# The final exam will take place on Friday May 11th from 8am 11am in Evans room 60.

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1 Mth 104: finl informtion The finl exm will tke plce on Fridy My 11th from 8m 11m in Evns room 60. The exm will cover ll prts of the course with equl weighting. It will cover Chpters 1 5, 7 15, 17 21, 23 34, 36, 37 of Ross. The finl will consist of ten questions, which will be of similr style to those on the midterms. This will give you slightly longer per question thn on the midterms. The exm will be closed book no textbooks, notebooks, or clcultors llowed. As on the midterms, you will be expected to be fmilir with the bsic defintions, nd know the key results, but it will not be necessry to remember the proof of every theorem by hert. Smple finl questions Questions 26 to 36 were used on the finl exm when the clss ws given in Spring Consider the power series x n n 3, x 3n 2n, x 2n!. n=0 For ech power series, determine its rdius of convergence R. By considering the series t x = ±R, determine the exct intervl of convergence. 2. () Prove by using the definition of convergence only, without using limit theorems, tht if (s n ) is sequence converging to s, then lim n s 2 n = s 2. (b) Prove by using the definition of continuity, or by using the ɛ-δ property, tht f (x) = x 2 is continuous function on R. 3. Let f be twice differentible function defined on the closed intervl [0, 1]. Suppose r, s, t [0, 1] re defined so tht r < s < t nd f (r) = f (s) = f (t) = 0. Prove tht there exists n x (0, 1) such tht f (x) = () Suppose tht n=0 n is convergent series. Define sequence (b n ) ccording to b n = 2n + 2n+1 for n N {0}. Prove tht n=0 b n converges. (b) Construct n exmple of series n=0 n tht diverges, but tht if (b n ) is defined s bove, then n=0 b n converges. 5. Suppose f is rel-vlued continuous function on R nd tht f () f (b) < 0 for some, b R where < b. Prove tht there exists n x (, b) such tht f (x) = 0.

2 6. Let f be rel-vlued function defined on n intervl [0, b] s { x for x Q, f (x) = 0 for x / Q. Consider prtition P = {0 = t 0 < t 1 <... < t n = b}. Wht re the upper nd lower Drboux sums U( f, P) nd L( f, P)? Is f integrble on [0, b]? 7. Let f be decresing function defined on [1, ), where f (x) 0 for ll x [1, ). Prove tht 1 f (x)dx converges if nd only if f (n) converges. [This is essentilly question sking you to prove generl form of the integrl test.] 8. Consider the function defined for x, y R s { 1 if x = y, d(x, y) = 0 if x = y. () Prove tht d defines metric on R. (b) Wht is the neighborhood of rdius 1/2 centered on 0? (c) Consider n rbitrry set S R. Is S open? Is S compct? 9. Let x = (x 1, x 2 ) nd y = (y 1, y 2 ) be in R 2. Consider the function () Prove tht d is metric on R 2. d(x, y) = x 1 y 1 + x 2 y 2. (b) Compute nd sketch the neighborhood of rdius 1 t (0, 0). 10. Consider function f defined on R which stisfies f (x) f (y) (x y) 2 for ll x, y R. Prove tht f is constnt function. 11. Suppose tht f is differentible on R, nd tht 2 f (x) 3 for x R. If f (0) = 0, prove tht 2x f (x) 3x for ll x Show tht if f is integrble on [, b], then f is integrble on every intervl [c, d] [, b]. 13. () Suppose r is irrtionl. Prove tht r 1/3 nd r + 1 re irrtionl lso. (b) Prove tht (5 + 2) 1/3 + 1 is irrtionl. 14. By using L Hôpitl s rule, or otherwise, evlute ( 1 lim x 0 sin x 1 x x 1 e x2 3x, lim x 0 ), lim x 0 x 3 sin x x.

