Mth 4200: Homework Problems Gregor Kovčič 1. Prove the following properties of the binomil coefficients ( n ) ( n ) (i) 1 + + + + 1 2 ( n ) (ii) 1 ( n ) ( n ) + 2 + 3 + + n 2 3 ( ) n ( n + = 2 n 1 n) n, ( n n) ( n ) 2 ( n ) 2 ( n ) ( ) 2 2n (iii) + + + = 0 1 n n = n2 n 1, HINT: In (ii), write the binomil coefficients in terms of fctorils. In (iii), consider the coefficient of x n in (1 + x) 2n. 2. For q 1, prove by induction tht 1 + 2q + 3q 2 + + nq n 1 = 1 (n + 1)qn + nq n+1 (1 q) 2. 3. If r is rtionl r 0 nd x is irrtionl, prove tht r + x nd rx re irrtionl. 4. Prove tht 3 is irrtionl. 5. Prove tht for ny rtionl root of polynomil with integer coefficients, n x n + n 1 x n 1 + + 1 x + 0, n 0, if written in lowest terms (i.e., with no common integer fctor) s p/q, tht the denumertor p is fctor of 0 nd the denomintor q is fctor of n. 1
6. If n is positive integer, show tht is positive integer. n = ( 1 + 5 ) n ( 1 5 ) n 2 n 5 HINT: Show tht n+2 n+1 n = 0. 7. If, b > 0 nd n > b n, where n is positive integer, prove tht > b. 8. Prove tht there is no rel number x such tht x 2 = 1. 9. Show tht for positive, b, nd c, (i) 2 + b 2 + c 2 b + bc + c, (ii) ( + b)(b + c)(c + ) 8bc. 10. Show tht (1 b 1 ) 2 + ( 2 b 2 ) 2 2 1 + 2 2 + b 2 1 + b 2 2, nd interpret this inequlity geometriclly. HINT: Show the result for the squre of this inequlity first, using the Cuchy-Schwrtz inequlity. 11. Show tht even positive integers form countble set. 12. Show tht the set of irrtionl numbers is uncountble. 13. (i) If z is complex number such tht z = 1, compute 1 + z 2 + 1 z 2. (ii) For complex nd b, show tht + b 2 + b 2 = 2( 2 + b 2 ), nd interpret this result geometriclly. 2
14. Find ll the solutions of the eqution z 3 = 1. Write the rel nd imginry prts of these roots in terms of frctions involving integers nd squre roots of integers. 15. If z = x + iy, x, y rel, show tht 1/z is lso of the form 1/z = u + iv, u, v rel. 16. If nd b re rel numbers, b 0, show tht + ib = ± + 2 + b 2 + i b + 2 + b 2, 2 b 2 nd explin why the expressions under the squre root signs re non-negtive. HINT: Assume (x + iy) 2 = + ib, nd find two equtions for x nd y. From these equtions, deduce (x 2 + y 2 ) 2 = 2 + b 2, nd then deduce the expressions for x 2 nd y 2. Finlly, be creful bout choosing the reltive signs of x nd y. 17. If n is positive integer, nd show tht ω = cos 2π n + i sin 2π n, 1 + ω h + ω 2h + + ω (n 1)h = 0 for ny integer h which is not multiple of n. Wht is the geometric interprettion of this equlity? Wht hppens if h is multiple of n? 18. Show tht the empty set is subset of every set. 19. (i) If A nd B re rbitrry sets, nd CA nd CB re their complements within some lrger set, prove tht CX CY = C(X Y ). (ii) Stte nd prove the sme result for n rbitrry number, n, of sets A 1,..., A n. 20. Interpret the crtesin product, R 2 = R R, of two sets of rel numbers, R, s plne. In other words, let R 2 = {(x, y) x, y R}. Let A, B, C R be the intervls A = {t R t < 3}, B = {t R t < 2}, C = {t R t < 1}. 3
(i) Drw the set A B C C. (ii) Drw the set ((A B) ((A B) C)) {(x, y) x 2 + y 2 < 1}. 21. Show tht, in C, definition of n open set equivlent to the one given in clss is: A C is open if every point z A is contined in squre S with sides prllel to the rel nd imginry xes such tht z S A. HINT: You cn inscribe squre into ny circle nd vice vers. 22. Construct countble infinity of open sets A n C, n = 1, 2, 3,..., such tht their intersection n=1a n is not open. 23. Show tht the squre S = {x + iy 0 < x < 1, 0 < y < 1} is n open subset of C. 24. Give n exmple of subset of the rel xis with precisely two cluster points. 