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legal research analysis

Questions and Answers of Legal Research Analysis

*8. (a) Prove that r1 cos(x2 + y2) dx 10 ..fi
converges uniformly on (-00,00).(b) Prove that fooo e-xy dx converges uniformly on [1,00).(c) Prove that foOOye-XYdx exists for each y E [0,00) and converges uniformly on any [a, b] C (0,00), but
1. For each of the following functions, prove that f is differentiable on its domain and compute D f.(a) f(x, y) = (sin x, xy, cos y). (b) f(s, t, u, v) = (st + u2, uv - s2).
2. Prove that I(x, y) = JiXYI is not differentiable at (0,0).
3. Prove that the following function is not differentiable at (0,0).° < II(x,y)11 < 7r(x,y) = (0,0)
4. Let r> 0, I: Br(O) --t R, and suppose that there exists an a > 1 such that II(z)1 ::; Ilzll'" for all z E Br(O).(a) Prove that 1 is differentiable at O.(b) What happens to this result when a = 1?
5. Prove that if a> 1/2, then I(x, y) = { ~xYI'" log(x2 + y2)is differentiable on R 2 •
6. Prove that is differentiable on R2 for all a < 3/2.
7. Prove that(x, y) =f (0,0)(x, y) = (0,0)(x, y) =f (0,0)(x, y) = (0,0)(x, y) =f (0,0)(x, y) = (0,0)is continuous on R 2 , has first-order partial derivatives everywhere on R 2 , but 1 is not
9. Let V be open in Rn, a E V, and 1 : V --t Rm.(a) Prove that Duf(a) exists for u = ek if and only if fXk (a) exists, in which case (b) Show that if f has directional derivatives at a in all
10. Let r > 0, (a,b) E R 2 , f : Br (a,b) -+ R, and suppose that the first-order partial derivatives fx and fy exist in Br(a,b) and are differentiable at (a, b).(a) Set ~(h) = f(a + h, b + h) - f(a +
2. Let V be open in Rn, let a E V, let f,g: V ---+ R3, and suppose that f and 9 are differentiable at a.(a) [CROSS-PRODUCT RULE] For the case n = 1, prove that fxg is differentiable at a and(f x
3. Prove (7) and (8) in Theorem 11.20.
4. [QUOTIENT RULE] Let f : Rn -> R be differentiable at a with f(a) f 0.(a) Show that for Ilhll sufficiently small, f(a+h) f 0.(b) Prove that D f(a)(h)/llhll is bounded for all h ERn \ {o}.(c) 1fT:=
5. For each of the following functions, find an equation of the tangent plane to z = f(x, y) at c.(a) f(x,y) = x3 siny, c= (0,0,0).(b) f(x,y) = x3y - xy3, c= (1,1,0).
7. Let 11. be the hyperboloid of one sheet, given by x2 + y2 - z2 = 1.(a) Prove that at every point (a,b, c) E 11.,11. has a tangent plane whose normal is given by (-a, -b, c).(b) Find an equation of
* 8. Compute the differential of the each of the following functions.(b) z = sin(xy).xy(c) z = 1 + x2 + y2'
*9. Let w = x2y + z. Use differentials to approximate Llw as (x, y, z) moves from(1,2,1) to (1.01,1.98,1.03). Compare your approximation with the actual value of Llw.
* 10. The time T it takes for a pendulum to complete one full swing is given by where 9 is the acceleration due to gravity and L is the length of the pendulum.If 9 can be measured with a maximum
*11. Suppose that 1 1 1 1- = - + - +-, w x y z where each variable x, y, z can be measured with a maximum error of p%.Prove that the calculated value of w also has a maximum error of p%.
2. Let r > 0, let a ERn, and suppose that 9 : Br(a) -+ Rm is differentiable at a.(a) If f : Br(g(a)) -+ R is differentiable at g(a), prove that the partial derivatives of h = fog are given by oh og
3. Let f,g: R -+ R be twice differentiable. Prove that u(x,y) := f(xy) satisfies au au xo-x -ya-y =0 , and v(x, y) := f(x - y) + g(x + y) satisfies the wave equation; i.e.,
4. Let u : R -+ [0,00) be differentiable. Prove that for each (x, y, z) -I- (0,0,0), F(x, y, z) := u( j x2 + y2 + z2)satisfies
5. Let t > 0, x E R.(a) Prove that u satisfies the heat equation; i.e., U xx - Ut = ° for all t > ° and x E R.(b) If a> 0, prove that u(x, t) -+ 0, as t -+ 0+, uniformly for x E [a, 00).