3 15. Let R. Consider the sequence (s n ) defined s { if n is odd, s n = 2 n if n is even. Compute lim sup s n nd lim inf s n. For wht vlue of does (s n ) converge? 16. Consider the function f : R 2 R defined s f (x 1, x 2 ) = 1 x x With respect to the usul Eucliden metrics on R nd R 2, prove tht f is continuous t (0, 0) nd t (0, 1). 17. () Clculte the improper integrl 1 for the cses when 0 < p < 1 nd p > 1. (b) Prove tht for ll p (0, ). 18. Prove tht if f is integrble on [, b], then 0 0 x p dx x p dx = d b lim f (x) dx = f (x) dx. d b 19. Let f (x) = x 2, nd define sequence (s n ) ccording to s 1 = λ nd s n+1 = f (s n ) for n N. Prove tht (s n ) converges for λ [ 1, 1], nd diverges for λ > Consider the three sets A = [0, 2] Q, B = {x 2 + x 1 : x R}, C = {x R : x 2 + x 1 < 0}. For ech set, determine its mximum nd minimum if they exist. For ech set, determine its supremum nd infimum. Detiled proofs re not required, but you should justify your nswers. 21. Let f n (x) = x x n on [0, 1] for n N. () Prove tht f n converges pointwise to limit f, nd determine f. (b) Prove tht f n does not converge uniformly to f.

4 (c) Find n intervl I contined in [0, 1] on which f n f uniformly. (d) Prove tht the f n re integrble, tht f is integrble, nd tht 1 0 f n 1 0 f. 22. Define for x R. () Clculte F(x) = x 0 (b) Sketch f nd F. 1 if x 1, f (x) = 2 if 1 < x 2, 0 if x > 2 f (t)dt for x R. (c) Compute F nd stte the precise rnge over which F exists. You my mke use of the second Fundmentl Theorem of Clculus. 23. () Let f nd g be continuous functions on [, b] such tht b there exists n x [, b] such tht f (x) = g(x). f = b g. Prove tht (b) Construct n exmple of integrble functions f nd g on [, b] where b f = g but tht f (x) = g(x) for ll x [, b]. b 24. Define the sequence of functions h n on R ccording to { n if x 1/2n, h n (x) = 0 if x > 1/2n. () Sketch h 1, h 2, nd h 3. (b) Prove tht h n converges pointwise to 0 on R/{0}. Prove tht lim n h n (0) =. (c) Let f be continuous rel-vlued function on R. Prove tht lim h n f = f (0). n (d) Construct n exmple of n integrble function g on R where lim h n g n exists nd is rel number, but does not equl g(0). 25. Consider the function on the intervl [0, ). f (x) = x 1 + x.

5 () Show tht lim x f (x) = 1, nd tht 0 f (x) < 1 for ll x [0, ). (b) Sketch f. (c) Clculte f, f, nd use them to construct the prtil Tylor series t x = 1 with the form (x 1) f T (x) = n f (n) (1). n! 2 n=0 (d) Show tht f T cn be written s qudrtic eqution with the form x 2 + bx + c, nd compute, b, nd c. (e) Add sketch of f T to the sketch of f. [Note: f T (1) = f (1) so the two curves should intersect t x = 1.] 26. Determine the rdius of convergence R of the power series f 1 (x) = n=0 x n n2 + 1, f 2(x) = ( 2) n x 2n n 2. By considering the series t x = ±R, determine the exct intervls of convergence. If you mke use of ny of the theorems for determining series properties, you should stte which ones you use. 27. Suppose (s n ) nd (t n ) re two sequences tht converge to s nd t respectively. Stte the definition of convergence, nd use it to prove crefully tht 3s n + t n 3s + t. Do not use the limit theorems for sequences. 28. The Fiboncci numbers re defined by F 0 = 0 nd F 1 = 1, nd F n+1 = F n + F n 1 for n N. Let the golden rtio be defined s ϕ = () Show tht ϕ 2 = 1 + ϕ. (b) Let f (n) = ϕn (1 ϕ) n 5. For n N, define H n to be the hypothesis tht both F n = f (n) nd F n 1 = f (n 1). Apply mthemticl induction to prove tht H n is true for ll n N, nd deduce tht F n = f (n) for ll n N {0}. [Hint: it is simpler to crry out the lgebr in terms of ϕ nd use the identity in (), s opposed to clculting explicitly in terms of (1 + 5)/2.] (c) Show tht F n+1 F n ϕ s n.