25. Is every finite subset of R or C closed? Justify your nswer. 26. Are the following subsets of the rel xis open, closed, or neither: (i) A = {x 2 < x < 3 or 4 < x < 5}. (ii) B = {x x(x 1)(x 2) 0}. (iii) C = {x 0 < x 2 1 3}. Justify your nswers. 27. Show tht closed bll in R or C is closed set. Wht geometric object is such bll in ech cse? 28. Drw the subsets A = C D, B = C D of the complex plne nd determine whether they re open, closed, or neither, where (i) C = {z z 1 < 2} nd D = {z z + 1 < 2}. (ii) C = {z z 1 2} nd D = {z z + 1 2}. (iii) C = {z z 1 < 2} nd D = {z z + 1 2}. 4
Justify your nswers. 29. (i) Construct closed subset of the rel xis with precisely three cluster points. (ii) Show tht the set {1/n n Z} is neither open nor closed. (iii) Show tht if you remove finite number of points from n open set, the remining set is still open. Then show by exmples tht this my or my not be the cse if you remove countble number of points? 30. Construct countble collection of open sets in R such tht their intersection is neither open nor closed. HINT: You my hve to use the fct tht countble union of countble collections is countble collection. 31. (i) Is every point of every open set E in C limit point of E? (ii) Sme question for closed sets in C. (iii) Do the nswers of (i) nd (ii) chnge with R in plce of C? 32. Let A denote subset of C. (i) Do A nd its closure Ā lwys hve the sme interior A? (ii) Do A nd its interior A lwys hve the sme closure Ā? Justify your nswers. Do these nswers chnge if C is replced by R? 33. Wht re the boundry points of Q in R? HINT: Tke it s fct tht ny neighborhood of ny number contins both rtionls nd irrtionls. 34. Show tht the union of two disjoint open or closed subsets of R or C is disconnected. 35. (i) Show tht every infinite set of rel numbers hs countble dense subset. (ii) Give n exmple of set A R such tht A Q is not dense in A. 5
36. Let E be the set of ll x [0, 1] whose deciml expnsion contins only digits 4 nd 7. Is E countble? Is E dense in [0, 1]? Is E open? Is E closed? Is E perfect, tht is, is every point of E limit point of E? Is E connected? Is E bounded? 37. Find the domins nd rnges of the following functions, nd determine whether they re one-to-one. If yes, find the inverse functions. (i) f(x) = x 2 + 1. (ii) f(x) = x 3. (iii) f(x + iy) = x yb + i(xb + y), where 2 + b 2 > 0. (iv) f(x + iy) = e x (cos y + i sin y). 38. Let f : A B be function mpping the set A into set B. If C, D A, show tht f(c D) f(c) f(d). Show by exmple tht this inclusion my be proper. 39. Let X λ = [0, 1] for ll λ [0, 1]. Show tht the uncountble Crtesin product λ [0,1] X λ equls the set of functions f : [0, 1] [0, 1]. Verify tht the xiom of choice holds in this cse. 40. Let n = 10 n /n!. (i) Is the sequence { n } monotonic? Is it monotonic from certin n onwrd? (i) Show tht the sequence converges. To wht limit? (iii) Give n estimte of the difference between n nd its limit. n! 41. Prove tht lim n n = 0. n HINT: Show tht 0 < n! n < 1, where [x] denotes the lrgest integer x. n 2 [n/2] 42. Prove tht ( 1 (i) lim n n + 1 2 (n + 1) + + 1 ) = 0, 2 (2n) 2 ( 1 1 (ii) lim n + + + 1 ) =, n n + 1 2n 6
