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finalexam

# finalexam - 18.100A Introduction to Analysis Practice Final...

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18.100A Introduction to Analysis Practice Final 3 hours Directions. You can use the textbook, but no other material. To justify the arguments, quote theorems , by name or number, verifying their hypotheses where necessary. Work problems totalling 100. You can include just parts of some problems. Problems will be graded in their numerical order, stopping at 100 or as close above it as possible, the total scaled back to 100 if it goes over. 1 1. (15: 5, 10) Let f ( x ) = . x 2 + n 2 1 a) Prove that f ( x ) is defined and continuous on ( −∞ , ). (Do not use part (b).) b) Prove f ( x ) is differentiable on ( −∞ , ). (Hint: Use intervals of the form [ a, a ].) 2. (15: 10,5) a) Let S be a compact subset of the open first quadrant Q = { ( x, y ) : x > 0 , y > 0 } . Prove there is a point x in S such that the ray connecting it to the origin has maximum slope (i.e., no other point in S gives a ray with bigger slope). b) Does (a) continue to be true if S is only assumed to be closed, not compact? If yes, show how to modify the proof in (a) to prove it; if no, give a counterexample.

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