4/18/2006 SECOND HOURLY SECOND 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 wor k 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 (30 points)Mark for each of the 20 questions the correct letter. No justifications are needed.1)T Ff(x, y) and g(x, y) = f(x2, y2) have the same critical po ints.2)T FIf a function f(x, y) = ax + by has a critical point, then f (x, y) = 0 for all(x, y).3)T FGiven 2 arbitrary points in the plane, there is a function f (x, y) which hasthese points as critical points and no other critical points.4)T FIf (x0, y0) is the maximum of f(x, y) on the disc x2+y2≤ 1 then x20+y20< 1.5)T FThere are no functions f(x, y) for which every point on the unit circle is acritical point.6)T FAn absolute maximum (x0, y0) of f(x, y) is also an absolute maximum off(x, y) constrained to a curve g(x, y) = c that goes through the point(x0, y0).7)T FIf f(x, y) has two local maxima on the plane, then f must have a localminimum on the plane.8)T FThere exists a function f(x, y) of two variables which has no critical pointsat all.9)T FIf fx(x, y) = fy(x, y) = 0 for all (x, y) then f(x, y) = 0 for all (x, y) .10)T F(0, 0) is a local maximum of the function f(x, y) = x2− y2+ x4+ y4.11)T FIf f(x, y) has a local maximum at the point (0, 0) with discriminant D > 0then g(x, y) = f(x, y) − x4+ y3has a local maximum at the point (0, 0)too.12)T FEvery critical point (x, y) of a function f(x, y) for which the discriminantD is not zero is either a local maximum or a local minimum.13)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 minimum.14)T FIn the second derivative test, one can replace the condition D > 0 , fxx> 0with D > 0, fy y> 0 to check whether a po int is a local minimum.15)T FThe function f (x, y) = (x4− y4) has neither a local maximum nor a localminimum at (0, 0 ) .16)T FIt is possible to find a function of two variables which has no maximum andno minimum.17)T FR20R20(x2+ y2) cos(x3+ y3) dxdy ≤ 32.18)T FR20Rx20f(x, y) dydx =R40R2√yf(x, y) dxdy.19)T FThe area of a polar region 0 ≤ r ≤ r(θ) isR2π0r(θ)2/2 dθ.20)T FIf R is the unit disc in the xy-plane, thenR RR−√1 − x2− y2dxdy =−2π/3.2Problem 2) (10 points)Match the integra ls with those obtained by changing the order o f integration. No justifi-cations are needed.Enter 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 3) (10 points)Which point on the surface g(x, y, z) =1x+1y+8z= 1 is closest to the origin?Problem 4) (10 points)3Find all extrema o f the function f(x, y) = x3+ y3− 3x − 12y + 20 on the plane andcharacterize them. Do you find a absolute maximum or absolute minimum among them?Problem 5) (10 points)Find all the critical points of f(x, y) =x55−x22+y33− y and indicate whether they arelocal maxima, local minima or saddle points.Problem 6) (10 points)What is the shape of the triangle with angles α, β, γ for whichf(α, β, γ) = log (sin(α) sin(β) sin(γ))is maximal?αβγProblem 7) (10 points)Evaluate the integralZ3π/4π/4Z2 sin(θ)1/ sin(θ)r drdθHint: There is not much to compute if you know how the region looks like.4Problem 8) (10 points)Evaluate the integralZπ0Zπxsin(y)ydydx .Problem 9) (10 points)Find the surface area of the surface parametrized by~r(u, v) = hu, v, 2u − viwith 1 ≤ u ≤ 2, −1 ≤ v ≤ 1.Problem 10) (10 points)Integrate the function f (x, y, z) = x2+y2over the solid bound by the planes z = 1, z = −1and the hyperbo lo id x2+ y2− z2=
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