6. Suppose that I is a nonempty, open interval and f : 1-+ Rm is differentiable on I. If f(I) ~ aBr(O) for some fixed r > 0, prove that f(t) is orthogonal to f'(t) for all tEl.
7. Let V be open in Rn, a E V, f: V -+ R, and let f be differentiable at a.(a) Prove that the directional derivative Duf(a) exists (see Exercise 9, p. 338), for each u E Rn such that Ilull = 1, and
8. Let z = F(x, y) be differentiable at (a,b) with Fy(a,b) -I- 0, and let I be an open interval containinga. Prove that if f : 1-+ R is differentiable ata, f(a) = b, and F(x, f(x)) = ° for all x E
9. Letf, 9 : R2 ---t R be differentiable and satisfy the Cauchy-Riemann equations, i.e., that af = ag and af = _ ag ax ay ay ax .If u(r, B) = f(r cos B, r sin B), and v(r, B) = g(r cos B, r sin B),
10. Let f : R2 ---t R be C2 on R2 and set u(r, B) = f(r cos B, r sin B). If f satisfies the Laplace equation, i.e., if prove for each r # 0 that
11.5 MEAN VALUE THEOREM AND TAYLOR'S FORMULA Using D f as a replacement for f', we guess that the multidimensional analogue of the Mean Value Theorem is f(x) - f(a) = Df(c)(x-a)for some c "between" x
11.30 Remark. The function f(t) = (cost,sint) is differentiable on R and satisfies f(21r) = f(O), but there is no c E R such that Df(c) = (0,0).PROOF. D f(t) = (- sin t, cos t) exists and is
11.31 THEOREM [MEAN VALUE THEOREM ON Rnj. Let V be open in Rn and suppose that f : V ---t Rm is differentiable on V. If x,a E V and L(x;a) ~ V, then for each u E R m, there is acE L(x;a) such that
1. Let f : R n ---t R. Suppose that for each unit vector u ERn, the directional derivative Duf(a+tu) exists for t E [0,1] (see Definition 11.19). Prove that f(a+u) - f(a) = Duf(a+ tu)for some t E
2. Suppose that r, ex are positive numbers, E is a convex subset of Rn such that E c Br(O), and there exists a sequence Xk E E such that Xk ---t 0 as k ---t 00. If f: Br(O) ---t R continuously
3. (a) Write out an expression in powers of (x + 1) and (y - 1) for f(x, y)x2 + xy + y2.(b) Write Taylor's Formula for f(x, y) = JX + /Y, a = (1,4), and p = 3.(c) Write Taylor's Formula for f(x, y) =
4. Suppose that f: R2 ---t R is CP on Br(xo, Yo) for some r > O. Prove that given (x, y) E Br(xo, yo), there is a point (c,d) on the line segment between(xo, yo) and (x, y) such that f(x,y) =
5. Let r > 0,a, bE R, f : Br(a,b) ---t R be differentiable, and (x, y) E Br(a, b).(a) Compute the derivative of g(t) = f(tx + (1 - t)a, y) + f(a, ty + (1 - t)b).(b) Prove that there are numbers c
6. [INTEGRAL FORM OF TAYLOR'S FORMULA]. Let pEN, V be an open set in Rn , x,a E V, and f: V -+ R be CP on V. If L(x;a) c V and h =x-a, prove that p-1 1 1 11 f(x) - f(a) = "'" _D(k) f(a;h) + (1-
7. Suppose that V is open in Rn , f: V -+ R is C2 on V, and fxJa) = ° for some a E V and all j = 1, ... , n. Prove that if H is a compact convex subset of V, then there is a constant M such that for
8. Suppose that V is an open subset of R2, (a,b) E V, and f : V -+ R is C3 on V. Prove that 4 127f lim -2 f(a + rcose, b + r sine) cos(2e) de = fxx(a,b) - fyy(a, b).r-+O trr 0