(iii) if n = 1 n + 1 n + 1 + + 1, its limit exists nd is contined between 1/2 nd 1. 2n HINT: Compre the sums with their lrgest nd smllest terms, respectively, when pproprite. For (iii), show tht { n } is monotonic. ( 1 43. Compute lim n 1 2 + 1 ) 2 3 + + 1. n(n + 1) HINT: Express ech term s n pproprite difference. 44. Prove tht the sequence 2, 2 2, 2 2 2,... converges, nd find its limit. 45. Prove tht convergence of { n } implies convergence of { n }. Is the converse true? If yes, prove it, if no, provide counterexmple. 46. Given sequence { n }, define its rithmetic mens by s n = 1 + + n. n (i) Show tht, if { n } converges, {s n } must converge to the sme limit. HINT: Write n = s n n = 1 n + 2 n +... + n 1 n. n n n Use the definition of convergence for { n } nd the boundedness of { n } to show tht n 0. (ii) Construct sequence { n } which does not converge lthough lim n s n = 0. 47. Let { n } = 1, 2, 3,... nd {b n } = b 1, b 2, b 3,... be two sequences. Show tht the set of cluster points of the sequence 1, b 1, 2, b 2, 3, b 3,... is the union of the sets of cluster points of the sequences { n } nd {b n }. HINT: Use subsequences. In prticulr, lso use the fct tht if some sequence converges, every one of its subsequences converges to the sme limit. 48. Let 1 nd b 1 be ny positive numbers, nd let 1 < b 1. Let 2 nd b 2 be defined by the equtions 2 = 1 b 1, b 2 = 1 + b 1. 2 7
Similrly, let nd, in generl, 3 = 2 b 2, b 3 = 2 + b 2, 2 n = n 1 b n 1, b n = n 1 + b n 1. 2 Prove tht (i) 1 < 2 < < n < < b n < < b 2 < b 1 nd deduce tht the sequences { n } nd {b n } converge. (ii) Show tht { n } nd {b n } converge to the sme limit. HINT: You my wnt to use the fct tht lim n x n 1 = lim n x n for every convergent sequence {x n } 49. (i) Show tht between ny two rel numbers there is n irrtionl number. HINT: Let the two numbers be < b. Mp the closed intervl [, b] onto [0, 1] by liner function to show tht [, b] is uncountble. Wht would hppen if [, b] contined nothing but rtionls? (ii) Show tht irrtionls re dense in R. 50. Prove tht the limit of the sequence 2, 2 + 2, equl to 2. 2 + 2 + 2,... exists nd is HINT: The generl term of this sequence stisfies the recursion reltion n+1 = 2 + n. 51. (i) Show in detil tht { n } converges to the limit A precisely when every one of its subsequences does. (ii) Suppose tht the sequences { n } nd {b n } converge to the sme limit, lim n = lim b n = n n x. Show tht ny sequence {c n } whose terms re k s nd b l s lso converges to x. 52. (i) Find lim sup nd lim inf of the sequence other cluster points? { 1 n + sin nπ 2 }. Does this sequence hve ny 8
(ii) Find lim sup nd lim inf of the sequence { n } defined by 1 = 1 2, 2m = 2m 1, 2m+1 = 1 2 2 + 2m. 53. (i) For ny two rel sequences { n } nd {b n }, prove tht lim sup n ( n + b n ) lim sup n provided the sum on the right is not of the form. n + lim sup b n, n (ii) Construct n exmple in which strict inequlity holds in (i). 54. Prove directly from the definition of the limit of function tht lim[f(x) + g(x)] = x lim f(x) + lim g(x), provided the limits on the right-hnd side exist. x x 55. Evlute the limit lim x 0 (1 cos x) 2 sin 2 x tn 2 x. 56. For the following functions, do the left-hnd nd/or right-hnd limits of f exist t x = 0? If either limit exists, wht is its vlue? 1, x < 0, (i) f(x) = 0, x = 0, 1, x > 0. (ii) f(x) = e 1/x. (iii) f(x) = sin 1 x. 57. Show tht the following functions re continuous: (i) x n, for ll x R. (ii) 1/x n for x 0. (iii) z 2 for ll z C. 58. Suppose f is rel function defined on R which stisfies lim[f(x + h) f(x h)] = 0 h 0 9
