申論題

1.The following problems may be answered in Chinese or English. You need to give all details in order to receive any credit (point). 1. Let g be a function defined at least on some open interval containing the number 0. We say that g is of smaller order than h, or that g(h) is little-o(h)
and write g(h) = o(h), iff g(h) is small enough compared with h that

Figure 1 shows that the graph of a differentiable function f and a line with slope m that passes through the point (x, f(x)). The vertical separation at x+h between the line with slope m and the graph of f has been labeled g(h).

(b) (13%) Show that, of all lines that pass through (x, f(x)), the tangent line is the line that best approximates the graph of f near the point (x, f(x)) byshowing that
g(h) = o(h) iff m = f'(x).