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The second derivative of a function f can be used to determine the concavity of the graph of f. [2] A function whose second derivative is positive is said to be concave up (also referred to as convex), meaning that the tangent line near the point where it touches the function will lie below the graph of the function.
Snap, [6] or jounce, [2] is the fourth derivative of the position vector with respect to time, or the rate of change of the jerk with respect to time. [4] Equivalently, it is the second derivative of acceleration or the third derivative of velocity, and is defined by any of the following equivalent expressions: = ȷ = = =.
the significance here is that only one component of ψ remains, so that 2D flow becomes a problem with only one dependent variable. The cross differentiated Navier–Stokes equation becomes two 0 = 0 equations and one meaningful equation. The remaining component ψ 3 = ψ is called the stream function.
Suppose that a function represents the position of an object at the time. The first derivative of that function is the velocity of an object with respect to time, the second derivative of the function is the acceleration of an object with respect to time, [28] and the third derivative is the jerk. [35]
The product of a step function with a number is also a step function. As such, the step functions form an algebra over the real numbers. A step function takes only a finite number of values. If the intervals , for =,, …, in the above definition of the step function are disjoint and their union is the real line, then () = for all .
Instead of just matching one derivative of () at =, this polynomial has the same first and second derivatives, as is evident upon differentiation. Taylor's theorem ensures that the quadratic approximation is, in a sufficiently small neighborhood of x = a {\textstyle x=a} , more accurate than the linear approximation.
The two rules presented above differ only in the way how the first derivative at the region end is calculated. The first derivative term in the Euler–MacLaurin integration rules accounts for integral of the second derivative, which equals the difference of the first derivatives at the edges of the integration region.
The Heaviside step function, or the unit step function, usually denoted by H or θ (but sometimes u, 1 or 𝟙), is a step function named after Oliver Heaviside, the value of which is zero for negative arguments and one for positive arguments. Different conventions concerning the value H(0) are in use.