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legal research analysis

Questions and Answers of Legal Research Analysis

2. For each of the following, evaluate Ie w.(a) C is the topological boundary of the rectangle [a, b] x [c, d], oriented in the counterclockwise direction, and w = (f(x) + y) dx + xy dy, where f :
3. For each of the following, evaluate IIs F ·nda, where n is the outward-pointing normal.(a) S is the topological boundary of the rectangle [0,1] x [0,2] x [0,3] and F(x, y, z) = (x + eZ , y + eZ ,
4. For each of the following, find IIs w, where n is the outward-pointing normal.(a) S is the topological boundary of the three-dimensional region enclosed by y = x2, Z = 0, z = 1, y = 4, and w =
5. (a) Prove that if E is a Jordan region whose topological boundary is a piecewise smooth curve oriented in the counterclockwise direction, then Area (E) = ~ f xdy-ydx.2 JaE (b) Find the area
6. (a) Show that Green's Theorem does not hold if continuity of P, Q is relaxed at one point in E. (Hint: Consider P = y/(x2 + y2), Q = -x/(x2 + y2), and E=B1(0,0).)(b) Show that Gauss's Theorem does
0. This exercise is used in Section 13.6. Suppose that V#-0 is an open set in R3 and F : V --+ R3 is C1 . Prove that divF(xo) = lim V; 1(; ( )) Jef F·ndO' r~O+ 0 r Xo JaBr(xo)for each Xo E V, where
8. Let F, G : R3 --+ R3 and f : R3 --+ R be differentiable. Prove the following analogues of the Sum and Product Rules for the "derivatives" curl and divergence.(a)(b)(c)(d)(e)\7 x (F + G) = (\7 x F)
[ill. This exercise is used in Section 13.6. Let E c R 3 . Recall that the gradient of a C1 function f : E --+ R is defined by grad f := \7 f := (fx, fy, fz).(a) Prove that if f is C2 at Xo, then
10. Let E be a set in Rm. For each u : E --+ R that has second-order partial derivatives on E, Laplace's equation is defined by m {)2U~u:= L () 2· x·j=l J(a) Show that if u is C2 on E, then ~u =
2. For each of the following, evaluate I Is curl F . nda.(a) 8 is the "bottomless" surface in the upper half space z 2 ° bounded by y = x2, Z = 1 - y, n is the outward-pointing normal, and F(x,y,z)
3. For each of the following, evaluate I Is F ·nda using Stokes's Theorem or Gauss's Theorem.(a) S is the sphere x2 + y2 + z2 = 1, n is the outward-pointing normal, and F(x,y,z) = (xz2,x2y-
4. For each of the following, evaluate Is w using Stokes's Theorem or Gauss's Theorem.(a) S is topological boundary of cylindrical solid y2 + z2 ~ 9, ° ~ x ~ 2, with outward-pointing normal, and w =
6. Let fl be a plane in R3 with unit normal nand Xo E fl. For each r > 0, let Sr be the disk in fl centered at Xo of radius rj i.e., Sr = Br(xo) n fl. Prove that if F : Bl (Xo) --+ R is C1 and aSr
7. Let S be an orient able surface with unit normal nand nonempty boundary as that satisfies the hypotheses of Stokes's Theorem.(a) Suppose that F : S --+ R3 \ {O} is C1, that as is smooth, and that
8. Let E be a two-dimensional region such that if (x, y) E E, then the line segments from (0,0) to (x,O) and from (x,O) to (x, y) are both subsets of E. If F: E -4 R2 is C1, prove that the following
9. Let 0. be a three-dimensional region and F : 0. -4 R3 be C1 on n. Suppose further that for each (x,y,z) En, both the line segments L((x,y,O); (x,y,z))and L((x, 0, 0); (x, y, 0)) are subsets of n.
10. Suppose that E satisfies the hypotheses of Gauss's Theorem and S satisfies the hypotheses of Stokes's Theorem.(a) If!: S -4 R is a C2 function and F = grad! on S, prove that I las (iF)· Tds =
11. Let F be Cl and exact on R2 \ {(O,O)} (see Exercise 8b).(a) Suppose that C1 and C2 are disjoint smooth simple curves, oriented in the counterclockwise direction, and E is a two-dimensional region
1. Using Definition 9.li, prove that the following limits exist.(a) Zk = (~, 1 - :2 ) .(b) ( k . 1) Zk = k + 1 ' sm k .(c) Zk = (log(k + 1) -log k, 2-k ) •
2. Using limit theorems, find the limit of each of the following vector sequences.(a)(b) Xk = (1,Sin1rk,Cos~).(c) Zk = (k- Jk2+k,kl/k,~).