9. Suppose that V is an open subset of R2, H = [a, b] x [0, c] C V, u: V -+ R is C2 on V, and u(xo, to) ~ ° for all (xo, to) E 8H.(a) Show that given E > 0, there is a compact set K c HO such that
1. For each of the following functions, prove thatf- 1 exists and is differentiable in some nonempty, open set containing (a, b), and compute D(f-1 )(a, b)(a) f(u, v) = (3u - v, 2u + 5v) at (a,
2. For each of the following functions, find out whether the given expression can be solved for z in a nonempty, open set V containing (0,0,0). Is the solution differentiable near (O,O)?(a) xyz +
3. Prove that there exist functions u(x, y), v(x, y), and w(x, y), and an r > 0 such that u, v, ware continuously differentiable and satisfy the equations u5 + xv2 - Y + w = 0 v5 + yu2 - X + w = 0 w4
4. Find conditions on a point (xo, Yo, un, va) such that there exist real-valued functions u(x, y) and v(x, y) that are continuously differentiable near(xo, Yo) and satisfy the simultaneous equations
5. Given nonzero numbers xo, Yo, un, va, So, to that satisfy the simultaneous equations(*) u2 + sx + ty = 0 v2 + tx + sy = 0 2s2x + 2t2y - 1 = 0 S2X - t2y = 0, prove that there exist functions u(x,
6. Let E = {(x, y) : 0 < y < x} and set f (x, y) = (x + y, xy) for (x, y) E E.(a) Prove that f is 1-1 from E onto {(s, t) : s > 20, t > O} and find a formula for f-1(S,t).(b) Use the Inverse Function
7. Suppose that f : R2 -+ R2 has continuous first-order partial derivatives in some ball Br(xo, Yo), r > o. Prove that if 6. j (xo, Yo) i= 0, then 8f11 (f( )) = 8h/8y(xo, Yo)uJ;lX xo, Yo tA.. ..J(.
[!J. This exercise is used in Section ell. 7. Let F : R3 --+ R be continuously differentiable at (a,b, c) with 'V F(a,b, c) :f:. O.(a) Prove that the graph of the relation F(x, y, z) = 0; Le., the
9. Suppose that f := (u, v) : R --+ R2 is C2 and (xo, Yo) = f(to).(a) Prove that if 'Vf(to):f:. 0, then u'(to) and v'(to) cannot both be zero.(a) If 'V f(to) :f:. 0, show that either there is a C1
1. Find all local extrema of each of the following functions.(a) f(x, y) = x2 - xy + y3 - y.(b) f(x, y) = sinx + cosy.(c) f(x, y, z) = eX+Y cos z.(d) f(x, y) = ax2 + bxy + cy2, where a f 0 and b2 -
2. For each of the following, find the maximum and minimum of f on H.(a) f (x, y) = x2 + 2x - y2 and H = {(x, y) : x2 + 4y2 :::; 4}.379(b) f (x, y) = x2 + 2xy + 3y2, and H is the region bounded by
3. For each of the following, use Lagrange multipliers to find all extrema of f subject to the given constraints.(a) f(x, y) = x + y2 and x2 + y2 = 4.(b) f(x, y) = x2 - 4xy + 4y2 and x2 + y2 = l.(c)
4. Let f : Rn ---+ Rm be differentiable ata, and 9 : Rm ---+ R be differentiable at b = f(a). Prove that if g(b) is a local extremum of g, then '\1(g 0 f)(a) = O.
5. Let V be open in R 2, (a,b) E V, and f : V ---+ R have second-order partial derivatives on V with fx(a,b) = fy(a,b) = O. If the second-order partial derivatives of f are continuous at (a,b) and
6. Let V be an open set in Rn, a E V, and f : V ---+ R be C2 on V. If f(a) is a local minimum off, prove that D(2) f(a) (h) 2: 0 for all h ERn.
