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Let v be a vectorial function of a real variable t, such that:

v(t) = ( f(t), g(t) )

I know that by definition v'(t) = ( f'(t), g'(t) )

In many books they give a geomtrical interpretation of this. They draw the curve (the values of f(t) in the x axis, the values of g(t) in y axis). They take two points in the curve (say v(t0) and v(t0 + h) ) and they draw the positions vectors of both.

They say: if you take smaller "h" then the vector v(t0 + h) - v(t0) gets nearer and nearer the tangent line that touches the curve in v(t0).

So they assume that if you take smallers "h" then the point v(t0 + h) will be nearer to v(t0) in the curve. Which is the basis of this assumption?

Thanks.