3. If Zk ----+ 0 in R n as k ----+ 00 and Yk is bounded in Rn , prove that Zk . Yk ----+ 0 as k ----+ 00.
4. Find convergent subsequences of( k 13k) Zk = ( -1) 'k' ( -1 )which converge to different limits. Prove your limits exist.
5. (a) Prove Theorem 9.4i and ii.(b) Prove Theorem 9.4iii and iv.(c) Prove Theorem 9.4v.
6. Prove Theorem 9.6.
7. Let E be closed and bounded in R, and suppose that for each x E E there is a nonnegative Coo function Ix such that Ix(x) > 0 and I~(y) = 0 for y ~ E. Prove that there is a nonnegative Coo function
9. Let E be a nonempty subset of Rn.(a) Show that a sequence x'" E E converges to some point a E E if and only if for every set U, which is relatively open in E and containsa, there is an N E N such
10. (a) Let E be a subset of Rn. A point a E Rn is called a cluster point of E if En Br(a) contains infinitely many points for every r > O. Prove that a is a cluster point of E if and only if for
(i) Find lim (3xy + 1, eY + 2).(x,y)-+(O,O)
(ii) Prove that the function f(x, y) = 1 + 2X2 + 3y2
1. For each of the following functions, find the maximal domain off, prove that the limit of f exists as (x, y) --4 (a, b), and find the value of that limit. (Note:You can prove that the limit exists
2. Compute the iterated limits at (0,0) of each of the following functions. Determine which of these functions has a limit as (x, y) --+ (0,0) in R2, and prove that the limit exists.(a) f( ) =
3. Prove that each of the following functions has a limit as (x,y) --+ (0,0).x3 _ y3(a) f(x'Y)=x2+y2' (x,y)-I=(O,O).(b)IxlQy4 f (x, y) = x2 + y4 'where 0: is ANY positive number.(x, y) -1= (0,0),
4. A polynomial on R n is a function of the form Nl Nn P(XI,X2, ... ,Xn ) = L ... L ajl, ... ,jnx{l ... x~n, iJ=O jn=O where aj" ... ,jn are scalars and NI' ... ' Nn are nonnegative integers. Prove
5. Prove Theorem 9.14i.
6. Prove Theorem 9.14ii.
7. Prove Theorem 9.14iii.
8. Prove Theorem 9.14iv.
1. Define f and 9 on R by f(x) = sin x and g(x) = x/lxl if x =f ° and g(O) = 0.(a) Find f(E) and g(E) for E = (0,11"), E = [0,11"], E = (-1,1), and E = [-1,1]'and explain some of your answers by
2. Define f on [0,00) and 9 on R by f(x) = Vx and g(x) = l/x if x =f ° and g(O) = 0.(a) Find f(E) and g(E) for E = (0,1), E = [0,1), and E = [0,1], and explain some of your answers by appealing to
3. Let A c Rn, let BeRm, let a E A, and let I : A \ {a} ~ B.(a) Suppose that A is open and b:= lim", ____ I(x) exists. If 9 is continuous at b, prove that lim go I(x) = g(b)."'----(b) If I is
4. Prove that is continuous on R 2 •{e-l/Ix-YI I(x,y) = 0 x#y x=y
5. Let B be a closed in Rn and I : B ~ Rm. Prove that the following are equivalent:(a) I is continuous on B.(b) 1-1 (E) is closed in Rn for every closed subset E of Rm.
6. Suppose that E ~ Rn and I : E ~ Rm.(a) Prove that I is continuous on E if and only if I-I (V) is relatively open in E for every open set V in R m.(b) Prove that I is continuous on E if and only if
is finite and there exists anxo E H such that 111(xo)11 = IIIIIH.(b) A sequence of functions Ik : H ~ Rm is said to converge uniformly on H to a function I : H ~ R m if and only if for every c > 0
8. Let n, mEN, E eRn, and suppose that D is dense in E; i.e., suppose that DeE and D = E. If f : D -+ Rm is uniformly continuous on D, prove that f has a continuous extension to E; i.e., prove that
9. [INTERMEDIATE VALUE THEOREM]. Let E be a connected subset of Rn. If f : E -+ R is continuous, f(a) 1- f(b) for some a,b E E, and y is a number that lies between f(a) and f(b), then prove that
1 *10 I. This exercise is used to prove *Corollary 11.35.(a) A set E
1. Identify which of the following sets are compact and which are not. If E is not compact, find the smallest compact set H (if there is one) such that E C H.(a) {l/k: kEN} U {a}.(b) {(x, y) E R2 : a
2. Let A, B be compact subsets of Rn. Prove that Au B and An B are compact.
3. Suppose that E ~ R is compact and nonempty. Prove that supE,inf E E E.
4. Let {V"}"EA be a collection of nonempty open sets in Rn that satisfies V" n V.a = 0 for all a -:f. f3 in A. Prove that A is countable. What happens to this result when "open" is omitted?
