MAS22 Alysis, Semester 2 Exercises Srh Whitehouse (Exercises lbelled * my be more demdig.) Chpter Problems: Revisio Questio () Stte the defiitio of covergece of sequece of rel umbers, ( ), to limit. (b) Stte the defiitio of the supremum of bouded set of rel umbers. (c) Prove tht y bouded mootoic icresig sequece coverges. Questio 2 Defie sequece ( ) itertively by =, + = +. 2 () Write dow the first few terms of this sequece. C you guess the limit? (b) Prove tht the sequece coverges to this limit. Questio 3 Let f : [, b] R be fuctio. () Stte wht is met by syig tht f is cotiuous t poit x (, b). (b) Stte wht is met by syig tht f is differetible t poit x (, b). (c) Prove tht if f is differetible t poit x the it is cotiuous t the poit x. (d) Give exmple to show tht the coverse of the sttemet i prt (c) is ot true.
Questio 4 Defie fuctio f : R R by f(x) = { x 2 x, x 2 x. Prove tht f is differetible. Write dow the derivtive. Does the secod derivtive of f t exist? Justify your swer. Chpter Problems: Series. Prove tht if the series = coverges, the s. (This is Propositio.3 i the otes; the proof ws left s exercise.) 2. Let k = k(k + )(k + 2) By expressig k i prtil frctios, fid expressio for the sum k= Prove tht the series k= k coverges d fid the limit. 3. Fid explicit expressio for the prtil sums of the series k= Deduce tht it does ot coverge. k ( l + ). k 4. Let ( ) be sequece of o-zero rel umbers d ssume tht the series = coverges bsolutely. Let (b ) be sequece of o-zero rel umbers. Suppose tht lim =. b Prove tht the series = b coverges bsolutely. 5. () By writig the terms i prtil frctios, prove tht the series coverges. = ( + ) 2
(b) By usig the result i the previous questio, prove tht = 2 coverges. (c) Prove tht the series = coverges wheever α 2. α 6. Usig y of the tests for covergece, or other resoig, which of the followig series coverge? Justify your swers. () (b) (c) (d) = 3 + 4 3 + 5 3 + 5 = = 4 ( ) cos 2 + 2 (e) = = (!) 2 (2)! 7. Show tht, if r <, the series = r coverges to ( r) 2. Hit: Show tht ( r)s = r r r, where s is the -th prtil sum of the series. 8. Show tht Hit: Let = t ( = 2 ( ) t 2 2 = π 4. ) ( d b = t t( b) = 2+ t t b + t t b. ) d use the idetity 9. For ech of the followig sttemets, sy whether it is true or flse, givig resos for your swers. () If lim =, the = coverges. 3
(b) If lim ( + 2 + + ) =, the = coverges. (c) If > for ll d the prtil sums s = + + re bouded, the = coverges. (d) If the series = is coverget, the so is = 2. Chpter 2 Problems: Itegrtio. Let r, s: R R be step fuctios. () Show tht the product rs: R R, defied by the formul (rs)(t) = r(t)s(t), is step fuctio. (b) Let < b. Prove tht ( 2 ( b ) ( b ) b r(t)s(t) dt) r(t) 2 dt s(t) 2 dt. (c) Let f, g : [, b] R be step fuctios. Defie ( b f = f(t) 2 dt Prove tht f + g f + g. ) 2 2. Cosider the fuctio f : [, ] R defied by f(x) = x 2. () Let N. Let r : [, ] R, r (t) = χ [,/) + ( ) 2 χ [/,2/) + +. ( ) 2 χ [( )/,). Show tht r (t) f(t) for ll t [, ], d clculte the itegrl r (t) dt, of the step fuctio r. You my use without proof the formul 2 + 2 2 + 3 2 + + 2 = ( + )(2 + ). 6 (b) Let N. Let s : [, ] R, s (t) = ( ) 2 χ (,/] + ( ) 2 2 χ (/,2/] + + χ (( )/,]. Show tht s (t) f(t) for ll t [, ], d clculte the itegrl s (t) dt, of the step fuctio s. 4
