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legal research analysis

Questions and Answers of Legal Research Analysis

1. (a) Let a::; band c ::; d be real numbers. Sketch a graph of the rectangle[a, b] x [c, dj := {(x, y) : x E [a, b], y E [c, dj}and decide whether this set is compact or connected. Explain your
2. (a) Sketch a graph of the set{(x, y) : x2 + 2y2 < 6, y ~ O}and decide whether this set is relatively open or relatively closed in the subspace {(x, y) : y ~ O}. Do the same for the subspace {(x,
3. Prove that the intersection of connected sets in R is connected. Show that this is false if "R" is replaced by "R 2 ."
5. Suppose that E c X is connected and E ~ A ~ E. Prove that A is connected.
6. Suppose that {Eoj"'EA is a collection of connected sets in a metric space X such that n"'EAE", #- 0. Prove that E= U E","'EA is connected.
[lJ. This exercise is used in Section 10.6. Let H ~ X. 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 subcover.
1 *10 I. This exercise is used to prove *Corollary 11.29.(a) A set E ~ Rn is said to be polygonally connected if and only if any two points a,b E E can be connected by a polygonal path in E; i.e.,
1. Let f(x) = sin X and g(x) = x/lxl if x =f 0 and g(O) = O.(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 appealing to results
2. Let f(x) = yX and g(x) = l/x if x =f 0 and g(O) = O.(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 results in this section.(b)
3. Let X be a metric space and f : X -t Y. Prove that f is continuous if and only iff- 1(C) is closed in X for every set C closed in Y.
4. Suppose that E ~ X and f : E -t Y.(a) Let E ~ X and f : E -t Y. Prove that f is continuous on E if and only if f-l(A) n E is relatively closed in E for all closed sets A in Y.(b) Suppose that f is
5. [INTERMEDIATE VALUE THEOREM]. Let E be a connected subset of a metric space X. If f : E -t R is continuous, f(a) =f f(b) for somea, bEE, and y is a number that lies between f(a) and f(b), then
6. Let X be metric space, Y be a Euclidean space, and H be a nonempty compact subset of X.(a) Suppose that I : H -+ Y is continuous. Prove that IIIIIH := sup 111(x)lly xEH is finite and there exists
7. Suppose that E is a compact subset of a metric space X.(a) If I,g : E -+ Rn are uniformly continuous, prove that 1+ 9 and I· 9 are uniformly continuous. Did you need compactness for both
8. Let X and Y be metric spaces, E
9 : X -+ Y such that g(x) = I(x) for all xED.
1. (a) For m = 1,2,3, let gm be the grid on [0,1] x [0,1] generated by where j = 1,2. For each of the following sets, compute V(E; gm).(o:)E={(X,Y)E[O,l]x[O,l]:x=O or y=O}.({3) E = {(x,y) E [0,1] x
~. This exercise is used in Section e12.6. Let E eRn. The translation of E by an x ERn is the set x + E = {y ERn: y = x + z for some z E E}, and the dilation of E by a scalar ex > 0 is the set exE =
(a) Prove that E is a Jordan region if and only if x + E is a Jordan region, in which case Vol (x + E) = Vol (E).(b) Prove that E is a Jordan region if and only if exE is a Jordan region, in which
[!]. This exercise is used in Section e12.5. Suppose that El , E2 are Jordan regions in Rn.(a) Prove that if El
5. Let E be a Jordan region in Rn.(a) Prove that EO and E are Jordan regions.(b) Prove that Vol (EO) = Vol (E) = Vol (E).(c) Prove that Vol (E) > 0 if and only if EO -=1= 0.(d) Let f : [a, bJ ......
7. (a) Prove that the boundary of an open ball Br(a) is given by 8Br(a) = {x: Ilx-all = r}.(b) Prove that Br (a) is a Jordan region for all a E Rn and all r ~ O.