5. Prove that if V is open in Rn, then there are open balls BI, B2 , ... such that V= U Bj .jEN Prove that every open set in R is a countable union of open intervals.
6. Let n E N.(a) A subset E of Rn is said to be sequentially compact if and only if every sequence Xk in E has a convergent subsequence Xkj whose limit belongs to E. Prove that every compact set is
7. Let H ~ Rn.(a) Prove that H is compact if and only if every cover {E"}"EA of H, where the E,,'s are relatively open in H, has a finite sub covering.(b) Use part (a), Exercise 6a, p. 276, and
1. Suppose that Ik : [a, bj -+ [0,00) for kEN and 00 f(x) := L fk(X)k=1 converges pointwise on [a, bj. If f and Ik are continuous on [a, bj for each kEN, prove that
2. Let E be closed and bounded in Rn and let g, fk,gk : E -+ R be continuous 287 on E with gk 2: 0 and h 2: h··· 2: Ik 2: 0 for kEN. If 9 = 2:~1 gk converges pointwise on E, prove that 2:%"=1 Ikgk
3. Suppose thatf, fk : R -+ [0,00) are continuous. Prove that if f(x) -+ 0 as x -+ ±oo and Ik i f everywhere on R, then fk -+ f uniformly on R.
4. For each of the following functions, find a formula for wf(t).f(x) = { ~ x E Q x ~ Q.(a)(b) f(x) = { ~ x2:0 x < O.) {Sin(l/X) x#O f(x =0 x = O.(c)
6. Show that if f : [a, bj -+ R is integrable and 9 : f([a, b]) -+ R is continuous, then go f is integrable on [a, bj. (Notice by Remark 3.34 that this result is false if 9 is allowed even one point
7. Using Theorem 7.10 or Theorem 9.30, prove that each of the following limits exists. Find a value for the limit in each case.(a) lim ln/2 sin x ~k k3 dx.k--+oo 0 4 - x(b) lim f1 x2 f (-k/ )
8. (a) Prove that for every E > ° there is a sequence of open intervals {hhEN that covers [0,1] n Q such that(b) Prove that if {h hEN is a sequence of open intervals that covers [0, 1], then there
9. Let E1 be the unit interval [0,1] with its middle third (1/3,2/3) removed; i.e., E1 = [0,1/3] U [2/3,1]. Let E2 be E1 with its middle thirds removed; i.e., E2 = [0,1/9] U [2/9,1/3] U [2/3,7/9] U
Assume that every point x E [0,1] has a binary expansion and a ternary expansion; i.e., there exist ak E {O, I} and bk E {O, 1, 2} such that(For example, if x = 1/3, then a2k-1 = 0, a2k = 1 for all k
Prove that there is a countable subset Eo of E such that f is 1-1 from E \ Eo onto [0,1]; i.e., prove that E is uncountable.(d) Extend f from E to [0,1] by making f constant on the middle thirds Ek -
1. Show that 00 n 2: k X X ---I-x k=n for Ixl < 1 and n = 0, 1, ....
2. Prove that each of the following series converges and find its value.00 (_I)k+l(a) 2: 'Irk k=l 00 3k 00(c ) "~ 7k-l. (d)~"2 ke- k •k=l k=O
3. Represent each of the following series as a telescopic series and find its value.00 1(a) 2: k(k + 1) .k=l 00 (k(k+2))(b) ~log (k+l)2 .(c) ~ t{f (1 - (~yk) , where jk = -1/(k(k + 1)) for kEN.
4. Find all x E R for which 00 2: 3(xk _Xk-1)(Xk +Xk- 1)k=l converges. For each such x, find the value of this series.
5. Prove that each of the following series diverges.00 1(a) 2: cos k2 ·k=l 00 ( l)k (b) {; 1- k 00 k + 1(c) 2:k2.k=l
6. (a) Prove that if I:~=l ak converges, then its partial sums Sn are bounded.(b) Show that the converse of part (a) is false. Namely, show that a series I:~l ak may have bounded partial sums and
7. Let {bk} be a real sequence and bE R.(a) Suppose that there is an N E N such that I b - bk I :::; M for all k 2: N. Prove that n N nb - 2: bk :::; 2: Ibk - bl + M(n - N)k=l k=l for all n > N.(b)
8. A series 2::'0 ak is said to be Cesaro summable to an L E R if and only if n-1 ( k) an := 2: 1 -:;;: ak k=O converges to L as n ~ 00.(a) Let Sn = 2:~:~ ak. Prove that nfor each n EN.(b) Prove that
9. (a) Suppose that {ad is a decreasing sequence of real numbers. Prove that if 2::'1 ak converges, then kak ~ 0 as k ~ 00.(b) Let Sn = 2:~=1 (_1)k+1 /k for n E N. Prove that S2n is strictly
