SECOND PRACTICE EXAM SECOND HOURLY Math 21a,Spring 2003Name:MWF10 Ken ChungMWF10 Weiyang QiuMWF11 Oliver KnillTTh10 Mark LucianovicTTh11.5 Ciprian Manolescu• Start by printing your name in the above box andcheck your section in the box to the left.• Do not detach pages from this exam packet or un-staple the packet.• Please write neatly. Answers which are illegible forthe grader can not be given credit.• No notes, books, calculators, computers, or otherelectronic aids can be allowed.• You have 90 minutes time to complete your work.• The hourly exam itself will have space for work oneach page. This space is excluded here in order tosave printing resources.1 202 103 104 105 106 107 108 109 10Total: 100Problem 1) TF questions (20 points) Circle the correct letter. No justifications are needed.The score for this question is the number of correct answers.T FA function f(x, y) on the plane for which the absolute minimumand the absolute maximum are the same must be constant.T FThe functions f(x, y) and g(x, y) = f(x, y) + 2002 do not have thesame critical points.T FThe sign of the Lagrange multiplier tells whether the critical pointof f(x, y) constrained to g(x, y) = 0 is a local maximum or a localminimum.T FThe gradient of a function f(x, y, z) is tangent to the level surfacesof fT FThe point (0, 1) is a local minimum of the function x3+(sin(y−1))2.T FFor any curve, the acceleration vector r00(t) of r(t) is orthogonal tothe velocity vector at r(t).T FIf Duf(x, y, z) = 0 for all unit vectors u, then (x, y, z) is a criticalpoint.T FRbaRdcx dxdy = (d2− c2)(b − a)/2, where a, b, c, d are constants.T FThe functions f(x, y) and g(x, y) = (f(x, y))2have the same criticalpoints.T FIf a function f(x, y) = ax + by has a critical point, then f(x, y) = 0for all (x, y).T Ffxyxyx= fyyxxxfor f (x, y) = sin(cos(y + x14) + cos(x)).T FThe function f(x, y) = −x2002− y2002has a critical point at (0, 0)which is a local minimum.T FIt is possible that for some unit vector u, the directional derivativeDuf(x, y) is zero even though the gradient ∇f(x, y) is nonzero.T FIf (x0, y0) is the maximum of f(x, y) on the disc x2+ y2≤ 1 thenx20+ y20< 1.T FThe linear approximation L(x, y, z) of the function f(x, y, z) = 3x+5y − 7z at (0, 0, 0) satisfies L(x, y, z) = f(x, y, z).T FIf f(x, y) = sin(x) + sin(y), then −√2≤ Duf(x, y) ≤√2.T FThere are no functions f(x, y) for which every point on the unitcircle is a critical point.T FAn absolute maximum (x0, y0) of f(x, y) is also an absolute maxi-mum of f(x, y) constrained to a curve g(x, y) = c that goes throughthe point (x0, y0).T FIf f(x, y) has two local maxima on the plane, then f must have alocal minimum on the plane.T FR RDf(x, y)g(x, y) dA = (R RDf(x, y) dA)(R RDg(x, y) dA) is truefor all functions f and g.2Problem 2) (10 points)Match the parametric surfaces with their parameterization. No justification is needed.I IIIII IVEnter I,II,III,IV here Parameterization(u, v) 7→ (u, v, u + v)(u, v) 7→ (u, v, sin(uv))(u, v) 7→ (0.2 + u(1 − u2)) cos(v), (0.2 + u(1 − u2)) sin(v), u)(u, v) 7→ (u3, (u − v)2, v)Problem 3) (10 points)Match the integrals with those obtained by changing the order of integration. No justificationsare needed.3Enter I,II,III,IV or V here. IntegralR10R11−yf(x, y) dxdyR10R1yf(x, y) dxdyR10R1−y0f(x, y) dxdyR10Ry0f(x, y) dxdyI)R10Rx0f(x, y) dydxII)R10R1−x0f(x, y) dydxIII)R10R1xf(x, y) dydxIV)R10Rx−10f(x, y) dydxV)R10R11−xf(x, y) dydxProblem 4) (10 points)Consider the graph of the function h(x, y) = e−3x−y+ 4.1. Find a function g(x, y, z) of three variables such that this surface is the level set of g.2. Find a vector normal to the tangent plane of this surface at (x, y, z).3. Is this tangent plane ever horizontal? Why or why not?4. Give an equation for the tangent plane at (0, 0).4Problem 5) (10 points)Find all the critical points of the function f(x, y) =x22+3y22− xy3. For each, specify if it is alocal maximum, a local minimum or a saddle point and briefly show how you know.Problem 6) (10 points)Minimize the function E(x, y, z) =k28m(1x2+1y2+1z2) under the constraint xyz = 8, where k2and 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 7) (10 points)Assume F (x, y) = g(x2+ y2), where g is a function of one variable. Find Fxx(1, 2) + Fyy(1, 2),given that g0(5) = 3 and g00(5) = 7.Problem 8) (10 points)Consider the region inside x2+ y2+ z2= 2 above the surface z = x2+ y2.a) Sketch the region.b) Find its volume.Problem 9) (10 points)Draw the gradient vector field of f(x, y) = xy − 2x together with a contour map of
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