Problem of the Week

PROBLEM 228

 One way to state the well-known Mean Value Theorem in calculus is the following:

If f (x) is a differentiable function, then f (a + h) = f (a) + hf '(a + lh), for some 0 < l < 1.

In words, it says that the slope of the secant line through the points (a, f (a)) and (a + h, f (a + h)) is the same as the slope of the tangent line at some point x = a + lh lying between x = a and x = a + h.  Unfortunately, one can rarely say much intelligent about the value of l.  But sometimes you can!

Suppose that f (x) is a quadratic polynomial function and x = a.  What is lim _h→0 l?

(For the more adventurous:  Let f (x) be an arbitrary many-times differentiable function with  f ''(a) ≠ 0.)

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ã2005 Alberto L. Delgado