10. Suppose that ak 2: 0 for k large and 2::'1 ak/k converges. Prove that 00.lim 2: .ak k =0.
1. Prove that each of the following series converges.00 k - 3(a) L k3 + k + 1·k=l(c) f lo;k, p> l.k=l 00 1 00 ( 1) (d) L.jk k-l· (e) L 10 + k k-e .k=l k 3 k=l
2. Prove that each of the following series diverges.00!fk(a) LT.k=l 00 1(b) {; 10gP(k + 1)' p> o.00 k2 + 2k + 3(c) L k3 _ 2k2 + J2.k=l 00 1(d) L k 10 P k ' P ~ l.k=2 g
3. Find all p 2: 0 such that the following series converges.00 1 L klogP(k + 1)· k=l
4. If ak 2: 0 is a bounded sequence, prove that 00'"' ak~ (k + l)p k=l converges for all p > 1.
5. Suppose that ak E [0, 1) and ak ---> 0 as k ---> 00. Prove that E%"=l arcsin ak converges if and only if E~l ak converges.
6. If E%"=l lak I converges, prove that~~~ kp k=l converges for all p 2: O. What happens if p < O?
7. Suppose that ak and bk are nonnegative for all kEN.(a) Prove that if E~l ak and E%"=l bk converge, then E%"=l akbk also converges.(b) Improve this result by replacing convergence of one of the
8. Prove Theorem 6.16ii and iii.
9. Suppose thata, bE R satisfy bja E R \ Z. Find all q > 0 such that 00 1 L (ak + b)qk k=l converges.
10. Suppose that ak -+ O. Prove that 2::~1 ak converges if and only if the series 2::;:'=1 (a2k + a2k+1) converges.
2. Decide, using results covered so far in this chapter, which of the following series converge and which diverge.00 k2 00 k'(a) Lk· (b) L 2~· (c) {00; (k2k+ +1 3 r (d) ~ (IT -~) k- 1.k=l IT k=l 00
3. Using Exercise 9, p. 135, prove that• 00 (_1)k x2k+1 smx = L (2k I)!k=O +00 (_1)kx2k and cos x = L (2k)!k=O for all x E [0, IT /2J.
4. Define ak recursively by a1 = 1 and k>1.Prove that 2:%"=1 ak converges absolutely.
5. Suppose that ak ~ 0 and a~/k ---7 a as k ---7 00. Prove that 2:~1 akxk converges absolutely for all Ixl < l/a if a =I- 0 and for all x E R if a = O.
6. For each of the following, find all values of pER for which the given series converges absolutely.00 1(a) ".,......,......;::-:~klogP k·k=2 00 1(d) {; Vk(kP - 1).00 1(b) L 10 P k·k=2 g
7. Suppose that akj ~ 0 for k,j EN. Set
8. (a) Suppose that L%:l ak converges absolutely. Prove that L%"=l lak IP converges for all p :::: l.(b) Suppose that L%:l ak converges conditionally. Prove that L%:l kPak diverges for all p > l.
9. (a) Let an > 0 for n EN. Set b1 = 0, b2 = log(a2/ar), and k = 3,4, ....Prove that if 1. an r= 1m--exists and is positive, then n (k-1) 00 lim log(a;(n) = lim L 1--- bk = Lbk = logr.n_oo n-oo n k=l
(b) Prove that if an E R \ {O} and lan+danl -+ r as n -+ 00, for some r > 0, then lanl1/ n -+ r as n -+ 00.*10. Let x ::; y be any pair of extended real numbers. Prove that if L%"=l ak is
1. Prove that each of the following series converges.00(a) L( _l)k (i - arctank) .k=l(c) f(~;)k, p>O.k=l(d) f sin~:x) x E R, p> O. (e) f _( _ l_)k _2 ·_4_· '.,-' (-,-2k--,-)--:k=l k2 1·3 ..
2. For each of the following, find all values x E R for which the given series converges.00 k(a) L xk .00 3k(b) L ~k .k=l k=l
3. Using any test covered in this chapter, find out which of the following series converge absolutely, which converge conditionally, and which diverge.Loo (-1)(-3) .. · (1 - 2k)(b) 1· 4 .. · (3k
4. [ABEL'S TEST] Suppose that 2:~1 ak converges and bk 1 b as k ---- 00. Prove that 2:~=1 akbk converges.
* 5. Prove that 00 L ak cos(kx)k=l converges for every x E (0,21l') and every ak 10. What happens when x = O?
*6. Suppose that ak 1 0 as k ---- 00. Prove that 00 L ak sin((2k + l)x)k=l converges for all x E R.
7. Show that under the hypotheses of Dirichlet's Test, 00 00 L akbk = L sk(bk - bk+1).k=l k=l

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