7. Leta, b,c, D, E be real numbers with c =I- O.(a) If DE > 0, find all extrema of ax + by + cz subject to the constraint z =Dx2 + Ey2. Prove that a maximum occurs when cD < 0 and a minimum when cD >
8. [IMPLICIT METHOD].(a) Suppose thatf, g : R3 - R are differentiable at a point (a,b, c), and f(a,b, c)is an extremum of f subject to the constraint g(x, y, z) = k, where k is a constant. Prove that
1. Using Definition 3.1, prove that each of the following limits exists.(a) lim x2 - x + 1 = 3.(b)(c)x2 -1 lim --=2.x_I x - 1 lim x3 + x + 1 = 3.x-I
2. Decide which of the following limits exist and which do not.Prove that your answer is correct.(a)(b)(c). 1 hm cos-.x-o X 1lim x sin -.x-o x 1. 1 1m--.x-I log x
3. Evaluate the following limits using results from this section. You may assume that sinx, 1 - cos x, and ?Ix converge to 0 as x -+ O.J(a)(b)(c)(d)(e)1. X2 + cos x 1m . x-o 2 - tan x 1. x2 + X - 2
5. Prove Theorem 3.9.
6. Prove Theorem 3.10.1. n· 1 1m x Slllx-->O X
[1]. This exercise is used in Sections 3.2 and 5.2. For each real function f define the positive part of f by f+(x) = If(x)l; f(x), x E Dom(f)and the negative part of f by r(x) = If(x)l; f(x), x E
(a) Prove that f+(x) ~ 0, f-(x) ~ 0, f(x) = f+(x) - f-(x), and If(x)1 =f+(x) + f-(x) hold for all x E Dom (f). (Compare with Exercise 1, p. 11.)(b) Prove that if L = lim f(x)x-->a exists, then f+ (x)
8. Suppose that f is a real function.(a) Prove that if L = lim f(x)x-->a exists, then If(x)1 ---; ILl as x ---; a.
(b) Show that there is a function such that as x ---;a, If(x)1 ---; ILl but the limit of f(x) does not exist.[!]. This exercise is used in Sections 3.2 and 5.2.
Letf, 9 be real functions, and for each x E Dom (f) n Dom (g) define(f V g)(x) := max{f(x), g(x)} and (f 1\ g)(x) := min{f(x), g(x)}.(a) Prove that and(f V g)(x) = (f + g)(x) + l(f - g)(x)1 2(f 1\
(i) Prove that f(x) = {x+l x-I x~o x
(ii) Prove that lim Vi = o. x--+o+
1. Using definitions (rather than limit theorems) Prove that lim f(x)x--+a+exists and equals L in each of the following cases.(a) f(x) = lxi/x, a = 0, and L = 1.(b) f(x) = -l/x, a = 0, and L =
2. Evaluate the following limits when they exist,(a)(b)(c)(d)(e)I, x + 1 1m 2 'x-+O+ X - 2x I, X3 - 3x + 2 1m 3 x-+l- X - 1 lim (x2 + 1) sinx, X-+7f+I, x 1m -, x-+O+ Ixl tanx lim x-+7f/2- X
3. Evaluate the following limits when they exist,(a)(b)(c)(d)I, 3X2 - 13x + 4 1m 2 'x-+oo 1 - x - x I, x2 + X + 2 1m 3 'x-+oo X - X - 2 I, x3 - 1 1m --, x-+-oo x2 + 2 lim arctan x, x-+oo
[You may assume that tan x ---. L as x ---.a, x E (-7f/2,7f/2), if and only if arctan x ---. a as x ---. L,](e)sinx lim -2-'x-+oo[!]. This exercise is used in many places. Recall that a polynomial of
(a) Prove that limx-->a xn = an for n = 0,1, ....
(b) Prove that if P is a polynomial, then lim P(x) = P(a)x-->a for every a E R.