(c) Use the bove to clculte the itegrl x 2 dx. 3. () Compute the itegrl (b) Show tht the itegrl does ot exist. x dx. x 2 dx 4. Which of the followig fuctios re Riem itegrble? Justify your swers. () The fuctio f : [, 2] R defied by f(x) = exp(si x) x 3 + 5. (b) The fuctio g : R R defied by g(x) = exp( x). (c) The fuctio h: [, ) R defied by h(x) = si x x 2. (d) The fuctio k : [, ] R defied by x < 2, 2 2 x < 3 4, 3 3 k(x) = 4 4 x < 7 8, 7 7 8 8 x < 5 6,. 5. () Defie step fuctios r, s: R R by r(t) = χ [,) + eχ [,2) + e 4 χ [2,3], s(t) = eχ [,] + e 4 χ (,2] + e 9 χ (2,3]. Evlute the itegrls. 3 r(t) dt, 3 s(t) dt. (b) Why is the fuctio f : [, 3] R defied by f(x) = e x2 itegrble? Riem 5
(c) Prove tht + e + e 4 3 e x2 dx e + e 4 + e 9. 6. Let f : R R be cotiuous fuctio, d let, b: R R be differetible fuctios. Prove tht d dx b(x) (x) 7. Defie fuctio L: (, ) R by f(y) dy = b (x)f(b(x)) (x)f((x)). L(x) = x t dt. Prove the followig directly from the defiitio of L. () L is differetible, d L (x) = x. (b) L(xy) = L(x) + L(y) for ll x, y >. (c) L(x/y) = L(x) L(y) for ll x, y >. 8. (*) Let f : (, ) R be cotiuous fuctio stisfyig the formul f(xy) = f(x) + f(y) for ll x, y >. () Prove tht f(x ) = f(x) for ll x > d R. [Hit: Prove this first for N, the Z, the Q, d filly exted to R by cotiuity.] (b) Prove tht f(x) = f(e) l(x) for ll x >. (c) For the bove fuctio L, prove tht L(x) = l x. 9. () Compute 2 x dx. (b) Is the fuctio f : [, ) R defied by the formul itegrble? Justify your swer. f(x) = x. Compute x 2 dx. 6
. Let f : [, b] R be Riem itegrble. Prove tht there is umber x [, b] such tht x f(t) dt = b x f(t) dt. 2. () Let f : [, b] R be Riem itegrble. Suppose there re m, M R such tht m f(x) M for ll x [, b]. Prove tht there is umber µ [m, M] such tht b f(x) dx = (b )µ. (b) Let f : [, b] R be cotiuous. Prove tht there is some ξ [, b] such tht b f(x) dx = (b )f(ξ). 3. Let f : [, b] R be cotiuous, d let g : [, b] [, ) be itegrble. Prove tht there is some ξ [, b] such tht b f(x)g(x) dx = f(ξ) b g(x) dx. Do we eed the ssumptio g(x)? Justify your swer. Chpter 3 Problems: Sequeces d series of fuctios. () Stte the defiitio of sequece of fuctios f : R R covergig uiformly to fuctio f : R R. (b) Prove tht if sequece (f ) of cotiuous fuctios f : R R coverges uiformly to fuctio f : R R, the the fuctio f is cotiuous. 2. Cosider the sequece of fuctios (f ), where f : [, π] R is defied by f (x) = si (x). Show tht (f ) coverges poitwise. Is (f ) uiformly coverget? Justify your swer. 3. For ech of the followig sequeces of fuctios (f ) determie the poitwise limit (if it exists) o the idicted itervl, d decide whether (f ) coverges uiformly to this limit. () f (x) = x /, x [, ]. (b) f (x) = { x, x x, x R. 7