6 29. Let f be continuous strictly incresing function defined on R. Consider subset S R nd let T = { f (x) : x S}. () If sup S is finite, prove tht sup T = f (sup S). (b) Suppose ( n ) is sequence such tht lim sup n is finite. Define the sequence (b n ) so tht b n = f ( n ) for ll n. Prove tht lim sup b n = f (lim sup n ). (c) Suppose the condition tht f is continuous is removed. Construct n exmple of strictly incresing function f defined on R, nd subset S R, where sup S is finite, but sup T = f (sup S). 30. Consider two dimensionl spce R 2, where n element x R 2 is written s x = (x 1, x 2 ). Let d E (x, y) = ((x 1 y 1 ) 2 + (x 2 y 2 ) 2 ) 1/2 be the usul Eucliden metric on R 2. () Prove tht d A (x, y) = min{ x 1 y 1, 2 x 2 y 2 } is not metric on R 2. (b) Prove tht d B (x, y) = mx{ x 1 y 1, 2 x 2 y 2 } is metric on R 2. Drw the neighborhood of rdius 1 t (0, 0). (c) Consider n rbitrry metric spce (X, d), nd mpping f : X R 2. Suppose tht f is continuous with respect to (X, d) nd (R 2, d E ). Prove tht it is lso continuous with respect to (X, d) nd (R 2, d B ). 31. Consider the function defined on [0, ) s { 0 if 0 x 1, f (x) = x if x > 1. () Compute F(x) = x 0 f (t)dt on [0, ). (b) Clculte F (x), stting the precise rnge over which it exists. You my mke use of the second Fundmentl Theorem of Clculus. (c) Prove tht neither f nor F is uniformly continuous on [0, ). 32. Let f (x) = x 2 (1 x) nd g(x) = f (x) for x R. () Plot f nd g. (b) By using the definition of differentibility, prove tht g(x) is differentible t x = 0, but not t x = 1. (c) Compute the derivtives g (n) (2) for n N, nd use them to write down the Tylor series of g t x = 2. Prove tht this series is equl to f (x) for ll x R. 33. Consider the function defined on the domin [0, ) s { x if 0 x 1, g(x) = 0 if x > 1. Define sequence of functions on the intervl [0, 1] ccording to f n (x) = ng(nx) for n N.

7 () Sketch f 1, f 2, nd f 3 on the intervl [0, 1]. (b) Prove tht f n converges pointwise to limit f, nd determine f. (c) Prove tht f n does not converge uniformly to f on [0, 1]. (d) Show tht 1 0 f n = 1/2 for ll n. Does 1 0 f n converge to 1 0 f? 34. () Suppose tht f is differentible function on (0, ), nd tht f (x) 0 s x. Define g(x) = f (x + 1) f (x). Use the Men Vlue Theorem to prove tht g(x) 0 s x. (b) Use the Intermedite Vlue Theorem to show tht for ll n N, the eqution p(x) = x 2n+1 4x + 1 hs t lest three rel roots. In ddition, prove tht these roots must be irrtionl. 35. () Use L Hôpitl s rule to evlute lim x 0 x e x e x, lim sin 2 x x 0 x 2. (b) Let f be rel-vlued function defined on n intervl (, b). Let x (, b), nd suppose f is twice differentible t x. Show tht the limit exists nd equls f (x). lim h 0 f (x + h) + f (x h) 2 f (x) h 2 (c) Construct n exmple of rel-vlued function f defined on n intervl (, b), where for some x (, b) the bove limit exists nd is finite, but f is not twice differentible t x. 36. () Let ( f n ) be sequence of integrble functions on [, b], nd suppose tht f n f uniformly on [, b]. Prove tht f is integrble on [, b] nd tht b b f = lim f n. n (b) By using integrtion by prts, prove tht 1 1/2 log x dx = log

8 (c) Recll tht for x < 1, log(1 + x) = x n ( 1) n+1. n By using this nd the results from prts () nd (b), prove tht log 2 = 1 1 n(n + 1)2 n.

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