anal - Analysis Problems Penn Math Jerry L Kazdan In the...

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Analysis Problems Penn Math Jerry L. Kazdan In the following, when we say a function is smooth , we mean that all of its derivatives exist and are continuous. These problems have been crudely sorted by topic – but this should not be taken seriously since many problems fit in a variety of topics. 1. A straight line is tangent to the cubic y = x 3 + bx 2 + cx + d . Let Q be the bounded region between the line and the cubic. Let be the (unique) line that is parallel to and is also tangent to thee same cubic. This also defines a bounded region, R , between and the cubic. a) Show that Q and R have the same areas. b) What else can you say about the relationship between Q and R ? 2. Find the elementary function having the following power series for | z | < 1. a ) 1 nz n b ) 0 z n n + 1 3. Determine the sum of each of the following numerical series. a ) n = 0 1 2 n n ! b ) n = 1 1 n 2 n c ) n = 0 2 n + 1 2 n d ) n = 0 ( n + 1 ) 2 2 n e ) n = 0 2 n ( 2 n ) ! 4. a) Find a function f : R R such that f ( n ) ( 0 ) = n for all n 0. Your answer should be expressed in terms of the elementary functions encountered in calculus (such as x , sin x , e x , etc.). b) The same question with f ( n ) ( 0 ) = n 2 , for each n . c) Let g ( x ) be a smooth function and say g ( n ) ( 0 ) = b n . Find a function f : R R such that f ( n ) ( 0 ) = nb n for all n 0. 5. Suppose f : R R is a smooth function such that for all x , 0 x 2 you know that f ( x ) = 0. Prove that f is constant for 0 x 2. Include very brief self-contained proofs of all the preliminary results you use (for instance, that a continuous function on a closed and bounded interval attains its maximum at some point on the interval). 1
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6. Let f ( x ) be a continuous real-valued function with the property f ( x + y ) = f ( x )+ f ( y ) for all real x , y . Show that f ( x ) = cx where c : = f ( 1 ) . [R EMARK : There is a very short proof if you assume f is differentiable]. 7. Assume that f is defined in a neighborhood U of x 0 R and has derivatives up to order 2 n - 1 for all x U with f ( x 0 ) = f ( x 0 ) = ··· = f ( 2 n - 1 ) ( x 0 ) = 0. Assume that f ( 2 n ) ( x 0 ) exists and is positive. Show that f has a local minimum at x 0 . 8. Let { b n } , n = 1 , 2 ,... be a sequence of real numbers with the property that b n + k b n + b k . Prove that lim n b n n exists. 9. Consider the sequence: x n + 1 = a x n Does it converge for a = 2? What about other values of a > 0? [If it converges, its limit is the intersection of y = a x with y = x .] 10. a) Let X j , j = 1 , 2 ,... be a sequence of points in R 3 . If X j + 1 - X j 1 j 4 , show that these points converge. b) Let { X j } be a sequence of points in R n with the property that j X j + 1 - X j < . Prove that the sequence { X j } converges. Give an example of a convergent se- quence that does not have this property. 11. Let f : R R be a continuous function with | f ( x ) - f ( y ) | ≤ c | x - y | for some c < 1 and all x , y . Show that f has fixed point (this is a point α with f ( α ) = α ) and that this fixed point is unique.
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