1. Using Exercise 1, p. 17, compute the upper and lower sums U(f, Ym), L(f, Ym)for mEN, where f(x, y) = xy and Ym is determined by for j = 1,2. Prove that
2. Let E be a Jordan region in Rn andf, 9 be integrable on E with Lf(X)dx= 5 and L9(X)dx= 2.(a) Find Lo (2f(x) - 3g(x)) dx, L (2f(x) - 3g(x)) dx, and h(2f (X) - 3g(x)) dx.(b) If h is integrable on E
3. Let Q := [0,1] x [0,1]' A := {(x, y) E Q : y ::; x}, B := {(x, y) E Q : y 2: x}, and let f be integrable on Q (hence, on A-see Exercise 4a) with JJA fdA = 4.(a) If JJQ fdA = 3, find JJB fdA, and
4. (a) Let El C E be Jordan regions in Rn. If f : E --+ R is integrable on E, prove that f is integrable on E1 .(b) If f is uniformly continuous on a Jordan region E, prove that f is integrable on
[!]. This exercise is used in Sections 12.4, 13.5, and 13.6.Let E be an open Jordan region in Rn and Xo E E. If I: E --+ R is integrable on E and continuous at Xo, prove that lim V< 1(; ( )) ( I(x)
6. (a) Suppose that E is a Jordan region in Rn and that Ik : E --+ R are integrable on E for kEN. If Ik --+ 1 uniformly on E as k --+ 00, prove that 1 is integrable on E and(b) Prove that lim ( A (x)
7. Let E be a Jordan region in Rn and let I, 9 : E --+ R be integrable on E.(a) Modifying the proof of Corollary 5.23, prove that Ig is integrable on E.(b) Prove that 1 V 9 and 11\ 9 are integrable
8. Let H be a closed, connected, nonempty Jordan region and let 1 : H --+ R be continuous on H.
(a) If 9 : H --+ R is integrable and nonnegative on H, prove that there is an Xo E H such that I(Xo) L g(x) d:i; = L I(x)g(x) d:i;.(b) If HO t- 0, prove that there is an open set V and a point Xo E V
9. Prove the following special case of Theorem 12.29i. If E is a closed nonempty Jordan region in Rn, Eo is a nonempty Jordan region of volume zero, and 1 : E --+ R is a bounded function that is
10. Suppose that V is open in Rn and 1 : V --+ R is continuous. Prove that if L I(x) d:i; = 0 for all nonempty Jordan regions E c V, then 1 = 0 on V.
11. Suppose that E is a Jordan region and 1 : E --+ R is integrable.(a) If I(E) ~ H, for some compact set H, and ¢ : K --+ R is continuous, prove that ¢ 0 1 is integrable on E.*(b) Show that part
1. Evaluate each of the following iterated integrals.(a) 1 1 1\x2+Y)dxdY. (b) 1111 Yxy+xdxdy. r/2 r/2(c) Jo Jo ycos(xy) dydx.
2. Evaluate each of the following iterated integrals. Write each as an integral over a region E, and sketch E in each case.[11x2+l(a) Jo x (x+y)dydx.(d) [lJl 11 ..jx3+zdzdxdy.Jo vy x 3
3. For each of the following, evaluate JE f(x) dx.(a) f(x,y) = xVfj and E is bounded by y = x and y = x2.(b) f(x, y) = x + y and E is the triangle with vertices (0,0), (0,1), (2,0).(c) f(x, y) = x
4. Compute the volume of each of the following regions.(a) E is bounded by the surfaces x + y + z = 3, z = 0, and x2 + y2 = 1.(b) E lies under the plane z = x + y and over the region in the xy plane
5. (a) Verify that the hypotheses of Fubini's Theorem hold when f is continuous on R.(b) Modify the proof of Remark 12.33 to show that Fubini's Theorem might not hold for a nonintegrablef, even if
6. (a) Suppose that !k is integrable on [ak, bk] for k = 1, ... , n, and set R = [all bl ] x... x [an, bn]. Prove that(b) If Q = [0,1]n and y:= (1,1, ... ,1), prove that
7. The greatest integer in a real number x is the integer [x] := n that satisfies n :::; x < n + 1. An interval [a, b] is called Z-asymmetric if b + a -=f. [b] + [a] + 1.(a) Suppose that R is a
8. Let E be a nonempty Jordan region in R2 and f : E ---- [0,00) be integrable on E. Prove that the volume of Q = {(x, y, z) : (x, y) E E, ° ~ z ~ f(x, y)}(as given by Definition 12.5) satisfies
9. Let R = [a, b] x [c, d] be a two-dimensional rectangle and f : R ---- R be bounded.(a) Prove that(L) fLfdA ~ (L) lb ((X) ld f(X,Y)dY) dx~ (U) lb ((X) ld f(x, y) dY) dx~ (U) fLfdA for X = U or X =
*10. [FUBINI'S THEOREM FOR IMPROPER INTEGRALS]. If a < b are extended real numbers, c < d are finite real numbers, f: (a,b) x [c, d] ---- R is continuous, and F(y) = lb f(x, y) dx converges uniformly
