[PDF] RealAnalysis Math 125A Fall 2012 Sample Final Questions

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Real Analysis

Math 125A, Fall 2012

Sample Final Questions

1.Definef:R→Rby

f(x) =x3

1 +x2.

Show thatfis continuous onR. Isfuniformly continuous onR?

Solution.

•To simplify the inequalities a bit, we write

x

31 +x2=x-x1 +x2.

Forx,y?R, we have

|f(x)-f(y)|=???? x-y-x1 +x2+y1 +y2???? ?x1 +x2-y1 +y2????

1 +x2-y1 +y2????

=????x-y+xy2-x2y(1 +x2)(1 +y2)???? xy|(1 +x2)(1 +y2)? |x-y| 1 2?

1 +x2+ 1 +y2(1 +x2)(1 +y2)?

|x-y| 12?

11 +y2+11 +x2?

|x-y|

•It follows that

Thereforefis Lipschitz continuous onR, which implies that it is uni- formly continuous (takeδ=?/2). 1

2.Does there exist a differentiable functionf:R→Rsuch thatf?(0) = 0

butf?(x)≥1 for allx?= 0?

Solution.

•No such function exists.

•We have

f ?(0) = limx→0? f(x)-f(0) x? The mean value theorem implies that for for everyx?= 0, there is some

ξstrictly between 0 andx(soξ?= 0) such that

f(x)-f(0) x=f?(ξ)≥1. •Since limits preserve inequalities, it follows that lim x→0? f(x)-f(0) x? ≥1, so we cannot havef?(0) = 0. 2

3.(a) Write out the Taylor polynomialP2(x) of order two atx= 0 for the

function⎷

1 +x. and give an expression for the remainderR2(x) in Taylor"s

formula⎷

1 +x=P2(x) +R2(x)-1< x <∞.

(b) Show that the limit lim x→0?

1 +x/2-⎷

1 +x x2? exists and find its value.

Solution.

•(a) The function and its derivatives are given by f(x) =⎷

1 +x, f(0) = 1,

f ?(x) =1

2(1 +x)-1/2, f?(0) =12,

f ??(x) =-1

4(1 +x)-3/2, f??(0) =-14,

f ???(x) =3

8(1 +x)-5/2.

•The Taylor polynomial and remainder are

P

2(x) =2?

k=01 k!f(k)(0)xk, R2(x) =13!f???(ξ)x3, whereξis between 0 andx, which gives

1 +x= 1 +12x-18x2+116(1 +ξ)-5/2x3

(b) For this part, we only need the Taylor polynomial of order one,

1 +x= 1 +12x-18(1 +ξ)-3/2x2

whereξis between 0 andx. Sinceξ→0 asx→0, it follows that lim x→0?

1 +x/2-⎷

1 +x x2? =18limξ→0(1 +ξ)-3/2=18. 3

4.(a) Supposefn:A→Ris uniformly continuous onAfor everyn?N

andfn→funiformly onA. Prove thatfis uniformly continuous onA. (b) Does the result in (a) remain true iffn→fpointwise instead of uni- formly?

Solution.

•(a) Let? >0. Sincefn→fconverges uniformly onAthere exists

N?Nsuch that

|fn(x)-f(x)|3for allx?Aandn > N. Choose somen > N. Sincefnis uniformly continuous, there exists

δ >0 such that

|fn(x)-fn(y)|3for allx,y?Awith|x-y|< δ.

Then, for allx,y?Awith|x-y|< δ, we have

which implies thatfis uniformly continuous onA. •(b) The result does not remain true iffn→fpointwise. For example, considerfn: [0,1]→Rdefined byfn(x) =xn. Thenfnis uniformly continuous on [0,1] because it is a continuous function on a compact interval, butfn→fpointwise where f(x) =?

1 ifx= 1.

The limitfis not even continuous on [0,1].

4

5.Definefn: [0,∞)→Rby

f n(x) =sin(nx)

1 +nx.

(a) Show thatfnconverges pointwise on [0,∞) and find the pointwise limit f. (b) Show thatfn→funiformly on [a,∞) for everya >0. (c) Show thatfndoes not converge uniformly tofon [0,∞).

Solution.

•(a) Ifx >0, then

1 +nx→0 asn→ ∞

sofn(x)→0. Also,fn(0) = 0 for everyn, sofn(0)→0. Thus,fn→0 pointwise on [0,∞).

•(b) We have

so given? >0 takeN= 1/aand then|fn(x)|< ?for alln > N, meaning thatfn→0 uniformly on [a,∞). •(c) If (fn) converges uniformly on [0,∞), then it must converge to the pointwise-limit 0. Letxn=π/(2n). Then f n(xn) =1

1 +π/2.

f n(x)≥?0, which means thatfndoes not converge uniformly to 0 on [0,∞). 5

00.20.40.60.81-0.5

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 x y Figure 1: Plot of the functionfn(x) = sin(nx)/(1 +nx) on [0,1] forn= 20 (green),n= 100 (red), andn= 500 (blue). Remark.The non-uniform convergence of the sequence nearx= 0 is illus- trated in the figure. We can also write the proof in terms of the sup-norm. Let ?f?a= sup x?[a,∞)|f(x)| denote the sup-norm offon [a,∞). Ifa >0, then na→0 asn→ ∞, sofn→0 uniformly on [a,∞). Ifa= 0, then ?fn?0≥1

1 +π/2for everyn?N,

so (fn) does not converge uniformly to 0 on [0,∞). 6

6.Suppose that

f(x) =∞? n=1sinnx n3, g(x) =∞? n=1cosnxn2. (a) Prove thatf,g:R→Rare continuous. (b) Prove thatf:R→Ris differentiable andf?=g.

Solution.

•(a) Since

?sinnx n3???? n=11n3<∞ cosnx n2??? n=11n2<∞, the WeierstrassM-test implies that both series converge uniformly (and absolutely) onR.

•Each term in the series is continuous, and the uniform limit of contin-uous functions is continuous, sof,gare continuous onR.

•(b) The series forgis the term-by-term derivative of the series forf. Since the series forgconverges uniformly, the theorem for the differen- tiation of sequences implies thatfis differentiable andf?=g. 7

7.LetP={2,3,5,7,11,...}be the set of prime numbers.

(a) Find the radius of convergenceRof the power series f(x) =? p?Px p=x2+x3+x5+x7+x11+... (b) Show that

Solution.

•(a) We write the series as

f(x) =∞? n=2a nxn where a n=?

1 ifnis prime,

0 ifnisn"t prime.

•Then

Therefore, if|x|<1 the series converges by comparison with the con- vergent geometric series?|x|n. Furthermore, if|x|>1, the terms in the series do not approach 0. So the radius of convergence of theseries isR= 1. p?Px n=2x n=x2∞? n=0x n=x2 1-x, which proves the result. 8

8.Let (X,d) be a metric space.

(a) Define the open ballBr(x) of radiusr >0 and centerx?X. (b) Define an open setA?X. (c) Show that the open ballBr(x)?Xis an open set.

Solution.

•(a) The open ball is defined by

B r(x) ={y?X:d(x,y)< r}. •(b) A setA?Xis open if for everyx?Athere existsr >0 such that B r(x)?A. •(c) Suppose thaty?Br(x). We have to show thatBr(x) contains an open ballBs(y) for somes >0. Choose s=r-d(x,y)>0. (Draw a picture!) Ifz?Bs(y), then by the triangle inequality meaning thatz?Br(x). Thus,Bs(y)?Br(x), which proves the result. 9quotesdbs_dbs14.pdfusesText_20

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