for every x R. Does this imply tht f is continuous. 59. Let f be continuous rel function defined on R, nd let A, B, nd C be the sets of ll x R such tht f(x) = 0, f(x) 0, nd f(x) > 0, respectively. Show tht A nd B re closed nd C is open. 60. Give n exmple of rel continuous function defined on R such tht (i) the imge of n open set is not open, (ii) the imge of closed set is not closed. 61. Let f be continuous rel function on R. ( ) (i) Is it necessrily true tht f lim sup x n = lim sup f(x n )? n n (ii) Is it true tht x being limit point of {x n } implies f(x) being limit point of f(x n )? (iii) Is it true tht the inverse imge under f of convergent sequence is necessrily convergent sequence? 62. (i) Let f be continuous on n intervl, I, nd let f(x) = 0 when x is rtionl. Show tht f(x) = 0 for ll x I. (ii) Let f nd g be continuous rel functions defined on n intervl, I. Let A I be dense in I. Show tht f(a) is dense in f(i). If f(x) = g(x) for ll x A, show tht f(x) = g(x) for ll x in I. (In other words, continuous function is determined by its vlues on dense subset of its domin.) 63. If f is rel continuous function defined on closed set A R, prove tht there exists continuous rel function g on R such tht g(x) = f(x) for ll x A. HINT: Let the grph of g be stright line on ech of the open intervls which constitute the complement of A. This is long nd difficult problem, nd to understnd wht the difficulty is, imgine, sy, A = n=1 [1/(2n + 1), 1/2n] [ 1, 0] {1}. 64. (i) Show tht if continuous function on n intervl tkes on only finite number of vlues, it must be constnt. 10
(ii) Let f = p + g, where p is polynomil of odd degree nd g is bounded continuous function on the line. Show tht there is t lest one solution of f(x) = 0. HINT: p(x) ttins rbitrrily lrge positive nd negtive vlues for lrge positive nd negtive x, respectively. 65. Determine which of the following functions is uniformly continuous on the indicted intervls. (i) x 3 on [ 1, 1]. (ii) x 3 on R. (iii) 1/x 2 on [1, 2]. (iv) 1/x 2 on (0, 2). 66. (i) Let f be rel uniformly continuous function on bounded set A R. Prove tht f is bounded. HINT: If it were not, you cn find sequence {x n } in A such tht f(x n+1 ) > f(x n ) +1. Find convergent subsequence {x nk } nd derive contrdiction by showing tht x nk x nl < 1/m for lrge enough k nd l, yet f (x nk ) f (x nl ) > 1. (ii) Give n exmple to show tht the conclusion of (i) is flse if boundedness of A is omitted from the hypothesis. 67. (i) If f nd g re uniformly continuous nd bounded rel functions on R, show tht fg is uniformly continuous. (ii) Give n exmple to show tht the conclusion of (i) is flse if boundedness of f nd g is omitted from the hypothesis. 68. A uniformly continuous function of uniformly continuous function is uniformly continuous. Stte this more precisely nd prove it. 69. (i) Let K = {0} {1/n n = 1, 2, 3,...}. Prove tht K is compct directly from the definition, without using the Heine-Borel theorem. (ii) Give n exmple of n open cover of the open intervl (0, 1) which hs no finite subcover. 70. Construct compct set of rel numbers whose limit points form countble set. 11
71. (i) Suppose f is uniformly continuous rel function defined on set A R. Prove tht {f(x n )} is Cuchy sequence in f(a) for every Cuchy sequence {x n } in A. HINT: The proof is bit similr to tht of 66 (i). (ii) Give n exmple to show tht this is not the cse if f is not uniformly continuous. 72. Let A be dense subset of n intervl, I, nd let f be uniformly continuous rel function defined on A. Show tht there exists unique continuous function g, defined on I, such tht g(x) = f(x) on A, by completing the following outline: (i) Let x I A. Use the results of problems 71 (i), 47, nd 51 to prove tht you cn define g(x) = lim f(x n ) for ny sequence {x n } in A with x n x, nd tht the vlue g(x) does n not depend on the choice of {x n }. (ii) To show tht g is continuous t ny x I, let {x n } be n rbitrry sequence in I with x n x. Use the result of (i) to show tht, given ɛ > 0, for ech x n, there is y n A such tht x n y n < 1/n nd g(x n ) f(y n ) < ɛ/2. Show tht lso y n x nd use this to rgue tht if n is lrge enough, g(x) g(x n ) < ɛ. (iii) For uniqueness, use the result of problem 62 (ii). 