1. Evaluate each of the following integrals.(a)(b)(c) 0:::; a < b.
2. For each of the following, find ffE fdA.(a) f(x, y) = cos(3x2 + y2) and E is the set of points satisfying x2 + y2/3 :::; l.(b) f(x,y) = yylx - 2y and E is bounded by the triangle with vertices
3. For each of the following, find fffE f dV.(a) f(x, y, z) = z2 and E is the set of points satisfying x2 + y2 + Z2 :::; 6 and z 2: x2 + y2.(b) f(x, y, z) = eZ and E is the set of points satisfying
4. (a) Prove that the volume bounded by the ellipsoid is 47rabc/3.(b) Leta, b,c, d be positive numbers and r2 < d2/(b2 + c2). Find the volume of the region bounded by y2 + Z2 = r2, x = 0, and ax + by
5. (a) Compute ffE ylx - yylx + 2ydA, where E is the parallelogram with vertices(0,0), (2/3, -1/3), (1,0), (1/3,1/3).(b) Compute ffE {!2x2 - 5xy - 3y2 dA, where E is the parallelogram bounded by the
6. Suppose that V is nonempty and open in Rn and f : V ---+ R n is continuously differentiable with f). f =f. 0 on V. Prove that lim Vol (f(Br(xo))) = If). (Xo) I r-->O+ Vol (Br(xo)) f for every Xo E
7. Show that Vol is rotation invariant in R2; i.e., if ¢ is a rotation on R2 (see Exercise 9, p. 241) and E is a Jordan region in R2, then Vol (¢(E)) = Vol (E).
8. (a) Compute the Jacobian of the change of variables from spherical coordinates to rectangular coordinates.(b) Assuming that Vol is translation and rotation invariant (see Exercise 3, p.393, and
9. Let fJj = (Vj1,"" Vjn) ERn, j = 1, ... , n, be fixed. The parallelepiped determined by the vectors fJj is the set P(fJ1, ... ,fJn ) := {hfJ1 + ... + tnfJn : tj E [0, I]}, and the determinant of
[!!]. This exercise is used in Section e12.6.(a) Prove that the improper integral Jooo e-x2 dx converges to a finite real number.(b) Prove that if 1 is the value of the integral in part (a), then
1. Iff, 9 : Rn -+ R, prove that spt (f g) ~ spt f n spt g.
2. Prove that if f,g E Cg
*3. Prove that if f is analytic on Rand f(xo) =f. 0 for some Xo E R, then f ~Cg
4. Suppose that V is a bounded, open set in Rn, and ¢ : V -+ Rn is 1-1 and continuously differentiable on V with ~~ =f. 0 on V. Let W = {Wj }jEN be an open covering of V and {¢j }jEN be a CP
5. Let V be open in Rn and V = {Vj}jEN, W = {WdkEN be coverings of V. If{ ¢j} jEN is a cP partition of unity on V subordinate to V and {'l/Jk} kEN is a CP partition of unity on V subordinate to W,
6. Show that given any compact Jordan region H c Rn, there is a sequence of C=functions ¢j such that e12.6(I ----::::=d=x= = ,Jrr. 10 y'-logx 1: e7rt-e' dt = r(ll').
4. Show that the volume of a four-dimensional ball of radius r is 1l'2r 4/2, and the volume of a five-dimensional ball of radius r is 81l'2 r 5/15.
