Problem Set 7. Calculus - Problem Seminar Fall 2013 Problem...

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Problem Seminar. Fall 2013. Problem Set 7. Calculus. Classical results. 1. Every continuous mapping of a circle into a line carries some pair of diametrically oppo- site points to the same point. 2. Mean value theorem. If f : [ a, b ] R is a differentiable function then there exists c ( a, b ) such that f 0 ( c ) = f ( b ) - f ( a ) b - a . 3. Leibniz formula. π 4 = 1 - 1 3 + 1 5 - 1 7 + . . . . 4. Gaussian integral. Z -∞ e - x 2 dx = π. Problems. 1. Put 94. A1. Suppose that a sequence a 1 , a 2 , . . . satisfies 0 < a n a 2 n + a 2 n +1 for all n 1 . Prove that the series n =1 a n diverges. 2. Put 87. B1. Evaluate Z 4 2 p ln(9 - x ) dx p ln(9 - x ) + p ln(3 + x ) . 3. Put 92. A2. Define C ( α ) to be the coefficient of x 1992 in the power series about x = 0 of (1 + x ) α . Evaluate Z 1 0 C ( - y - 1) 1 y + 1 + 1 y + 2 + . . . + 1 y + 1992 dy. 4. Put 91. B2. Suppose f and g are non-constant, differentiable, real-valued functions on R . Furthermore, suppose that for each pair of real numbers x and y , f ( x + y ) = f ( x ) f ( y ) - g ( x ) g ( y ) , g ( x + y ) = f ( x ) g ( y ) + g ( x ) f ( y ) . If f 0 (0) = 0 prove that ( f ( x )) 2 + ( g ( x )) 2 = 1 for all real x . 5. Put 94. B3. Find the set of all real numbers k with the following property: For any positive, differentiable function f that satisfies
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Unformatted text preview: ( x ) for all x , there is some number N such that f ( x ) > e kx for all x > N . 6. Put 92. A4. Let f be an infinitely differentiable real-valued function defined on the real numbers. If f ± 1 n ² = n 2 n 2 + 1 , n = 0 , 1 , 2 ,... compute the values of the derivatives f ( k ) (0) for all positive integers k . 7. Put 93. B4. The function K ( x,y ) is positive and continuous for ≤ x ≤ 1 , ≤ y ≤ 1 , and the functions f ( x ) and g ( x ) are positive and continuous for ≤ x ≤ 1 . Suppose that Z 1 f ( y ) K ( x,y ) dy = g ( x ) and Z 1 g ( y ) K ( x,y ) dy = f ( x ) for all ≤ x ≤ 1 . Show that f ( x ) = g ( x ) for ≤ x ≤ 1 . 8. Put 96. A6. Let c > be a constant. Give a complete description, with proof, of the set of all continuous functions f : R → R such that f ( x ) = f ( x 2 + c ) for all x ∈ R ....
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