73. Let rel function f on R be continuous, nd let it stisfy the eqution f(x + y) = f(x) + f(y) for ll x, y R. Show tht f(x) = cx for some constnt x. HINT: First, find the vlues of f(x) for rtionls. 74. Provide n lterntive proof of the Theorem: A continuous function on compct set is uniformly continuous, by completing the detils of the following rgument. If K is compct nd f is not uniformly continuous on K, then for some ε > 0 there re sequences {x n } nd {t n } in K such tht x n t n 0 but f(x n ) f(t n ) ε. Use the fct tht ny sequence in K must hve convergent subsequence to obtin contrdiction. 75. If E is nonempty subset of C, define the distnce from x C to E by ρ E (x) = inf x z. z E (i) Prove tht ρ E (x) = 0 if nd only if x Ē (the closure of E). (ii) Prove tht ρ E (x) is uniformly continuous function on C by showing tht ρ E (x) ρ E (y) x y 12
for ll x, y C. HINT: ρ E (x) x z x y + y z, so tht ρ E (x) x y + ρ E (y). (iii) Suppose K nd F re disjoint subsets of C, K is compct, F is closed. Prove tht there exists δ > 0 such tht z w > δ if z K, w F. HINT: ρ F is continuous positive function on K. (iv) Show tht the conclusion of (iii) my fil for two disjoint closed sets if neither is compct. 76. Let I = [0, 1], the closed unit intervl. Suppose f is continuous function of I into I. Prove tht f(x) = x for t lest one x I. 77. Cll mpping from R to R open if f(a) is n open set whenever A is. Prove tht every continuous open mpping is monotonic. 78. Let [x] denote the lrgest integer contined in x, tht is, [x] is the integer such tht x 1 < [x] x. Wht kind of discontinuities do the functions [x] nd x [x] hve? Describe them in s much detil s you cn. 79. Find nd clssify the discontinuities of the following functions (i) f(x) = e 1/x + sin 1 x, (ii) f(x) = 1 1 e 1/x. 80. Prove tht f(x) = lim n [ ] lim (cos m n!πx)2m = { 0, x irrtionl, 1, x rtionl. Wht kind of discontinuities does the function f hve? 81. Suppose nd c re rel numbers, c > 0, nd f is defined on [ 1, 1] by { x sin( x c ), x 0 f(x) = 0, x = 0. 13
Show the following sttements: (i) f is continuous if nd only if > 0. (ii) f (0) exists if nd only if > 1. (iii) f is bounded if nd only if 1 + c. (iv) f is continuous if nd only if > 1 + c. 82. Let f n denote the n-th iterte of f, f 1 (x) = f(x), f 2 (x) = f(f 1 (x)),..., f n (x) = f(f n 1 (x)). Express f n in terms of f. Show tht if f (x) b for ll x, then n f n(x) b n. 83. Let f be defined for ll rel x, nd suppose tht f(x) f(t) (x t) 2 for ll rel x nd t. Show tht f is constnt. 84. If C 0 + C 1 2 + C 2 3 + + C n 1 n + C n n + 1 = 0, where C 0,... C n re rel constnts, prove tht the eqution C 0 + C 1 x + C n 1 x n 1 + C n x n = 0 hs t lest one rel root between 0 nd 1. 85. Suppose f is defined nd differentible for every x > 0 nd f (x) 0 s x. Prove tht f(x + 1) f(x) 0 s x. 86. Suppose g is rel function on R with bounded derivtive (sy g M). Fix ɛ > 0 nd defined f(x) = x + ɛg(x). Prove tht f is one-to-one if ɛ is smll enough. Determine the set of dmissible vlues of ɛ s depending on M. 87. Suppose (i) f is continuous for x 0, (ii) f (x) exists for x > 0, (iii) f(0) = 0, 14
