1873_solutions

# Q such that f n c q for every positive integer n now

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Unformatted text preview: en there exists a sequence of polynomials that converges uniformly to g on the interval 1, 1 . This part follows at once from part b, in the light of the fact that if g is the function defined in part b then for each x 1, 1 we have 377 gx f1 d. x2  1 1 x2  |x |. Use Scientific Notebook to calculate some nth Maclaurin polynomials of the function f defined in part a. For each chosen value of n, if f n is the nth Maclaurin polynomial, and h n x  f n 1 x 2 for each x, ask Scientific Notebook to sketch the graph of the function h together with the graph of the absolute value function and observe graphically that the sequence h n is converging uniformly to the absolute value function on the interval 1, 1 . The case n  35 is illustrated in the following figure. 1 0.8 0.6 0.4 0.2 -1 -0.5 0 0.5 1 This exercise is of considerable importance because it may be used as the starting point for a major theorem known as the Stone-Weierstrass . You can find an elementary presentation of the Stone-Weierstrass theorem in Rudin reference starting with Corollary 7.27. Exercises on the Trigonometric Functions 1. Given any real number x, prove that sin x  0 if and only if x is an integer multiple of . Prove that cos x  0 if and only if x is an odd multiple of /2. Prove that if n is any integer then cos n  1 n . Solution: We know that sin   sin 0  0 and that sin x  0 whenever 0  x  . We also know that whenever x and y are numbers and sin x  sin y  0 we have sin x y  sin x cos y sin y cos x  0 and from this fact we see that sin n  0 for every integer n. Now suppose that x is any number for which the equation sin x  0 holds. If we define n to the the greatest integer that does not exceed the number x/ then we have 0 x n   and since sin x n  0 we conclude that x  n. A similar argument may be used to show that n n is an integer . 2 Finally, if G is the set of integers n for which the equation cos n  1 n is true then we know that 1 and that the sum and difference of any two members of G must belong to G. Therefore the equation cos n  1 n holds for every integer n. x R cos x  0  2. Prove that if  is any real number then the equation sin x    sin x holds for every real number x if and only if  is an even multiple of . Solution: We already know that the equation sin x  2n  sin x holds for every number x and every integer n. Now suppose that  is a number for which the equation sin x    sin x 378 G holds for every real number x. Since sin   sin 0    sin 0  0 we know that  is an integer multiple of . We can therefore express  as n for some integer n. Since 1  sin   sin   n  cos n  1 n 2 2 we know that n is even. 3. Prove that the restriction of the function sin to the interval /2, /2 is a strictly increasing function from /2, /2 onto the interval 1, 1 . Prove that if the function arcsin is now defined to be the inverse function of this restriction of sin then for every number u 1, 1 we have u 1 arcsin u  Þ dt. 0 1 t...
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