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1. (15 points) Let A be the unit ball B1(0) in R³. Compute

1

^cos(x

⁺

^y

⁺

z) dxdydz.

2016. 10.21

2. Let

f

be a real-valued function on R which has period 21!' and is Riemann integrable on [-1r, 1!']. We define its Fourier coefficients

11w 11w

an : = - f(x) cos(nx)dx (n = 0, 1, 2, ... ) and bn : = - f(x) sin(nx)dx (n = 1, 2, ... ).

1T' -w 1l' -w

(1) (10 points) Show that f(x)²is Riemann integrable on [-1!', 1!'].

00

(2) (15 points) Show that the series ~a6

+ L

(a~+ b~) converges.

n=1

3. (1) (15 points) Let {an}~=1 ^and^{bn}~=1 ^be^sequencesof real numbers and let En = b1

+ · · · +

bn (n E N). Suppose that an ~ 0 as n --t oo and that there exists

00

.M

>

0 such that IBnl ~ M for every n EN. Show that the series

L

^anbnconverges.

n=1 (2) (10 points) Show that the function series

f (-

¹^)n-¹^sin(nx)

n=1 n converges uniformly on [-K, K] if IKI

<

1!'.

4. Definition. Let F be a set of real-valued functions on a set X. F is uniformly bounded if there exists M

>

0 such that If ( x) I ~ M for every x E X and every f E F.

(20 points) Let {Fn}~=1 ^be^asequence of convex functions on [-2, 2] and let fn = Fnl[-1,1] (n E N). Suppose that { Fn In E

N}

is uniformly bounded. Show that there exists a subsequence of {fn}~=1 which is uniformly convergent on [-1, 1].

5. (15 points) Let

f

= (JI, ... Jn) : U --t Rn be a C 1 map from an open set U in Rn, and let g : V --t U be a continuous map from an open set V in Rn. Suppose that

det (

~~:

⁽

x)) ^=I

^{0 for}^every

x

^E^U,

and that f(g(x)) = x for every x E V. Show that g is C1.

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