(iv) f is monotoniclly incresing. Put g(x) = f(x) x, x > 0, nd prove tht g is monotoniclly incresing. HINT: Differentite g nd use the men-vlue theorem. 88. Suppose f is continuous on [, b] nd ɛ > 0. Prove tht there exists δ > 0 such tht f(t) f(x) f (x) t x < ɛ whenever 0 < t x < δ, x, t b. HINT: Use the uniform continuity of f nd the men-vlue theorem. 89. Let f be continuous rel function on R, of which it is known tht f (x) exists for ll x 0 nd tht f (x) 3 s x 0. Does it follow tht f (0) exists? HINT: Use the men-vlue theorem crefully to show tht it does. 90. Suppose f is rel function on (, ). Cll x fixed point of f if f(x) = x. (i) If f is differentible nd f (t) 1 for ll rel t, show tht f hs t most one fixed point. HINT: Men-vlue theorem. (ii) Show tht the function f defined by f(t) = t + 1 1 + e t hs no fixed point lthough 0 < f (t) < 1 for ll rel t. (iii) However, if there is constnt A < 1 such tht f (t) A for ll rel t, prove tht fixed point x of f exists, nd tht x = lim n x n, where x 1 is n rbitrry number nd for n = 1, 2, 3,.... x n+1 = f(x n ) (vi) Show tht the process described in (iii) cn be visulized by the zig-zg pth (x 1, x 2 ) (x 2, x 2 ) (x 2, x 3 ) (x 3, x 3 ) (x 3, x 4 ). 15
91. Second Men-Vlue Theorem: Let f nd g be continuous on [, b] nd differentible on (, b), with g() g(b). Prove tht there exists point c (, b) such tht f(b) f() g(b) g() = f (c) g (c). HINT: Apply the men-vlue theorem to the function [f(b) f()]g(x) [g(b) g()]f(x). 92. If f(x) = x 3, compute f (x), f (x) for ll rel x, nd show tht f (0) does not exist. 93. Suppose F is continuous in neighborhood of x. Show tht F (x + h) + F (x h) 2F (x) lim h 0 h 2 = F (x). HINT: Replce x, f(x), nd g(x) in problem 91 by t, F (x+t)+f (x t), nd t 2, respectively, nd let = 0 nd b = h. You cn compute the remining limit either directly or by L Hospitl s rule. 94. Let < b nd f(x) = { (x ) 2 (x b) 2, x [, b] 0, otherwise. Show tht f is continuously differentible function tht is non-zero exctly on the intervl (, b). 95. Let A be closed subset of R. Construct continuously differentible rel function defined on R tht vnishes exctly on A. 96. Show tht the function f(x) = { e 1/x 2, x 0, 0, x = 0, hs ll the derivtives t x = 0, nd tht they re ll equl to 0 nd continuous. HINT: Use the fct tht lim t t α e t = 0 for ll α. 97. Suppose tht f 0, f is continuous on [, b], nd for ll x [, b]. 16 f(x) dx = 0. Prove tht f(x) = 0
98. Compute Recll tht sec x = 1/ cos x. ( 1 lim 1 + sec 2 π ) 2π nπ + sec2 + + sec2. n n 4n 4n 4n 99. Let f nd p be continuous nd p(x) > 0 on [, b]. Prove tht there exists ξ [, b] such tht f(x)p(x) dx = f(ξ) p(x) dx. 100. Prove tht if the rel-vlued function f is integrble on the intervl [, b] then so is f 2. Using the identity (f + g) 2 = f 2 + 2fg + g 2, prove tht the product of two integrble functions is integrble. HINT: First show tht f 2 (x) f 2 (t) 2M f(x) f(t), where M = P = {x 0,..., x n } is prtition of [, b], let M i (f) = nd deduce tht if x, t [x i 1, x i ], then sup f(x), m i (f) = inf f(x), x [x i 1,x i ] x [x i 1,x i ] f 2 (x) f 2 (t) 2M[M i (f) m i (f)]. sup f(x). x [,b] If 101. Let y(x) be continuously differentible function on the intervl [, b]. Show tht the length of the curve (x, y(x)) for < x < b is given by the expression by completing the following outline: 1 + [y (x)] 2 dx Let = x 0 x 1... x n = b be ny prtition of the intervl [, b]. Let S be the polygonl curve with corners t the points (x k, y(x k )) for k = 1,..., n. (i) Let M = mx x [,b] y (x). Show tht, given ɛ > 0, if x i < ɛ/2m, then S pproximtes y(x) to within ɛ. 17
