# project1 - 1 = 0 prove that the equation a a 1 x a 2 x 2 a...

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Group project 1: MAC 2313 (Sec 3196) Due on 15th October,2010. Please write in detail your solution in a separate paper clearly.You also need to present the problem in class on the due date. Total Points: 30 Problem 1. Assume a > - 1 and b > - 1. Find the value of the following limit. lim n →∞ n b - a 1 a + 2 a + ..... + n a 1 b + 2 b + .... + n b Problem 2. We say two polynomials f ( x ) and g ( x ) are relatively prime if there are no non constant polynomials that divide both f ( x ) and g ( x ). Show that a polynomial f ( x ) has no multiple roots if and only if f ( x ) and f 0 ( x ) are relatively prime. Problem 3. Note that 1 4 6 = 1 2 but 1 4 1 4 = 1 2 1 2 .Prove that there exists infinitely many pairs of positive real numbers α and β such that α 6 = β but α α = β β . Problem 4. Let a 0 , a 1 , ...a n be real numbers with the property that a 0 + a 1 2 + a 2 3 + .... + a n

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Unformatted text preview: + 1 = 0 prove that the equation a + a 1 x + a 2 x 2 + ..... + a n x n = 0 (1) has atleast one solution in the interval (0 , 1). Problem 5. Suppose that f is a continuous function on [0 , 2] such that f (0) = f (2). Show that there is a real number ξ ∈ [1 , 2] with f ( ξ ) = f ( ξ-1). Problem 6. The axes of two right circular cylinders of radius a intersect at a right angle. Find the volume of the solid of intersection of the cylinders. Problem 7. Let f be a real valued function such that f,f ,f 00 are all continuous on [0 , 1]. Prove that the series ∞ X k =1 f ( 1 k ) is convergent if and only if f (0) = f (0) = 0....
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