Midterm 1

# Midterm 1 - Andruy Tynyuk Math Studies Midterm J Cummings...

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Math Studies Midterm J. Cummings and D. Handron This midterm is due at 1159pm on Friday 25 February 2005. You may not collaborate. Please contact us if you find a typo or want a question clarified. 1. (Spivak, 2-23) Let A = { ( x, y ) R 2 | x < 0 , or x 0 and y = 0 } . (a) If f : A R and D 1 f = D 2 f = 0, show that f is constant. (b) Find a function f : A R such that D 2 f = 0 and D 1 f exists on A, but f is not independent of the second variable. 2. Let A be a symmetric n × n matrix. Define a function f : R n × R n R by f ( x, y ) = x T Ay. (a) Find the derivative Df . (b) Show that if α : R R n is a path satisfying f ( α ( t ) , α ( t )) = 1, then f ( α ( t ) , α ( t )) = 0. 3. (Rudin, p. 240 #14) Define f ( x, y ) = 0 , ( x, y ) = (0 , 0) , x 3 x 2 + y 2 else . (a) Prove that D 1 f and D 2 f are bounded functions in R 2 . (Hence f is continuous). (b) Let u be any unit vector in R 2 . Show that the directional derivative D u ( f )(0 , 0) exists and that its absolute value is at most 1. (c) In spite of this, prove that f is not differentiable at (0 , 0).

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