(ii) Show tht the length of the polygonl curve S equls n [ ] 2 y(xk ) y(x k 1 ) l(s) = 1 + (x k x k 1) x k x k 1 k=1 = n k=1 1 + [y (ξ k )] 2 (x k x k 1 ). for some ξ k [x k 1, x k ], k = 1,..., n. (iii) Let mx x k x k 1 0 to complete the proof. 102. Let p nd q be positive rel numbers such tht Prove the following sttements: 1 p + 1 q = 1. (1) (i) If u 0 nd v 0, then Equlity holds if nd only if u p = v q. uv up p + vq q. HINT: Compute the res between the curve y = x p 1 nd the two coordinte xes, strting t the origin nd ending t x = u nd y = v, respectively. Use (1). (ii) If f nd g re integrble on [, b], f 0, g 0, nd then f p (x) dx = g q (x) dx = 1, f(x)g(x) dx 1. (Problems 100 nd 107 gurntee tht ll these functions re integrble.) (iii) If f nd g re integrble on [, b], then { f(x)g(x) dx } 1/p { 1/q f(x) p dx g(x) dx} q. (2) This is Hölder s inequlity. When p = q = 2 it is clled the Cuchy-Schwrtz inequlity. 18
103. For u integrble on [, b] define { 1/2 u 2 = u(x) dx} 2. Assume f, g, nd h to be integrble on [, b], nd prove the tringle inequlity f h 2 f g 2 + g h 2 s consequence of the Cuchy-Schwrtz inequlity from problem 102 (iii). HINT: First squre it. 104. With the nottion s in problem 103, suppose f is integrble on [, b] nd ɛ > 0. Prove tht there exists continuous function g on [, b] such tht f g 2 < ɛ. HINT: For suitble prtition P = {x 0,..., x n } of [, b], define g(t) = x i t f(x i 1 ) + t x i 1 f(x i ) x i x i if x i 1 t x i. Argue tht f(t) g(t) < M i m i on [x i 1, x i ], where M i = sup f(x), m i = inf f(x), x [x i 1,x i ] x [x i 1,x i ] nd tht 0 [f(t) g(t)] 2 2M(M i m i ), where M = sup f(x). x [,b] 105. Suppose f is continuously differentible on [, b], with f() = f(b) = 0, nd f 2 (x) dx = 1. Prove tht nd tht xf(x)f (x) dx = 1 2, [f (x)] 2 dx x 2 f 2 (x) dx > 1 4. HINT: While it is esy to show in the lst inequlity, > requires you to solve simple differentil eqution. 19
106. Show without n explicit vrible chnge tht x/ 1+x 2 0 dt x = du 1 t 2 1 + u. 2 Wht does this eqution sy bout inverse trigonometric functions? 0 107. Prove the following tht continuous function of n integrble function is integrble. In prticulr, prove the Theorem: Suppose f is integrble on [, b], m f M, φ is continuous on [m, M] nd h(x) = φ(f(x)) on [, b]. Then h is integrble on [, b]. HINT: Choose ɛ > 0 nd 0 < δ < ɛ such tht φ(s) φ(t) < ɛ if s t < δ nd s, t [m, M]. Argue tht for some prtition P = {x 0,..., x n } of [, b], S + (f, P ) S (f, P ) < δ 2. (3) Let M i nd m i, nd Mi nd m i, re the suprem nd infim of f nd h on [x i 1, x i ], respectively. Divide 1,..., n into two clsses: i A if M i m i < δ, i B if M i m i δ. Show tht, for i A, Mi m i ɛ, nd for i B, Mi m i 2K, with K = φ(t). Use (3) to show tht δ i B x i < δ 2, mx x [m,m] nd conclude tht S + (h, P ) S (h, P ) ɛ(b ) + 2Kδ. 108. Suppose f is bounded on [, b], nd f 2 is integrble. Does it follow tht f is integrble? Does the nswer chnge if we ssume tht f 3 is integrble? HINT: The result of problem 107 should help. 109. If p q > 0, use the Cuchy-Schwrtz inequlity to prove tht log p q p q pq. 110. Use the properties of the geometric progression to show tht x x2 2 + x3 3 x2n 2n for 0 < x < 1 nd ny integer n 1. < log(1 + x) < x x2 2 + x3 3 + x2n+1 2n + 1 20