(c) f (x) = e x /x, x (, ). (d) f (x) = e x2, x [, ]. (e) f (x) = e x2, x R. 4. For ech of the followig sequeces of fuctios (g ) fid the poitwise limits, d determie whether they coverge uiformly o [, ], d o [, ). () g (x) = x/. (b) g (x) = x /( + x ). (c) g (x) = x /( + x ). 5. For ech of the followig sequeces of fuctios (h ), where h : [, ] R, fid the poitwise limits, d determie whether they coverge uiformly. () h (x) = ( x/) 2. (b) h (x) = x x. (c) h (x) = k= xk. 6. Defie f : R R by f (x) = + cos x 2 + si 2 x. () Fid the poitwise limit of the sequece of fuctios (f ). (b) Show tht the sequece (f ) coverges uiformly. (c) Clculte 7 lim 2 f (x) dx. 7. () Let N. Show tht we c defie cotiuous fuctio f : [, ] R by f (x) = x =, x / l x < x <, x =. (Note: you oly eed check cotiuity t x = d x =.) (b) Does the sequece (f ) uiformly coverge to limit? Justify your swer. If you wish, you my ssume without proof tht ech fuctio f is mootoe icresig. (c) Prove tht lim f (x) dx =. 8
8. Compute the limits e t4 lim dt, justifyig y procedures you use. 2 lim t 2 ((si t)/) dt, 9. () Fid sequece of differetible fuctios (f ), where f : [, ] R, tht coverges poitwise to fuctio f : [, ] R, but such tht the sequece of derivtives (f ) does ot coverge to f. (b) Is it possible to hve sequece of differetible fuctios (f ), where f : [, ] R, tht coverges uiformly to fuctio f : [, ] R, but such tht the sequece of derivtives (f ) does ot coverge to f? Justify your swer.. Which of the followig fuctios re uiformly cotiuous? Justify your swers. () The fuctio f : (, 2] (, 2] defied by f(x) = /x. (b) The fuctio g : [, 2] [, 2] defied by g(x) = /x. (c) The fuctio h: [, ) [, ) defied by h(x) = /x.. By usig the Weierstrss M-test or otherwise, for ech of the folowig series, determie whether it coverges uiformly o R d whether it coverges uiformly o [, ]. () (b) (c) (d) = = 2 + x 2 ( ) x 2+ (2 + )! = si(x) si(x) = 2. () Show the series = x coverges uiformly for x [, ] wheever < <. (b) Does the series coverge uiformly o [, )? Expli. 9
3. Prove tht there is fuctio f : R R defied by f(x) = d tht this fuctio is cotiuous. = si x 2 Chpter 4 Problems: Applictios I. Fid the rdius of ech of covergece of the followig power series. () (b) (c) (d) (e) = 2 2 + x 2x + 4x 2 8x 3 + 6x 4 + x 2 + x 4 + x 6 + x 8 + = ( ) x 2 (2)! (x) 3 = 2. Let = x be power series with rdius of covergece R. Is it true tht x t dt = + x+ wheever x < R? = = Justify your swer usig y theorems you eed from the course. 3. By itegrtig the power series betwee r d r, show tht ( ) + r log r = wheever < r <. Justify ech step. x = x ) = 2 (r + r3 3 + r5 5 +
4. () Express the fuctio + x 2 s power series. Wht is the rdius of covergece? (b) Use prt () d questio 2 to fid power series covergig to the fuctio rct(x). (c) Use prt (b) to fid series covergig to π 4. 5. Write dow series with rdius of covergece. 6. Use the power series for exp(x) to prove tht the umber e is irrtiol. Chpter 5 Problems: Sequeces i R k. () Let (x ) be sequece i R k with limit x. Let α R. Prove tht the sequece (αx ) coverges to αx. (b) Let (x ) d (y ) be sequeces i R k with limits x d y respectively. Prove tht the sequece (x + y ) coverges to x + y. 2. For ech of the followig sequeces i R 2 determie whether they coverge. If so, fid the limit. () = (/, 2 ). (b) b = (, / 2 ). (c) c = (/, / 2 ). (d) d = (/, ( ) /). (e) e = 2 (cos, 2 + si ). (f) f = (/, ( ) ). 3. For ech of the followig sequeces i R 2 determie whether they coverge. If so, fid the limit. () The sequece (x, y ) defied itertively by x =, y = 2, d x + = y, y + = x. (b) = (( + ) ), l. (c) The sequece (u, v ) defied itertively by u =, v = d u + = 2 (u + v ), v + = 2 (u v ).
