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HARVARD MATH 21A - Second Hourly Practice III

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7/22/2010 SECOND HOURLY PRACTICE III Maths 21a, O.Knill, Summer 2010Name:• Start by printing your name in the above box.• Try to answer each question on the same page as the question is asked. If needed, usethe back or th e next empty page for work. If you need additional paper, write your nameon it.• Do not detach pages from this exam packet or unstaple the packet.• Please write neatly. Answers which are illegible for the grader can not be given credit.• No notes, books, calculators, computers, or other electronic aids can be allowed.• You have 90 minutes time to complete your work.1 202 103 104 105 106 107 108 109 1010 10Total: 1101Problem 1) (20 points)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 FThe value of the function f(x, y) = sin(−x + 2y) at (0.001, −0.002) can bylinear approximation be estimated as −0.003.4)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).5)T FIf the velocity vector ~r′(t) of the planar curve ~r(t) is orthogonal to thevector ~r(t) for all times t, then the curve is a circle.6)T FThe gradient of f(x, y) is normal to the level curves of f.7)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) u nder the constraint f(x, y) =f(x0, y0).8)T FIf ~u is a unit vector tangent at (x, y, z) to the level surface of f(x, y, z) thenthe directional derivative satisfies Duf(x, y, z) = 0.9)T FIf ~r(t) = hx(t), y(t)i and x(t), y(t) are polynomials, then the tangent line isdefined at all points.10)T FThe vector ~ru(u, v) is tangent to the surface parameterized by ~r(u, v) =hx(u, v), y(u, v), z(u, v)i.11)T FThe second derivative test allows to check whether an extremum found withthe Lagrange multiplier method is a maximum.12)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.13)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.14)T FIf f(x, y) has two local maxima on the plane, then f must have a localminimum on the plane.15)T FGiven a unit vector v, define g(x) = Dvf(x). If at a critical point, for allvectors v we have Dvg(x) > 0, then f is a local maximum.16)T FIf x4y + sin(y) = 0 then y′= 4x3/(x4+ cos(y)).17)T FThe critical points of F (x, y, λ) = f(x, y) −λg(x, y) are solutions to the La-grange equations when extremizing the function f(x, y) under the constraintg(x, y) = 0.18)T FThe volume under the graph of f (x, y) = x2+y2inside th e cylinder x2+y2≤1 isR10R2π0r3dθdr.19)T FThe surface area of the unit sphere is 4π.20)T FThe area of a disc of radius 2r is 4 times larger than a disc of radius r.2Problem 2) (10 points)Match the regions with the corresponding double integrals.a0.00.20.40.60.81.00.20.40.60.81.0b0.00.20.40.60.81.00.20.40.60.81.0c0.00.20.40.60.81.00.20.40.60.81.0d0.00.20.40.60.81.00.20.40.60.81.0e0.00.20.40.60.81.00.20.40.60.81.0f0.00.20.40.60.81.00.20.40.60.81.0Enter a,b,c,d,e or f Integral of f(x, y)R10R√xx2f(x, y) dydxR10R√y0f(x, y) dxdyR10R1y2f(x, y) dxdyEnter a,b,c,d,e or f Integral of f(x, y)R10R√1−x20f(x, y) dydxR10R1(1−x)2f(x, y) dydxR10R√1−x2(1−x)2f(x, y) dydxProblem 3) (10 points)a) Use the technique of linear approximation to estimate f(log(2) + 0.001, 0.006) for f(x, y) =e2x−y. (Here, log means the natural logarithm).b) Find the equation ax + by = d for the tangent line which goes through the point (log(2), 0).Problem 4) (10 points)Find a point on the surface g(x, y, z) =1x+1y+8z= 1 for which the distance to the origin is alocal minimum.Problem 5) (10 points)3Find all extrema of the function f(x, y) = x3+ y3−3x −12y + 20 on the plane and characterizethem. Do you find a absolute maximum or absolute minimum amoung them?Problem 6) (10 points)Find the surface area of the ellipse cut from the plane z = 2x+2y+1 by the cylinder x2+y2= 1.Problem 7) (10 points)Find the tangent plane to the surface f(x, y, z) = x3y − xy2+ 3z = 6 at the point (1, 1, 2).Problem 8) (10 points)You find yourself in the desert at the point A = (a, 1),completely dehydrated and almost dead. You want toreach the point B = (b, 1) as fast as possible but you cannot reach it without water. There is an lake inside theellipsoid g(x, y) = x2+ 2y2= 1. The amount of ”effort”you need to go from a point (x, y) to a point (u, v) isassumed to be (x − u)2+ (y − v)2(this is justified bythe fact that if you walk for a long time, you walk lessand less efficiently so that walking twice as long will takeyou 4 times as much effort). Find the path of least effortwhich connects A with X = (x, y) and then with B.a) Which function f(x, y) do you extremize? The parameters a, b are constants.b) Write down the Lagrange equations.c) Solve the Lagrange equations in the case a = −1, b = 1.Problem 9) (10 points)a) (5 points) Integrate f(x, y) = x2−y2overthe unit disk {x2+ y2≤ 1}.b) (5 points) An evil integral!Z10Z√1−θ20r2drdθ .4Problem 10) (10 points)xy01234556677-1-2-3-4-5-6-7-8RSTUVWXYa) (4 points) Circle the point at which the magnitude of the gradient vector ∇f is greatest.Mark exactly one point. Justify your answer.R S T U V W X Yb) (3 points) Circle the points at which the partial derivative fxis strictly positive. Mark anynumber of points on this question. Justify your answers.R S T U V W X Yc) (3 points) We know that the directional derivative in the direction (1, 1)/√2 is zero at oneof the following points. Which one? Mark exactly one point on this question.R S T U V W X


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