I am trying to intuitively grasp the geometric meaning of introducing the third order term delta (Yo^3) to the graph of finite difference initially set only to second order. Till second order, I can imagine that Newton formula has given addition of initial height, slope into distance and (constant curvature X distance) X distance; bracketed term being interpreted as difference in slope. But now with third order introduction it should give additional height Delta (Yo^3) X D X D X D. It is this term that is eluding me to have clear geometric interpretation although vaguely I can feel it. It can feel that the last multiplication (seeing it in reverse order) in this term will bring about slope difference * d = change in height, but what are the previous two multiplications precisely doing geometrically? I cannot figure out with precision.
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