stationary points are inflection points (not a maximum or minimum). The function's second derivative , if it exists, can sometimes be used to determine whether a stationary point is a maximum or minimum.
In the case of a single parameter, parametric equations are commonly used to express the trajectory of a moving point, in which case, the parameter is often, but not necessarily, time, and the point describes a curve, called a parametric curve. In the case of two parameters, the point describes a surface, called a parametric surface.
In the statistical analysis of time series, autoregressive–moving-average (ARMA) models are a way to describe a (weakly) stationary stochastic process using autoregression (AR) and a moving average (MA), each with a polynomial. They are a tool for understanding a series and predicting future values.
A spacetime diagram is a graphical illustration of locations in space at various times, especially in the special theory of relativity.Spacetime diagrams can show the geometry underlying phenomena like time dilation and length contraction without mathematical equations.
The curve of fastest descent is not a straight or polygonal line (blue) but a cycloid (red).. In physics and mathematics, a brachistochrone curve (from Ancient Greek βράχιστος χρόνος (brákhistos khrónos) 'shortest time'), [1] or curve of fastest descent, is the one lying on the plane between a point A and a lower point B, where B is not directly below A, on which a bead slides ...
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