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Unformatted text preview: Below are a few practice problems for the first midterm. They are intended to be slightly longer and/or harder than the exam problems (the theory being if you can master these problems then the exam should be a piece of cake). 1. Show that f (x) = is not continuous at x = 0 but g(x) = x sin(1/x) x = 0 0 x=0 sin(1/x) x = 0 0 x=0 is continuous at 0 (Hint: the sine function is bounded by 1 in absolute value). 2. Prove that there is at least one x R such that 2x = 2 - 3x. 3. Given an example of a function f : R R such that f 2 is continuous but f is not. 4. Show that if f : [0, ) R is continuous and
x lim f (x) = L < , (1) then f is bounded. Give a counterexample to show that the hypothesis (??) is necessary. 5. Show that if f : R R satisfies the inequality |f (x)| x2 for all x R, then f (0) = 0, f is differentiable at x = 0, and f (0) = 0. 1 ...
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This note was uploaded on 03/18/2012 for the course MAT MAT 125A taught by Professor Bremer during the Winter '10 term at UC Davis.
- Winter '10