x du HINT: First verify tht log(1 + x) = 0 1 + u. 111. Show tht for n = 1, 2, 3,..., the number S n = 1 + 1 2 + 1 3 + + 1 n log n is positive, tht it decreses s n increses, nd hence tht the sequence of these numbers converges to limit, γ, between 0 nd 1. HINT: Verify tht n+1 dx n x < 1 n n < dx n 1 x, nd use it to show tht S n is bounded between 0 nd 1 nd monotoniclly decresing. 112. Evlute the following limits (i) lim h 0 (1 + hx) 1/h, (ii) lim n n(x 1/n 1) if x > 0, log(1 + x) (iii) lim. x 0 x e (1 + x) 1/x (iv) lim. x 0 x n (v) lim n log n (n1/n 1). 113. Suppose f(x)f(t) = f(x + t) for ll rel x nd t. (i) Assuming tht f is differentible nd not zero, prove tht where c is constnt. f(x) = e cx, (ii) Prove the sme thing ssuming only tht f is continuous. HINT: In (ii), proceed s in problem 73. 114. Compute the rclength of the prbol y = x 2 /2 between the origin nd the point whose bsciss is x = > 0. 21
115. For positive integer n, prove tht the integrl x 2n+1 e x2 dx cn be computed in terms of elementry functions. HINT: Find suitble recursion formul. No need to compute the integrl explicitly. 116. (i) Consider the indefinite integrl dx x2 + 2bx + c. Depending on the sign of the coefficient 0 nd the discriminnt b 2 c, show tht this integrl leds to log, rcsin, rccosh, or rcsinh. Describe the corresponding substitutions nd results in detil. (ii) Show tht the substitution t = 1/x trnsforms n integrl of the type dx x Ax 2 + 2Bx + C into n integrl of the type discussed in prt (i). 117. Derive ll the solutions of the differentil eqution y = 6yy + y tht vnish t x ± together with ll their derivtives. HINT: At some convenient point, in order to integrte the eqution once more, you should multiply it by y. You cn lso ssume tht y > 0. The solution is y = 1 (x x 0 ) 2 sech2. 2 (You will get no points t ll if you just plug this solution into the eqution nd check it works!) 118. Compute the indefinite integrls (i) dx (x 2 1), 2 (ii) x dx (x 2 + x + 1). 2 22
119. Let R(, ) denote n expression which is rtionl in both its rguments. (i) Show tht the integrl R(cos x, sin x) dx cn be trnsformed into n integrl of rtionl function of the vrible t = tn x/2. (ii) Show tht the integrl R(cosh x, sinh x) dx cn be trnsformed into n integrl of rtionl function of the vrible t = tnh x/2. (iii) Show tht the integrl R(x, 1 x 2 ) dx cn be trnsformed into n integrl of the type discussed in prt (i) by the substitution x = cos u. (iv) Show tht the integrls R(x, x 2 ± 1) dx cn be trnsformed into integrls of the type discussed in (ii) by the substitutions x = sinh u nd x = cosh u, respectively. 120. Evlute the integrl s function of cos x, sin x,, nd b. dx cos x + b sin x 121. (i) Show tht the Fresnel integrls, converge. F 1 = sin ( x 2) dx, F 2 = 0 0 cos ( x 2) dx, (ii) Show tht the integrl 0 2x sin ( x 4) dx converges even though its integrnd becomes unbounded s x. HINT: Use pproprite substitutions. 122. Let f be continuous rel function on [0, ), let nd let lim x 0 f(x) = L. Prove tht 0 f(αx) f(βx) dx x 23 f(x) dx converge for every > 0, x
converges for ll positive α nd β nd hs the vlue L log β α. 123. Show tht Γ(n) = 1 0 ( log 1 x) n 1 dx. 124. Let < 0 nd b > 0 nd let F be bounded on [, 0) nd (0, b]. The Cuchy principl vlue integrl of F on [, b] is defined s [ ɛ ] P F (x) dx = lim F (x) dx + F (x) dx. ɛ 0+ ɛ (i) Compute P 1 1 dx x. (ii) If < 0 < b nd f is continuously differentible on [, b], show tht exists. P f(x) x dx HINT: Add nd subtrct f(0) in the numertor. 125. The integrl f(x) dx is sid to converge bsolutely if tht if n integrl converges bsolutely, then it converges. f(x) dx converges. Prove 126. Let p(x) nd q(x) be polynomils nd let q(x) 0 for x >. Show tht p(x) q(x) dx converges bsolutely if nd only if their degrees stisfy the inequlity deg(p) deg(q) 2. 127. Show tht cos x 0 1 + x dx = sin x 0 (1 + x) dx, 2 nd tht one of these integrls converges bsolutely, but the other does not. 24