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5.3. This problem works out a special case of part of the theorem in class that the composition of two Cr functions is also Cr. Let Z = f(x,y) (x,y)=(8(t),h(t)), where t, X, y, and Z are all scalar variables. Suppose that all three of the functions f(x,y), g(t), and h(t) are C², , and that g(0) = h(0) = 0. Since the functions are C², we obtain these power series at the origin: X = 2 = 2 Z = =f+fxx+fyy + fax2 +2fory+fyxy2 + oll (x,y) | 12) 2 In these formulas I am using abbreviations like this: = &t = For the exercise, you should compose the functions and prove that Z = f + (fx8t + f,ht) + fxx82 + 2fxy8tht + fyrhh2 + fx8tt + fyhup + o(1t12). 2 as t 0. Many terms are swept up into the total error o(1+12) at the end. Prove that these terms are in fact o(1t12). (Note: You do not have to list every term separately in your reason- ing. Instead, you can group the terms that disappear for the same reason.) 5.4. Recall one formulation of the inverse rule for a differentiable function f : R R, that = 1 dx (x) when y = f (x) and the denominator on the right is not zero. Let U², , V² C R 2 be open sets, let F : U² V² be a C¹ function with a C¹ inverse F-1:V² U², , and let y = f(x) with X = (x1,x2) and y = (y1,y2). Find an example of all of that such that 0x1 1 dy1 (y) = ? 0y1 (x) 0x1 is not true, not even when the denominator is non-zero.

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