Homework for MATH 4604 (Advanced Calculus II) Spring 2017

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1 Homework for MATH 4604 (Advanced Calculus II) Spring 2017 Homework 14: Due on Tuesday 2 May 55. Let m, n P N, A P R mˆn and v P R n. Show: L A pvq 2 ď A 2 v Let n P N and let A P R nˆn. Let I n denote the n ˆ n identity matrix. Assume that I n A 2 ď 1{2. Show that A is invertible. 57. Let m, n P N and let g : R m R n. Assume 0 m P dom rdgs. Assume gp0 m q 0 n. Assume, for all j P t1,..., mu, pb j gqp0 m q 0 n. Show that g P Omnp1q. Homework 13: Due on Tuesday 25 April 50. Let V and W be finite dimensional vector spaces. Show (1) QpV, W q Ď O V W p2q, and (2) rqpv, W qs X rov W p2qs t0 V W u. 51. Let m P N, q P R, v P R m, S P R mˆm sym. Define C P CpR m, Rq and L P LpR m, Rq and Q P QpR m, Rq by Cpxq q and Lpxq px H v V q 11 and Qpxq ppx H Sx V q 11 q{p2!q. Let f : C ` L ` Q P P ď2 pr m, Rq. Let I : t1,..., mu. Show (1) fp0 m q q, P I, pb i fqp0 m q v i and j P I, pb i B j fqp0 m q S ij. 52. Let f : R R and δ ą 0. Let I : p δ, δq. Assume that f 1 is defined on I, i.e., that I Ď dom rf 1 s. Show that there exists c : I Ñ I such that c P Op1q and such that, for all x P I, we have rfpxqs rfp0qs rf 1 pcpxqqsx. 53. Let f : R R, g : R 2 R. P R, fptq gpt, 0q. P R, f 1 ptq pb 1 gqpt, 0q. 54. Let f : R 2 R. Assume 0 2 P dom rd 2 fs. Show p R f P O21p2q. Hint: Show: for all x, y P R, pr f px, yq fpx, yq pfp0 2 qq ` ppb 1 fqp0 2 qqx ` ppb 2 fqp0 2 qqy ` r 1{p2!q s r ppb 1 B 1 fqp0 2 qqx 2 ` 2ppB 1 B 2 fqp0 2 qqxy ` ppb 2 B 2 fqp0 2 qqy 2 s.

2 Then show: at 0 2, 0 p R f B 1 p p R f q B 2 p p R f q B 1 B 1 p p R f q B 1 B 2 p p R f q B 2 B 2 p p R f q. Homework 12: Due on Tuesday 18 April 47. Let m P N and let W be a finite dimensional vector space. Let f : R m W. Let p P R m and let i P t1,..., mu. Show: pb i fqpp ` q B i pfpp ` qq. 48. Let V, W and X be finite dimensional vector spaces and let p P V. Let f : V W, g : V X. Show that D p ppf, gqq pd p f, D p gq. 49. Let m, n P N, let f : R m R n, let k P N 0 and let p P R m. Assume, for all i P t1,..., mu, that B i f is both defined near p and bounded near p. Show both of the following: (1) f T p P O mn p1q and (2) f is continuous at p. Hint: For (1), simply capitalize all the Os appearing in the proof of the corresponding result about O proved during class. For (2), combine (1) with O mn p1q Ď Omn to see that fp T P Omn. Homework 11: Due on Tuesday 11 April 44. Let V, W and X be vector spaces, let f : V W, g : W X and let p P dom rg fs. Show that pg fq T p pg T fppq q pf T p q. Hint: Given v P V. We wish to show: pg fq T p pvq pg T fppq qppf T p qpvqq. To keep the notation from getting messy, I suggest defining q : fppq and w : f T p pvq. Then compare pg fq T p pvq rgpfpp ` vqqs rgpfppqqs with pg T fppq qppf T p qpvqq g T q pwq rgpq ` wqs rgpqqs. 45. Let U and V be finite dimensional vector spaces, let f : U V and let p P dctrfs. Assume dom rfs is a nbd of p in U. Show f T p P OUV. Note: f T p rfpp ` qs rf C p s. Unassigned HW: Let U and V be finite dimensional vector spaces, let f : U V and let p P dlinrfs. Show that fp T P O UV p1q. Note: Paige showed us, in class, how to do the Unassigned HW above.