5. Verify (49).6. For n > 2, prove that the volume of the n-dimensional ellipsoid is 2 n/2 Vol(E)= al···an ll' .nr(n/2)
7. For n > 2, prove that the volume of the n-dimensional cone C = {(Xl,'" ,Xn ) : (h/r)Jx~ + ... + X~ ~ Xl ~ h}is
8. Find the value of for each kEN.2hrn - 11l'(n-I)/2 Vol (C) = n(n _ l)r((n - 1)/2)'
9. If 1 : BI (0) --+ R is differentiable with 1(0)=0 and 11V'I(x)ll~l for x E BI (0), prove that the following exists and equals O.lim ( II(xW dx.k--+oo 1 B1 (0)
10. (a) Prove that r is differentiable on (0,00) with r'(x) = 100 e-ttX-Ilogtdt.* (b) Prove that r is Coo and convex on (0, 00 ) .
1. Let 1/J(t) = (asint,acost), O"(t) = (acos2t,asin2t), 1= [0, 27r), and J = [0,7r).Sketch the traces of (1/J, I) and (0", J). Note the "direction of flight" and the"speed" of each parametrization.
2. Let a,b E Rm, b f 0, and set 1>(t) = a + tb. Show that G = 1>(R) is a smooth unbounded curve that contains a and a +b. Prove that the angle between 1>(tI) -1>(0) and 1>(t2) -1>(0) for any tI, t2 f
3. Let I be an interval and 1 : I -t R be continuously differentiable with I/(OW + 11'(0)12 f 0 for all 0 E I. Prove that the graph of r = 1(0) (in polar coordinates) is a smooth CI curve in R 2.
*4. Show that the curve y = sin(l/x), 0 < x :S 1, is not rectifiable. Thus show that Theorem 13.17 can be false if G is not an arc.
5. Sketch the trace and compute the arc length of each of the following.(a) 1>(t) = (et sint, et cost, et ), t E [0, 27r].(b) y3 = x2 from (-1,1) to (1,1).(c) 1>(t) = (t2, t2, t2), t E [0,2].(d) The
6. For each of the following, find a (piecewise) smooth parametrization of G and compute fegds.(a) G is the curve y = J9 - X2, x ~ 0, and g(x, y) = xy2.(b) G is the portion of the ellipse x2/a2 +
7. Let G be a smooth arc and gk : G ~ R be continuous for n E N.(a) If gk ~ 9 uniformly on G, prove that fe gk ds ~ fe gds as k ~ 00.* (b) Let {gd be pointwise monotone and let gk ~ 9 pointwise on G
If 9 is continuous on ¢(I), prove that fe gk ds ~ fe 9 ds as k ~ 00.S. Let (¢,1) be a parametrization of a smooth arc in Rm, and let T : J ~ R be a C1 function, 1-1 from J onto I. If T'(U) =f °
9. [FOLIUM OF DESCARTES]. Let G be the piecewise smooth curve ¢(h U I 2), where h = (-00, -1), h = (-1,00), and( 3t 3t2)¢(t)= l+t3 'I+t3 •Show that if (x, y) = ¢(t), then x3 + y3 = 3xy. Sketch G.
10. The absolute curvature of a smooth curve with parametrization ('!/J, I) at a point Xo = '!/J(to) is the number. B(t)I\;(xo) = hm O( ) , t--+to (. t when this limit exists, where B(t) is the angle
11. Let G be a smooth C2 arc with parametrization (¢, [a,b]), and let s = let) be given by (2). The natuml pammetrization of G is the pair (v, [0, L]), where v(s) = (¢ol-l)(S) and L = L(G).(a)
1. For each of the following, sketch the trace of (¢, R), describe its orientation, and verify that it is a subset of the surface S.(a) ¢(t) = (3t, 3 sin t, cos t), S = {(x, y, z) : y2 + 9z2 =
3. For each of the following, compute Ic w.(a) C is the polygonal path consisting of the line segment from (1,1) to (2,1)followed by the line segment from (2,1) to (2,3), and w = y dx + x dy.(b) C is
4. (a) Let c E R, 0> 0, and set T(U) = ou + c for U E R. Prove that if (¢, I) is a smooth parametrization of some curve, if J = T- 1 (1), and if 'if; = ¢ 0 T, then('if;, J) is orientation
5. Let (¢, I) be a smooth parametrization of some arc and T be a C1 function, 1-1 from J onto I, that satisfies T'(U) > 0 for all but finitely many u E J.If 'if; = ¢ 0 T, prove that 1 F(¢(t)) .