(c) Suppose that P is a polynomial and P(a) > O. Prove that P(x)/(x-a) ----; 00 as x ----; a+, P(x)/(x -a) ----; -00 as x ----; a-, but does not exist.1. P(x)lm-x-->a X - a
5. Prove that (sin(x + 3) - sin3)/x converges to 0 as x ----; 00.
6. Prove that y'I- cosx/sinx ----; V2/2 as x ----; 0+.
7. Prove the following comparison theorems for real functions.(a) If f(x) ~ g(x) and g(x) ----; 00 as x ----;a, then f(x) ----; 00 as x ----; a.(b) If f(x) :::; g(x) :::; h(x) and L:= lim f(x) = lim
8. Suppose that f: [a, 00) ----; R for some a E R. Prove that f(x) ----; L as x ----; 00 if and only if f (xn ) ----; L for any sequence Xn E (a, 00) that converges to 00 as n ----; 00.
9. Suppose that f : [0,1]----; Rand f(a) = limx-->a f(x) for all a E [0,1]. Prove that that f(q) = 0 for all q E Q n [0,1] if and only if f(x) = 0 for all x E [0,1].
10. [CAUCHY] Suppose that f : N ----; R. If lim f(n + 1) - f(n) = L, n-->oo prove that limn-->oo f(n)/n exists and equals L.
Prove that the function X = !!. E Q (in reduced form)q x~ Q.is continuous at every irrational in the interval (0, 1) but discontinuous at every rational in (0,1).
1. For each of the following, prove that there is at least one x E R that satisfies the given equation.(a) ex =x2.(b) eX = cos x + 1.(c) 2x = 2 - x.
2. Use limit theorems to show that the following functions are continuous on [0,1].(a) 2 f(x) = xeX + 5.(b) f(x) = 1 - x.l+x(c) f(x} ~ {:XSin~ x=t'=O x = O.(d) f(x) = Jf=X.
3. If f : [a, b] ---+ R is continuous, prove that SUPxE[a,b]lf(x)1 is finite.
4. Suppose that f is a real-valued function of a real variable. If f is continuous at a with f (a) < M for some MER, prove that there is an open interval I containing a such that f (x) < M for all x
6. Let xi=O x = O.(a) Prove that f is continuous on (0,00) and (-00,0) but discontinuous at O.(b) Suppose that g : [0,2/7f] ---+ R is continuous on (0, 2/7f) and that there is a positive constant C >
7. Suppose that a E R, that I is an open interval containinga, that f,g : 1---+ R, and that f is continuous at a.(a) Prove that g is continuous at a if and only if f + g is continuous at a.(b) Make
8. Suppose that f : R ---+ R satisfies f(x + y) = f(x) + f(y) for each x, y E R.(a) Show that f(nx) = nf(x) for all x E Rand n E Z.(b) Prove that f(qx) = qf(x) for all x E Rand q E Q.(c) Prove that f
Suppose that f : R ---+ (0,00) satisfies f(x + y) = f(x)f(y). Modifying the outline in Exercise 8, show that if f is continuous at 0, then there is an a E (0,00)such that f(x) = aX for all x E R.
10. If f : R ---+ R is continuous and lim f(x) = lim f(x) = 00, x--+oo x--+-oo prove that f has a minimum on R; i.e., there is an Xm E R such that f(xm ) = inf f(x) < 00.
1. Using Definition 3.35, prove that each of the following functions is uniformly continuous on (0,1).(a) f(x) = x3 .(b) f(x) = x2 - x.(c) f(x) = x sin 2x.
2. Prove that each of the following functions is uniformly continuous on (0,1).(a)(b)(c)(d)(e)x3 -1 f(x) = --1' x-. 1 f(x) = xsm-.x f(x) is any polynomial.f(x) = sin x .x
3. Find all real a such that xC< sin(l/x) is uniformly continuous on the open interval(0,1).
4. (a) Suppose that f: [0,00) -t R is continuous and there is an L E R such that f(x) -t L as x -t 00. Prove that f is uniformly continuous on [0,00).(b) Prove that f (x) = 1/ (x2 + 1) is uniformly
5. (a) Let I be a bounded interval. Prove that if f : I -t R is uniformly continuous on I, then f is bounded on I.(b) Prove that (a) may be false if I is unbounded or if f is merely continuous
6. Suppose that 0: E R, E is a nonempty subset of R, and j, g : E ~ Rare uniformly continuous on E.(a) Prove that j + g and o:j are uniformly continuous on E.(b) Suppose that j, g are bounded on E.
7. Let E

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