3 46. Let U, V, W and X be vector spaces, let P BpV, W, Xq and let m, n P N 0. Show: rp m pu, V qs rp n pu, W qs Ď rp m`n pu, Xqs. Hint: Given P P P m pu, V q and Q P P n pu, W q. We wish to show: P Q P P m`n pu, Xq. We wish to show DH P SM m`n pu, Xq such P U, Hpu,..., uq pp Qqpuq. Choose F P SM m pu, V q such P U, F pu,..., uq P puq. Choose G P SM n pu, W q such P U, Gpu,..., uq Qpuq. Define H 0 P M m`n pu,..., U, Xq by H 0 pt 1,..., t m, u 1,..., u n q rf pt 1,..., t m qs rgpu 1,..., u n qs. Let H : SymrH 0 s. Homework 10: Due on Tuesday 4 April 38. Let X be a topological space, let W be a finite dimensional vector space, let φ : X W, let x P X and let } } P N pw q. Show: r lim φ 0 W s ñ r lim }φ} 0 s. x x 39. Let W be a finite dimensional vector space and let ε P ORW p1q. εphq Show that lim hñ0 h 0 ε W, i.e., show that lim 0 W. 0 id R 40. Let V and W be finite dimensional vector spaces, let f : V W and let x P dlinrfs. Show: x P dctrfs, i.e., show: f is continuous at x. Hint: Let L : D x f and let R : fx T L. Then L P LpV, W q and R P OV W p1q. It follows that L and R are both continuous at 0 V. So, since fx T L ` R, we see that fx T is continuous at 0 V. Use this to show that f is continuous at x. 41. Let P N prq be the absolute value function. Show: 0 R dlinr s. 42. Let V be a finite dimensional vector space, let f : V R and let x P dlinrfs. Assume that D x f 0 V R. Show that f does not have a local semi-max at x. That is, show, for any nbd U in V of x, that there exists y P U such that fpyq ą fpxq.

4 Hint: Let U be given, and we seek y. Let L : D x f and let R : f T x L. Then L P LpV, Rq and R P OV Rp1q. Show that you can choose v P V such that Lpvq ą 0. Show that you can choose h ą 0 small enough so that all that of the following works. Let y : x ` hv. Then y P U, and we wish to show: fpyq ą fpxq. We have Rphvq ă h rlpvqs{100. Then f T x phvq h rlpvqs ` rrphvqs ą 99 h rlpvqs { 100 ą 0. Then rfpyqs rfpxqs f T x phvq ą 0, so fpyq ą fpxq, as desired. 43. Let V and W be finite dimensional vector spaces, C P CpV, W q. Let X : LpV, W q. Show DC 0 V X. That is, P V, D u C 0 X. Note: 0 X 0 V W. Homework 9: Due on Tuesday 28 March 34. Let V and W be finite dimensional vector spaces. Prove that P 1 pv, W q Ď O V W p1q. 35. Let V and W be finite dimensional vector spaces. Prove that rp 1 pv, W qs X rov W p1qs t0 V W u. 36. Let f : R R. Assume that f 1 P Op3q and that fp0q 0. Show that f P Op4q. Hint: Using the Choice MVT, show that there exists c P Op1q such that, for all x «0, we have fpxq rf 1 pcpxqqs x. 37. Define f : R Ñ R by fpxq x 3. Show that rd 2 fs 11 f 1 p2q. Homework 8: Due on Tuesday 21 March 31. Let V and W be finite dimensional vector spaces. Let P N pv q, ε : V W, p ą 0. Assume 0 V P dom rεs and εp0 V q 0 W. Show: ˆ ε p ε P OV W ppq q ô P Oˆ p V W. 32. Let V and W be finite dimensional vector spaces. Let P N pv q, α : V W, p ą 0. Assume 0 V P dom rαs and αp0 V q 0 W. Show: ˆ α p α P O V W ppq q ô P Oˆ p V W. 33. Let V and W be finite dimensional vector spaces. Prove that poˆ V R q poˆ V W q Ď Oˆ V W.

5 Homework 7: Due on Tuesday 7 March 26. Let R, S and T be sets. Let f : R Ñ S and let g : S Ñ T. Let U Ď T. Show: f pg puqq pg fq puq. 27. Let S and T be sets. Let f : S ĂÑą T. Let U Ď S. Show: f puq pf 1 q puq. 28. Let S and T be sets. Let f : S ĂÑ T. Let U Ď S. Show: f pf puqq U. 29. Let S and T be sets. Let f : S Ñą T. Let U Ď T. Show: f pf puqq U. 30. Let V be a vector space, P N pv q, C ą 0. Let x P V, r ą 0. Show: B px, rq B C px, Crq. Homework 6: Due on Tuesday 28 February 22. Let V be a vector space and let P SBF ě0 pv q. Show P P V, cx c x, and 0 V Let S be a set, let n P N, let V be a vector space and let f : S n Ñ V. Let g : Symrfs : S n Ñ V. Show both (i) g is symmetric, i.e., for all x 1,..., x n P S, for all σ P Σ n, we have gpx 1,..., x n q gpx σ1,..., x σn q, and (ii) the diagonal restrictions of f and g are the equal to one another, i.e., for all x P S, we have gpx, x,..., xq fpx, x,..., xq. 24. Let V be a vector space. Show: rp 2 pv qsrp 3 pv qs Ď P 5 pv q. Hint: Given fpvq Bpv, vq and gpvq T pv, v, vq, we wish to show that rfpvqsrgpvqs Qpv, v, v, v, vq. (You need to set up all the quantifications.) We know that rfpvqsrgpvqs rbpv, vqsrt pv, v, vqs. Let Let Q : SymrQ 0 s. Q 0 pv, w, x, y, zq rbpv, wqs rt px, y, zqs. 25. Let V, W be vector spaces. Show: rp 2 pw qs rp 3 pv, W qs Ď P 6 pv q.

