4130hw7 - f-1 may not be C 1 if f ( x ) = 0 for some x U ....

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4130 HOMEWORK 7 Due Tuesday April 13 (1) Let D R . Let f,g : D R and let a be a cluster point of D . Suppose lim x a f ( x ) = L and lim x a g ( x ) = M . Show that lim x a f ( x ) g ( x ) = LM . (2) Section 5.2.4 Exercise 3. (3) Section 5.2.4 Exercise 4. (4) Let U be an open subset of R and suppose f : U U is a C 1 bijection and that f 0 ( x ) 6 = 0 for all x U . (a) Use the Inverse Function Theorem to show that the inverse function f - 1 : U U is also C 1 . (b) Give an example to show that
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Unformatted text preview: f-1 may not be C 1 if f ( x ) = 0 for some x U . (5) Section 5.4.6 Exercise 22(b). (Hint: let y = p 1 x 1 + + p n x n . For each 1 i n , estimate f ( x i ) using the Taylor expansion of f about the point y , applying the Lagrange Remainder Theorem from page 188.) 1...
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