(d) The sequece defied itertively by u =, v = d u + = 2 (u + v ), v + = u v. 4. (*) For ech of the followig sequeces i R 2 determie whether they coverge. () The sequece = (( + ) (, ) ). (b) The sequece (x, y ) defied itertively by x = 2 3, y = 3, d 2 = +, y + = x + y. x + x y Hit: prove tht the sequece (x ) is mootoic decresig d bouded below, d the sequece (y ) is mootoic icresig d bouded bove. 5. Cll sequece ( ) i R 2 bouded if there is costt C such tht C for ll. Prove tht every bouded sequece i R 2 hs coverget subsequece. 6. Which of the followig sets re ope? Justify your swer. () The itervl [, ) i R. (b) The set {x R x } i R. (c) The squre (, ) (, ) i R 2. (d) The lie R {} i R 2. 7. Which of the followig subsets of R 3 re ope? Justify your swers. A = {(x, y, z) R 3 x 2 + y 2 + z 2 > }. B = {(x, y, z) R 3 x >, y >, z > }. C = {(x, y, z) R 3 x = y = }. D = {(x, y, z) R 3 x 2 + y 2 + z 2 = }. 8. We cll set A R k closed if for y sequece ( ) i A tht coverges i R k, the limit of the sequece lso lies i A. Prove tht the followig sets re closed. () The sets R k d the empty set. (b) The oe-poit set {x} for some x R k. 2
(c) The squre [, ] [, ] i R 2. 9. () Prove tht subset A R k is closed if d oly if the complemet R k \A is ope. (b) Give exmple of set tht is either ope or closed.. Let z C be complex umber. Write z = x + iy. Wht c we sy bout covergece of the sequece (x, y ) whe z < d whe z >?. () Let ( ) be sequece i R k with limit. Prove tht y subsequece of ( ) lso hs limit. (b) Suppose sequece ( ) hs subsequece tht coverges. Does the sequece ( ) itself hve to coverge? Prove tht it does or give couterexmple. Chpter 6 Problems: Fuctios of severl vribles. Defie f : [, 2] [, ] R 2 by Fid d sketch the imge of f. f(x, y) = ( y, x 2 + y). 2. Defie f : R 2 R by f(x, y) = x 2 + y 2. () Fid d sketch the pre-imges f (), f (), d f ( ). (b) Fid d sketch the pre-imge f ((, ]). 3. () Prove, usig the chrcteristio of cotiuity i terms of limits (Propositio 6.6), tht the fuctio f : R 2 R defied by f(x, y) = xy is cotiuous. (b) Prove, usig the chrcteristio of cotiuity i terms of limits (Propositio 6.6), tht if two fuctios f, g : R 2 R re cotiuous, the the product fg : R 2 R defied by (fg)(x, y) = f(x, y)g(x, y), x, y R is cotiuous. (This will give ltertive proof to tht give i Propositio 6.9 which ws direct from the ɛ d δ defiitio.) 4. Usig y results you eed from the lectures, prove tht the fuctio f : R 2 R defied by the formul is cotiuous. f(x, y) = si(e xy ) 3
5. Let f : R 2 R be cotiuous. Let x, y R 2. Suppose tht there is some c R such tht f(x) < c < f(y). Prove tht there is some z R 2 such tht f(z) = c. [Hit: You will eed the itermedite vlue theorem for cotiuous rel fuctios here (Semester, Theorem 4.3.). Thik bout defiig fuctio g by g(t) = f(tx + ( t)y). ] 6. (*) Prove tht cotiuous fuctio f : [, b] [c, d] R is bouded, d tkes o its mximum d miimum vlues. 7. Defie fuctio f : R 2 R by { xy f(x, y) = x 2 +y (x, y) (, ), 2 (x, y) = (, ). () Prove tht the fuctios g, h: R R defied by g(x) = f(x, ) d h(x) = f(, x) re cotiuous. (b) Is the fuctio f cotiuous t (, )? Justify your swer. Chpter 7 Problems: Applictios II. () Let t >. Show tht we c defie cotiuous fuctio f t : [, ] R by x = f t (x) = x t l x < x < t x = (b) Stte wht it mes to sy tht sequece of fuctios (g ), where g : [, ] R, coverges uiformly to fuctio g : [, ] R. Wht c you sy bout the sequece (f / ) where f t is s i prt ()? (See questio 7 of Chpter 3.) (c) Prove tht lim t f t (x) dx =. You my ssume tht ech f t is mootoe icresig. (d) Let h: [, ] R R be fuctio. Stte coditios o h which esure tht the equtio holds. (e) Prove tht for ll t >. d dt h(x, t) dx = x t l x h (x, t) dx t dx = l(t + ) 4