7/28/2011 SECOND HOURLY PRACTICE VI Maths 21a, O.Knill, Summer 2011Name:• Start by printing your name in the above box.• Try to answer each question on th e same page as the question is asked. If needed, usethe back or the next empty page for work. If you need additiona l paper, write your n a m eon it.• Do not detach pages from this exam packet or unstaple the packet.• Provide details to all computat io n s except for problems 1-3.• Please writ e neatly. Answers which are illegible for the grader can not be given cr edit.• No notes, books, calculators, comput er s, or ot h er 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) True/False questions (20 points)1)T FIt is possibl e that (1, 1) is a local maximum for the function f and 1 =fxx= −fyy.2)T F(0, 0) is a local maximum of the function f(x, y) = 5 − x8− y8.3)T FIf the curvature of a curve is zero everywhere, then it is a line.4)T FIf the Lagrange multiplier λ is negative then the critical po int under con-straint is a saddle point.5)T FThe arc length of a curve on [0, 1] can be obtained by integrating up thecurvature of the curve al o n g the i nterval [0, 1].6)T FIf D is the discrimin ant at a critical point and Dfxx> 0 then we eitherhave a saddle point or a lo ca l maximum.7)T FThe function f (x, y) = sin(y)x2sin(y2) satisfies the partial differential equa-tion fxyyxyxy= 0.Problem 2) (10 points)Match the regions with the corresponding polar double integralsAB CD E F2Enter A-F Integral of f(r, θ)Rπ/20Rπ/20f(r, θ)r drdθRπ/20Rθ0f(r, θ)r drdθRπ/20Rπ/2−θ0f(r, θ)r drdθEnter A-F Integral of f(r, θ)Rπ/20Rπ/2θf(r, θ)r drdθRπ/20Rπ/2π/2−θf(r, θ)r drdθRπ/20Rπ/2π/4f(r, θ)r drdθProblem 3) (10 points)The following statements are not complete. Fill in from the pool of words below.statement Fill in the l et t er s statementThe surface area does on the parametrization.√48 can be est im a t ed by at x = 7. The result is 7-1/14.The discrim i n a nt D is if the point is a saddle point.For a Lagrange minimum, ∇g is to ∇f .Arc length is ap p r oximated by a sum if the curve is smo o t h .The gradient ∇f is to the surface f = c.O not dependL linear approximationI negativeD not dependM tangentO parabolaE perpendicularL parallelE orthogonalR RiemanProblem 4) (10 points)The green near on e of th e holes i n the Cambridge Fresh pond golf cou r se has the heightf(x, y) = x3+ y3− 3x2− 3y23Find local maxima, local minima or saddle points of this function. Near which point willgolf ball s most likely end up, if balls like to roll to lower areas.Problem 5) (10 points)A torus can be obtained by rotating a circleof radius b around a circle of radius a. Thevolume of such a t o r u s is 2π2ab2and the sur-face area is 4π2ab. If we want to find thetorus which has minimal surface area whilethe volume with fixed packing 2π2a(b2+ 1)is fixed 2π2, we need to extremi ze the func-tion f(a, b) = 4π2ab under the constrainta + ab2= 1. Find the optimal a, b.Problem 6) (10 points)a) Find the ar c length of the curve ~r(t) = ht2, 2t3/3, 1i from t = −1 to t = 1.4b) What is th e curvature of the curve at time t = 1? The formula for the curvature isκ(t) =|~r′(t) ×~r′′(t)||~r′(t)|3Problem 7) (10 points)A right angle triangle has the side lengths x = 0.999 and y = 1.00001. Estimate the valueof the hypotenuse f(x, y) =√x2+ y2using li n e ar approximation.Problem 8) (10 points)Oliver got a diagmagnetic kit, where strong magnets produce a force field in which pyrolyticgraphic flots. The gr avitational field produces a well of the form f(x, y) = x4+y3−2x2−3y.Find all critical points of this function and classify them. Is there a global minimum?Right picture credit:
View Full Document
Unlocking...