lliJ. This exercise is used in Section 13.5. Let f : [a, b] --+ R be C1 on [a, b]with f'(t) =f. 0 for t E [a,b]. Prove that the explicit curve x = f-1(y), as y runs from f(a) to f(b), is orientation
7. Let V =f. 0 be open in R2. A function F : V --+ R 2 is said to be conservative on V if and only if there is a function f : V --+ R such that F = '\l f on V. Let(x, y) E V and let F = (P, Q) : V
*8. Let f : [0, 1] ~ R be increasing and continuously differentiable on [0,1] and let T be the right triangle whose vertices are (0, f(O)), (1, f(O)), and (1, f(l)).If c represents the hypotenuse of
2. For each of the following, find a (piecewise) smooth parametrization of Sand of as, and compute I Is 9 dO".(a) S is the portion of the surface z = x2 - y2 that lies above the xy plane and between
3. Find a parametrization (cp, E) of the ellipsoid x2 y2 z2 2a +b2 +2c "=1 that is smooth off the topological boundary aE.
4. (a) Suppose that E is a two-dimensional region and S = {(x, y, z) E R3 : (x, y) E E and z = o}. Prove that Area (E) = lis dO"Let I: [a, b] -+ R be a CP function, let C be the curve in R2
5. Suppose that 'ljJ(B) and ¢(E) are CP surfaces, and 'ljJ = ¢ 0 T, where T is a C1 function from B onto Z.(a) If ('ljJ, B) and (¢, E) are smooth and T is 1-1 with ~r =I- 0 on B, prove that f Is
6. Let I: B3(0,0) -+ R be differentiable with IIV'/(x,y)ll:::; 1 for all (x,y) E B3(0,0). Prove that if 8 is the paraboloid 2z = x2 + y2, 0 :::; Z :::; 4, then f Is I/(x, y) - 1(0,0)1 da :::; 407r.
7. Let ¢(E) be a CP surface and (xo, Yo, zo) = ¢(uo, vo), where (uo, vo) E EO.If N (uo, vo) =I- 0, prove that ¢( E) has a tangent plane at (xo, Yo, zo).
8. Let 'ljJ(B) be a smooth surface. Set E = II'ljJull, F = 'ljJu . 'ljJv, and G = II'ljJvll. Prove that the surface area of 8 is IE vE2G2 - F2 d(u,v).
9. Suppose that 8 is a C1 surface with parametrization (¢, E) that is smooth at (xo, Yo, zo) = ¢(uo, vo). Let ('ljJ,1) be a parametrization of a C1 curve in E that passes through the point (uo,vo)
1. For each of the following, find a (piecewise) smooth parametrization of as that agrees with the induced orientation, and compute las F . T ds.(a) S is the truncated paraboloid y = 9 - x2 - z2, Y 2
2. For each of the following, compute lIs F . n der.(a) S is the truncated paraboloid z = x2 + y2, 0 ::; Z ::; 1, n is the outwardpointing normal, and F(x,y,z) = (x,y,z).(b) S is the truncated half
3. For each of the following, compute I Is w.(a) 8 is the portion of the surface z = x4 + y2 that lies over the unit square[0,1] x [0,1], with upward pointing normal, and w = x dydz+y dz dx+z dx
4. Suppose that 'Ij;(B) and ¢(E) are CP surfaces, and 'Ij; = ¢ 0 T, where T is a Cl function from B onto E.(a) If ('Ij;, B) and (¢, E) are smooth, and T is 1-1 with AT > 0 on B, prove for all
5. Let E be the solid tetrahedron bounded by x = 0, y = 0, z = 0, and x+y+z = 1, and suppose that its topological boundary, T = 8E, is oriented with outward pointing normal. Prove for all C1
6. Let T be the topological boundary of the tetrahedron in Exercise 5, with outward pointing normal, and 8 be the surface obtained by taking away the slanted face from T; i.e., 8 has three triangular
7. Suppose that 8 is a smooth surface.(a) Show that there exist smooth parametrizations (¢j, Ej ) of portions of 8 such that 8 = Uf=l¢j(Ej ).(b) Show that there exist nonoverlapping surfaces 8j
1. For each of the following, evaluate Ie F . T ds.(a) C is the topological boundary of the two-dimensional region in the first quadrant bounded by x = 0, y = 0, and y = J4"=X2, oriented in the

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