6 Hint: Given gpwq Bpw, wq and fpvq T pv, v, vq, we wish to show that gpfpvqq Spv, v, v, v, v, vq. (You need to set up all the quantifications.) We know that gpfpvqq BpT pv, v, vq, T pv, v, vqq. Let Let S : SymrS 0 s. S 0 pu, v, w, x, y, zq Bp T pu, v, wq, T px, y, zq q. Homework 5: Due on Tuesday 21 February 18. For all p P p0, 8s, define B p : tx P R 2 s.t. x p ď 1u. Then, for all p P p0, 8q, we have B p tps, tq P R 2 s.t. s p ` t p ď 1u. Also, B 8 tps, tq P R 2 s.t. maxt s, t u ď 1u. Graph B 1{2, B 1, B 2, B 3, B 4, B Show, for all x P R 2, that x 1 ě x 2 ě x 8 ě x 1 {100. That is, show, for all s, t P R, that s ` t ě? s 2 ` t 2 ě maxt s, t u ě p s ` t q{ Find the largest C ą 0 such P R 2, x 8 ě C x 1. That is, find the largest C ą 0 such t P R, maxt s, t u ě Cp s ` t q. 21. Let a, b, c P R. Assume a ě 0. Assume, for all x P R, that ax 2 ` 2bx ` c ě 0. Show (i) pa 0q ñ pb 0q, (ii) ac b 2 ě 0. and Hint for (ii): Replacing x :Ñ b{a in the assumption, we see that ap b{aq 2 ` 2bp b{aq ` c ě 0. Homework 4: Due on Tuesday 14 February 13. Let m, n P N, L P LpR m, R n q, v P R m. Show that plpvqq V rls v V. (Note: We have Lpvq P R n, so plpvqq V P R nˆ1. Also, rls P R nˆm. Also, v P R m, so v V P R mˆ1.) 14. Let m, n P N. Show that the two maps L ÞÑ rls : LpR m, R n q Ñ R nˆm and A ÞÑ L A : R nˆm Ñ LpR m, R n q are inverses.

7 15. Let m, n P N, let u P R m, let v P R n and let A P R nˆm. Show that B A pu, vq pv H A u V q Let m, n P N. Show that the two maps B ÞÑ rbs : BpR m, R n q Ñ R nˆm and A ÞÑ B A : R nˆm Ñ BpR m, R n q are inverses. 17. Let n P N. Show: P SBFpR n q, rbs P Rsym nˆn P R nˆn sym, B A P SBFpR n q. and Homework 3: Due on Tuesday 7 February 8. Let X be a topological space, let f : R X and let a, b P R. Show that lim f pa ` q lim f. b a`b 9. Let X and Y be topological spaces. Let f, g : X Y. Assume that f Ď g; that is, assume, for all x P dom rfs, that both x P dom rgs and fpxq gpxq. Let a P LP X pdom rfsq. Assume Y is Hausdorff. Show that lim a f lim a g. 10. Let X and Y be topological spaces, φ : X Y, p P X and q P Y. Assume that φ Ñ q near p. Define ψ : pdom rφsq Y tpu Ñ Y by # φpxq, if x p; ψpxq q, if x p. Show that ψ : X Ñ Y is continuous at p. 11. Define r : Rzt0u Ñ R by rpxq 1{x. Show, for all x P Rzt0u, that r 1 pxq 1 x Let f, g : R R. Let a P LP R pdom rf{gsq. Show that pf{gq 1 paq rgpaqsrf 1 paqs rfpaqsrg 1 paqs rgpaqs 2. Hint: Define r : Rzt0u Ñ R by rpxq 1{x. Then f{g f pr gq. Homework 2: Due on Tuesday 31 January

8 4. Let f : R R, I Ď ddrfs. Assume that I is an interval. Let S : f 1 piq. Assume that S ě 0. Show that f is semi-increasing on I. That is, show, for all a, b P I, that r p a ă b q ñ p fpaq ď fpbq q s. 5. Let f : R R, I Ď ddrfs. Assume that I is an interval. Let S : f 1 piq. Assume that S ă 0. Show that f is strictly decreasing on I. That is, show, for all a, b P I, that r p a ă b q ñ p fpaq ą fpbq q s. 6. Let δ ą 0. Let U : p δ, δq. Define β : U Ñ R by βphq maxt0, hu. Show that β Ñ 0 near 0 in R. 7. Show that O O Ď O. That is, P P O, αε P O. Homework 1: Due on Tuesday 24 January 1. Let g : R R and let L P R. Assume g is defined near ă 0. Assume g ď 0 near ă 0. Assume g Ñ L near 0. Show L ď Let m, b P R. Define f : R Ñ R by fpxq mx ` b. Show f 1 C m R. 3. Let f : R R and let a P ddrfs. Assume f is defined near a in R. Assume f has a local semi-max at a in R. Show f 1 paq 0.