Course 121, , Test III (JF Hilary Term)

Transcription

1 Course 2, 989 9, Test III (JF Hilry Term) Fridy 2d Februry 99, pm Aswer y THREE questios. Let f: R R d g: R R be differetible fuctios o R. Stte the Product Rule d the Quotiet Rule for differetitig f.g d f/g respectively, d prove these rules, usig stdrd properties of its. 2. () Stte Rolle s Theorem. (b) Let f: R R be 5-times differetible fuctio. Let, b d c be rel umbers stisfyig < b < c. Suppose tht f() = f () = f(b) = f (b) = f(c) = f (c) =. Prove tht there exists some s stisfyig < s < c for which f (5) (s) =. (c) Stte the Me Vlue Theorem, d show how it my be derived from Rolle s Theorem. 3. () Stte Cuchy s Me Vlue Theorem, d show how l Hôpitl s Rule my be derived from it. (b) Evlute the followig its, usig l Hôpitl s Rule: t si si si t t 3 2t t 5, t t 5 t 3 9t 2 + 5t + 25, cos(t 2 ) t si t 4. () Stte d prove Tylor s Theorem.

2 Course 2, 989 9, Test IV (JF Hilry Term) Fridy 2d Mrch 99, pm Attempt questio d 2 other questios. Let f: [, b] R be bouded fuctio defied o the itervl [, b]. () Defie the cocept of prtitio of the itervl [, b]. Give the defiitio of the lower sum L(P, f) d the upper sum U(P, f) of f for the prtitio P. (b) Defie the lower Riem itegrl L f(t) dt d the upper Riem itegrl U f(t) dt of f o the itervl [, b]. Defie precisely wht is met by syig tht the fuctio f is Riem-itegrble o [, b], d defie the Riem itegrl of Riem-itegrble fuctio o [, b]. (c) Let f: [, ] R be defied by f(t) = t 2. Clculte L(P, f) d U(P, f), where P deotes the prtitio of [, ] ito subitervls of legth / (i.e., P = {t, t,..., t }, where t i = i/ for i =,,..., ). Hece show tht L f(t) dt = U f(t) dt = 2 3. [You my use, without proof, the followig idetities: i= i = 2( + ), i= i 2 = 6( + )(2 + ).] 2. () Let f: [, b] R be cotiuous fuctio o the itervl [, b], d let F (x) = x f(t) dt, ( x b). Prove tht F (x) = f(x) for ll x stisfyig < x < b. (b) Fid the derivtive of the fuctio g: R R defied by g(x) = x 4 t 2 e t2 dt. 3. () Let f be fuctio tht is k times differetible d whose kth derivtive is cotiuous o some ope itervl cotiig the rel umbers d + h. Usig the rule for itegrtio by prts, show tht where k f( + h) = f() + r k (, h) = hk (k )! h! f () () + r k (, h), ( t) k f (k) ( + th) dt. 2

3 (b) Show tht log( + h) = for ll h stisfyig < h <. k + k ( ) ( ) h 4. Let f, f 2, f 3,... be sequece of cotiuous rel-vlued fuctios o itervl [, b], d let f be rel-vlued fuctio o [, b]. () Defie wht is met by syig tht the fuctios f coverge uiformly to f o the itervl [, b] s +. (b) Suppose tht the fuctios f coverge uiformly to f o [, b] s +. Prove tht + f (t) dt = f(t) dt. (c) Give exmple of sequece f, f 2, f 3,... of cotiuous rel-vlued fuctios o itervl [, b] d cotiuous rel-vlued fuctio f o [, b] such tht eve though + f (t) dt f(t) dt, f (t) = f(t) for ll t [, b]. + 3

4 Course 2, 989 9, Test V (JF Triity Term) Fridy 2th April 99, pm Attempt questio d 2 other questios. Test the followig ifiite series for covergece: () z +! (z C), (b) 3 si 2, (c) =2 ( ) log, (d) 4 + cos 2, (e)! () Defie wht is met by syig tht sequece z, z 2, z 3, z 4,... of complex umbers is Cuchy sequece. (b) Prove tht every coverget sequece of complex umbers is Cuchy sequece. (c) Prove the every Cuchy sequece of complex umbers is bouded. (d) Prove tht every Cuchy sequece of complex umbers is coverget. [You my use, without proof, the Bolzo-Weierstrss Theorem, which sttes tht every bouded sequece of complex umbers hs coverget subsequece.] 3. () Prove tht the ifiite series is diverget (b) By usig the sme method s i (), or otherwise, prove tht the ifiite series =2 log is diverget 4. () Stte the Altertig Series Test, d prove tht y ifiite series stisfyig the coditios of this test is coverget. (b) Does the ifiite series 2 cos π + si 2 π 2 stisfy the coditios of the Altertig Series Test? Is this ifiite series coverget? 4

5 Course 2, 989 9, Test VI (JF Triity Term) Fridy th My 99, pm Attempt questio d 2 other questios. Determie which of the followig subsets of the complex ple re ope d which re closed: () {z C : z + 2 < 7}, (b) {z C : z + 2 > 7}, (c) {z C : z d Re z }, (d) {z C : z d Re z < }, (e) {z C : exp z + z 3 < 7}. [Briefly justify your swers.] 2. () Prove tht sequece z, z 2, z 3,... of complex umbers coverges to some complex umber l if d oly if, give y ope set U which cotis l, there exists some turl umber N such tht the poit z j belogs to U for ll j stisfyig j N. (b) Usig (), or otherwise, show tht if F is closed set i the complex ple, d if z, z 2, z 3, z 4,... is ifiite sequece of complex umbers belogig to F which coverges to some complex umber l the l F. 3. () Let K be closed bouded subset of the complex ple d let f: K C be cotiuous fuctio o K. Prove tht there exists some o-egtive rel umber C such tht f(z) C for ll z K. 4. Let (b) Let K be closed bouded subset of the complex ple d let f: K C be cotiuous fuctio o K. Let w be complex umber with the property tht f(z) w for ll z K. Prove tht there exists some rel umber δ > such tht f(z) w δ for ll z K. = umbers. z be power series whose coefficiets,, 2,... re complex () Defie the rdius of covergece of this power series. (b) Prove tht the power series = z coverges if z < R, but diverges if z > R, where R is the rdius of covergece of the power series. c Dvid R. Wilkis