4/18/2006 SECOND HOURLY FIRST PRACTICE Math 21a, Spring 2006Name:MWF 10 Samik BasuMWF 10 Joachim KriegerMWF 11 Matt LeingangMWF 11 Veronique GodinTTH 10 Oliver KnillTTH 115 Thomas Lam• Start by printing your name in the above box and checkyour section in the box to the left.• Do not detach pages from this exam packet or unstaplethe packet.• Please write neatly. Answers which are illegible for thegrader can not be given credit.• No notes, books, calculators, computers, or other elec-tronic aids can be allowed.• You have 90 minutes time to complete your work.• The hourly exam itself will have space for work on eachpage. This space is excluded here in order to save print-ing resources.1 202 103 104 105 106 107 108 109 1010 10Total: 1101Problem 1) TF questions (20 points)Mark for each of the 20 questions the correct letter. No justifications are needed.1)T F(1, 1) is a local maximum of the function f(x, y) = x2y −x + cos(y).2)T FIf f is a smooth function of two variables, then the number of critical pointsof f inside the unit disc is finite.3)T FIf (1, 1) is a critical point for the function f (x, y) then (1, 1) is also a criticalpoint for the function g(x, y) = f(x2, y2).4)T FThere is no function f (x, y, z) of three variables, for which every point onthe unit sphere is a critical point.5)T FIf (x0, y0) is a maximum of f(x, y) under the constraint g(x, y) = g(x0, y0),then (x0, y0) is a maximum of g(x, y) under the constraint f(x, y) =f(x0, y0).6)T FIf ~u is a unit vector tangent at (x, y, z) to the level surface of f(x, y, z) thenDuf(x, y, z) = 0.7)T FThe vector ~ru− ~rvis tangent to the surface parameterized by ~r(u, v) =(x(u, v), y(u, v), z(u, v)).8)T FIf (1, 1, 1) is a maximum of f under the constraints g(x, y, z) = c, h(x, y, z) =d, and the Lagrange multipliers satisfy λ = 0, µ = 0, then (1, 1, 1) is a criticalpoint of f.9)T FIf (0, 0) is a critical point of f(x, y) and the discriminant D is zero butfxx(0, 0) > 0 then (0, 0) can not be a local maximum.10)T FLet (x0, y0) be a saddle point of f (x, y). For any unit vector ~u, there arepoints arbitrarily close to (x0, y0) for which ∇f is parallel to ~u.11)T FA function f (x, y) on the plane for which the absolute minimum and theabsolute maximum are the same must be constant.12)T FThe sign of the Lagrange multiplier tells whether the critical point of f(x, y)constrained to g(x, y) = 0 is a local maximum or a local minimum.13)T FThe point (0, 1) is a local minimum of the function x3+ (sin(y − 1))2.14)T FThe integralR10Rπ/40R2π01 dθdφdρ is the volume of the ice cream cone ob-tained by intersecting x2+ y2≤ z2with x2+ y2+ z2≤ 1.15)T FThe formulaR10Ry0f(x, y) dxdy =R10Rx0f(x, y) dydx holds for all functionsf(x, y).16)T FThe surface area of a surface does not depend on the parameterization ofthe surface.17)T FThe formula for surface area isR RR|~ru·~rv| dudv.18)T FThe integralR10R10f(x, y) dxdy is the volume under the graph of f and sonon-negative.19)T FR2−2R3−3(x2+ y2) sin(y) dxdy = 0.20)T FWhen changing to cylindrical coordinates, we include a factor ρ2sin(φ).2Problem 2) (10 points)Match the parametric surfaces S = ~r(R) with the corresponding surface integralR RSdS =R RR|~ru×~rv| dudv. No justifications are needed.I IIIII IVEnter I,II,III,IV here Surface integralR RR|~ru×~rv| dudv =R10R10√1 + 4u2+ 4v2dvduR RR|~ru×~rv| dudv =R10R10√3 dvduR RR|~ru×~rv| dudv =R2π0Rπ0sin(v)q1 + cos(v)2dvduR RR|~ru×~rv| dudv =R2π0Rπ0sin(v) dvdu3Problem 3) (10 points)Find and classify all the critical points of the function f (x, y) = xy(4 − x2− y2).Problem 4) (10 points)Find the area of the moon-shaped region outside the disc of radius 1 and inside the Cardiodr = 1 + cos(θ).Problem 5) (10 points)Minimize the function E(x, y, z) =k28m(1x2+1y2+1z2) under the constraint xyz = 8, wherek2and m are constants.Remark. In quantum mechanics, E is the ground state energy of a particle in a box withdimensions x, y, z. The constant k is usually denoted by ¯h and called the Planck constant.Problem 6) (10 points)A beach wind protection is manufactured as follows. There is a rectangular floor ACBDof length a and width b. A pole of height c is located at the corner C and perpendicularto the ground surface. The top point P of the pole forms with the corners A and C onetriangle and with the corners B and C an other triangle. The total material has a fixedarea of g(a, b, c) = ab + ac/2 + bc/2 = 12 square meters. For which dimensions a, b, c is thevolume f(a, b, c) = abc/6 of the tetrahedral protected by this configuration maximal?4Problem 7) (10 points)A region R in the xy-plane is given in polar coordinatesby r(θ) ≤ θ for θ ∈ [0, π/2]. Find the double integralZZRsin(√x2+ y2)√x2+ y2(π/2 −√x2+ y2)dx dy .RProblem 8) (10 points)Find the volume of the solid bound by x2+ y2+ z2= 4 and x2+ y2+ z2= 9 above thecone x2+ y2= z2and in the first octant x ≥ 0, y ≥ 0, z ≥ 0.Problem 9) (10 points)FindR R RRz2dV , where R of the solid obtained by intersecting {1 ≤ x2+ y2+ z2≤ 4}with the double cone {z2≥ x2+ y2}.Problem 10) (10 points)Consider the region inside x2+ y2+ z2= 2 above the surface z = x2+ y2.a) Sketch the region.b) Find its
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