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Unformatted text preview: integer n such that f = O(x^n). f(x) = (2 x^3 + 3 x log x) (x+2)^{10} f(x) = (x^2 + x^2 log x) (x  log x)^3 f(x) = sqrt{x+1}  sqrt{x} A. n = 13. The x^3 and x^10 terms dominate. n = 6. The x^2 log x "rounds up" to x^3. There are two ways to do this. One is to factor out and use the binomial theorem: f(x) = sqrt(x) ( sqrt(1+1/x)  1 ) = sqrt(x) O(1/x) = O(x^(1/2)) The other is to multiple "top and bottom" by sqrt(x+1) + sqrt(x): f(x) = (sqrt{x+1}  sqrt{x}) (sqrt(x+1) + sqrt(x)) / (sqrt(x+1) + sqrt(x)) = 1 / (sqrt(x+1) + sqrt(x)) = O(x^{1/2}) Either way, the smallest integer n is n=0. 3. What is the set of all real numbers n such that x + sqrt{x} = O(x^n)? Make sure you find all of them, and that you don't include any that aren't there. A. Any real strictly more than 1 will work, whereas 1 will not work....
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 Spring '08
 STRAIN
 Math

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