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《数学通报》82年第2期与第8期,相继发表了两篇论述二次曲线弦的中点及其应用的文章。二次曲线弦的中点的一个主要问题,是弦的斜率如何用它的中点坐标表示。本文应用微分中值定理给出一般二次曲线弦的斜率公式。一、微分中值定理的一个特例我们知道,二元函数的微分中值定理是:设函数f(x,y)在闭区域D上有定义且连续,而且在区域D内部有连续偏导数f′_x,f′_y。那末,对于定义域中两点M(x,y)、M_1(x+△x,y+△y),有公式△f(x,y)=f′x(x+θ△x,y+θ△y)△x+f′y(x+θ△x,y+θ△y)△y其中θ∈(0.1)区间。一般地说,我们很难定θ具体的数值。仅在少数的情况下,可以确定它。下面证明当f(x,y)是二元二次函数时,微分中值定理中的θ是1/2。
In the 8th and 8th editions of the “Mathematical Bulletin,” Issue 82, two articles on the midpoints of the secondary curve strings and their applications were published. A major problem with the midpoint of the quadratic curve chord is how the slope of the chord is represented by its midpoint coordinates. In this paper, the differential midpoint theorem is used to give the slope formula of the general quadratic curve. 1. A special case of the differential mean value theorem We know that the differential mean value theorem of a binary function is: Let the function f(x, y) be defined and continuous in the closed region D, and there is a continuous partial derivative f in the region D. ’_x,f’_y. Then, for the two points M(x,y) and M_1(x+Δx,y+Δy) in the definition domain, there is a formula Δf(x,y)=f′x(x+θΔx,y+θΔ y) Δx+f′y(x+θΔx,y+θΔy)Δy where θ∈(0.1) intervals. In general, it is difficult to determine the specific value of θ. In only a few cases, it can be determined. The following shows that when f(x,y) is a binary quadratic function, θ in the differential mean value theorem is 1/2.