11/18/2008 SECOND HOURLY FIRST PRACTICE Math 21a, Fall 2008Name:MWF 9 Chung-Jun John TsaiMWF 10 Ivana BozicMWF 10 Peter Gar fieldMWF 10 Oliver KnillMWF 11 Peter Gar fieldMWF 11 Stefan HornetMWF 12 Aleksander SuboticTTH 10 Ana CaraianiTTH 10 Toby GeeTTH 10 Valentino TosattiTTH 11:30 Ming-Tao ChuanTTH 11:30 Valentino Tosatti• Start by printing your name in the above boxand check your section in the box to the left.• Do not detach pages from this exam packetor unstaple the packet.• Please write neatly. Answers which are illeg-ible for the grader can not be given credit.• No notes, books, calculators, computers, orother electronic aids can be allowed.• You have 90 minutes time to complete yourwork.• The hourly exam itself will have space forwork on each page. This space is excludedhere in order to save printing resources.1 202 103 104 105 106 107 108 109 1010 10Total: 110Problem 1) True/False questions (20 points), no justifications needed1)T FThe directional derivative D~vf is a vector perpendicular to ~v.2)T FUsing linearization of f(x, y) = xy we can estimate f(0.9, 1.2) ∼ 1 − 0.1 +0.2 = 1.1.3)T FGiven a curve ~r(t) on a surface g(x, y, z) = 1, thenddtg(~r(t)) = 0.4)T FGiven a function f (x, y) such that ∇f(0, 0) = h2, −1i. ThenDh0,−1if(0, 0) = 0.5)T F~r(u, v) = h u cos(v), u sin(v), vi is a surface of revolution.6)T FIf (1, 1) is a critical point for the function f(x, y) then (1, 1) is also a criticalpoint f or the function g(x, y) = f(x2, y2).7)T FIf f(x, y) has a local maximum at (0, 0) the it is possible that fxx(0, 0) > 0and fyy(0, 0) < 0.8)T FThe integralRx0Ry01 dxdy computes the area of a region in the plane.9)T FThe function f(x, y) = x2+ y4has a local minimum at (0, 0).10)T FThe integralR10R10x2+ y2dxdy is the volume of the solid bounded by the 5planes x = 0, x = 1, y = 0, y = 1, z = 0 and t he para boloid z = x2+ y2.11)T FThere exists a region in the plane, which is neither a type I integral, nor atype II integral.12)T FFubini’s theorem assures thatR10Rx0f(x, y) dydx =R10Ry0f(x, y) dxdy.13)T FThe function f (x, y) = sin(x) cos(y) satisfies the partial differential equationfxx+ fyy= 0.14)T FLet L(x, y) be the linearization of f(x, y) = sin(x(y + 1)) at (0, 0). Then,the level curves of L(x, y) consist of lines.15)T FFor any smooth function f(x, y), the inequality |∇f | ≥ |fx+ fy| is true.16)T FAny differentiable function f(x, y) which satisfies the partial differentialequation || ∇f ||2= 0 is constant.17)T FIf x + sin(xy) = 1, dy/ dx =−(1+y cos(yx))(x cos(xy)).18)T FThe directional derivative Dvf(1, 1) is zero if v is a unit vector tangent tothe level curve of f which goes through (1, 1).19)T FIf (a, b) is a maximum of f(x, y) under the constraint g(x, y) = 0, thenthe Lagrange multiplier λ there has the same sign as the discriminant D =fxxfyy− f2xyat (a, b).20)T FIf Dh1/√2,1/√2if(1, 2) = 0 and Dh−1/√2,1/√2if(1, 2) = 0, then (1, 2) is a criticalpoint.Problem 2) (10 points)Match the regions with t he corresponding double integralsa0.00.20.40.60.81.00.20.40.60.81.0b0.00.20.40.60.81.00.20.40.60.81.0c0.00.20.40.60.81.00.20.40.60.81.0d0.00.20.40.60.81.00.20.40.60.81.0e0.00.20.40.60.81.00.20.40.60.81.0f0.00.20.40.60.81.00.20.40.60.81.0Enter a,b,c,d,e or f Integral of Function f(x, y)R10Rxx/2f(x, y) dydxR10Ry0f(x, y) dxdyR10Rx/20f(x, y) dydxR10R1y/2f(x, y) dxdyR10Rx0f(x, y) dydxR10R11−xf(x, y) dydxProblem 3) (10 points)Let g(x, y, z) = x2+ 2y2− z −3.a) (5 points) Find the equation of the tangent plane to the level surface g(x, y, z) = 0 atthe point (x0, y0, z0) = (2, 0, 1).b) (5 points) The surface in a) is the gr aph z = f(x, y) of a function of two variables. Findthe tangent line to t he level curve f (x, y) = 1 at the point (x0, y0) = (2, 0).Problem 4) (10 points)a) (5 points) Use the technique of linear approximation to estimate f(π/2 + 0.1, 2.9) forf(x, y) = (10 sin(x) − 5y2+ 8)1/3.b) (5 points) Find the unit vector at (π/2, 3), in the direction where the function increasesfastest.Problem 5) (10 points)The pressure in the space at the position (x, y, z) is p(x, y, z) = x2+ y2− z3and thetrajectory of an observer is the curve ~r(t) = ht, t, 1/ti.a) (2 points) State the chain rule which applies in this situation.b) (4 points) Using the chain rule in a) compute the rate of change of the pressure theobserver measures at time t = 2.c) (4 points) At which time t does the observer go in the direction, in which the pressuredecreases most?Problem 6) (10 points)The coffee chain Astrbucks1has branches at (0, 0), (0, 3) and (3, 3 ) (JFK street, Churchstreet, and Broadway) near Harvard square. A caffeine a ddicted [politically correct: loving]mathematician wants to rent an apartment at a location, where the sum of the squareddistances f(x, y) to all those shops is a local minimum. The function isf(x, y) = (x−0)2+(y−0)2+(x−0 )2+(y−3)2+(x−3 )2+(y−3)2= 27−6x+3x2−12y+3y2.a) (5 points) Where does the mathematician have to live to locally minimize f(x, y)?b) (3 points) For every local minimum answer: Is this local minimum a global minimum?c) (2 points) Is there a global maximum to this problem? If yes, give it. If no, why not?1This problem was sponsored by Astrbucksc.Problem 7) (10 points)Find all the critical points of f(x, y) = 3xy + x2y + xy2and classify them as saddle points,local maxima or local minima.Problem 8) (10 points)A solid cone of height h and with base radius r has the volume f(h, r) = hπr2/3 andthe surface area g(h, r) = πr√r2+ h2+ πr2. Among all cones with fixed surface areag(h, r ) = π use the Lag r ange method to find the cone with maximal volume.Problem 9) (10 points)Marsden and Tromba pose in their textbook the following riddle:Suppose w = f(x, y) and y = x2. By the chain rule∂w∂x=∂w∂x∂x∂x+∂w∂y∂y∂x=∂w∂x∂x∂x+ 2x∂w∂yso that 0 = 2x∂w∂yand so∂w∂y= 0.a) Find an explicit example of a function f(x, y), where you seethe a r gument is fa lse.b) What is flawed in the above application of the chain rule?Problem 10) (10 points)Evaluate the double integralZ ZRqx2+ y2dxdywhere R is the region bounded by the positive x- axes, the spiralcurve ~r(t) = ht cos(t), t sin(t)i, 0 ≤ t ≤ 2π and the circle with radius2π.To place a high-impact advertisement here, phone 1-800-Go21aAd
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