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Exercise 10: The minimization of a one-dimensional function by Newton's method.

Consider minimizing the one-dimensional function

by Newton's method. You might want to transform this problem into a root-finding one: . Newton's method was originally devised for solving nonlinear equations and then extended for minimization. Roughly speaking, the Newton techniques we learned above for minimizing can be adapted to solve nonlinear equations of form .

While good graphics skills can be very effective for minimizing or finding the roots of one-dimensional problems, we cannot expect to solve such problems systematically when many variables are involved and for repeated applications. The following exercises will give you practical experience in solving such problems using graphics and suitable programs.