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HARVARD MATH 21A - p-1

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FIRST PRACTICE EXAM FOR FINAL Math 21a, Spring 03Name: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.• Try to answer each question on the same page asthe question is asked. If needed, use the back ornext empty page for work. If you need additionalpaper, write your name on it.• 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. Justify youranswers.• No notes, books, calculators, computers or otherelectronic aids are allowed.• You have 180 minutes time to complete your work.1 202 103 104 105 106 107 108 109 1010 1011 1012 1013 10Total: 140Problem 1) TF questions (20 points) Circle the correct letter. No justifications are needed.T FThe length of the curve r(t) = (sin(t), t4+ t, cos(t)) on t ∈ [0, 1] isthe same as the length of the curve r(t) = (sin(t2), t8+ t2, cos(t2))on [0, 1].T FThe parametric surface r(u, v) = (5u −3v, u −v −1, 5u −v −7) isa plane.T FAny function u(x, y) that obeys the differential equation uxx+ ux−uy= 1 has no local maxima.T FThe scalar projection of a vector a onto a vector b is the length ofthe vector projection of a onto b.T FIf f(x, y) is a function such that fx−fy= 0 then f is conservative.T F(u × v) · w = (u × w) · v for all vectors u, v, w.T FThe equation ρ = φ/4 in spherical coordinates is half a cone.T FThe function f(x, y) =(xx2+y2if (x, y) 6= (0, 0)0 if (x, y) = (0, 0)is continuous atevery point in the plane.T FR10Rx01 dydx = 1/2.T FLet a and b be two vectors which are perpendicular to a given planeΣ. Then a + b is also perpendicular to Σ.T FIf g(x, t) = f(x −vt) for some function f of one variable f(z) theng satisfies the differential equation gtt− v2gxx= 0.T FIf f(x, y) is a continuous function on R2such thatR RDf dA ≥ 0for any region D then f(x, y) ≥ 0 for all (x, y).T FAssume the two functions f (x, y) and g(x, y) have both the criticalpoint (0, 0) which are saddle points, then f + g has a saddle pointat (0, 0).T FIf f(x, y) is a function of two variables and if h(x, y) = f(g(y), g(x)),then hx(x, y) = fy(g(y), g(x))g0(y).T FIf we rotate a line around the z axes, we obtain a cylinder.T FThe line integral of F(x, y) = (x, y) along an ellipse x2+ 2y2= 1 iszero.T FIf u(x, y) satisfies the transport equation ux= uy, then the vectorfield F(x, y) = hu(x, y), u(x, y)i is a gradient field.T F3 grad(f) =ddtf(x + t, y + t, z + t).T FR10R2π/110Rπ0ρ2sin(φ) dφdθdρ = 4π/33.T FIf F is a vector field in space and f is equal to the line integral ofF along the straight line C from (0, 0, 0) to (x, y, z), then ∇f = F.x 4 =2Problem 2) (10 points)Match the equations with the curves. No justifications are needed.I IIIII IVEnter I,II,III,IV here Equationr(t) = (sin(t), t(2π − t))r(t) = (cos(5t), sin(7t))r(t) = (t cos(t), sin(t))r(t) = (cos(t), sin(6/t))3Problem 3) (10 points)In this problem, vector fields F are written as F = (P, Q). We use abbreviations curl(F ) =Qx−Pyand div(F ) = Px+Qy. When stating curl(F )(x, y) = 0 we mean that curl(F )(x, y) = 0vanishes for all (x, y). The statement curl(F ) 6= 0 means that curl(F )(x, y) does not vanish forat least one point (x, y). The same remark applies if curl is replaced by div.Check the box which match the formulas of the vectorfields with the corresponding pictureI,II,III or IV. Mark also the places, indicating the vanishing or not vanishing of curl and div. Ineach of the four lines, you should finally have circled three boxes. No justifications are needed.Vectorfield I II III IV curl(F ) = 0 curl(F ) 6= 0 div(F ) = 0 div(F ) 6= 0F(x, y) = (0, 5)F(x, y) = (y, −x)F(x, y) = (x, y)F(x, y) = (2, x)I IIIII IV4Problem 4) (10 points)a) Find the scalar projection of the vector v = (3, 4, 5) onto the vector w = (2, 2, 1).b) Find the equation of a plane which contains the vectors h1, 1, 0i and h0, 1, 1i and containsthe point (0, 1, 0).Problem 5) (10 points)Find the surface area of the ellipse cut from the plane z = 2x+2y+1 by the cylinder x2+y2= 1.Problem 6) (10 points)Sketch the plane curve r(t) = (sin(t)et, cos(t)et) for t ∈ [0, 2π] and find its length.Problem 7) (10 points)Let f(x, y, z) = 2x2+3xy+2y2+z2and let R denote the region in R3, where 2x2+2y2+z2≤ 1.Find the maximum and minimum values of f on the region R and list all points, where saidmaximum and minimum values are achieved. Distinguish between local extrema in the interiorand extrema on the boundary.Problem 8) (10 points)Sketch the region of integration of the following iterated integral and then evaluate the integral:Zπ0 Z√π√zZx0sin(xy)dydx!dz .Problem 9) (10 points)Evaluate the line integralRCF · dr, where F = (x +exsin(y), x + excos(y)) and C is the right handed loopof the lemniscate described in polar coordinates as r2=cos(2θ).Problem 10) (10 points)5Evaluate the line integralZCF · dr,where C is the planar curve r(t) = (t2, t/√t + 2), t ∈ [0, 2] and F is the vector field F(x, y) =(2xy, x2+ y). Do this in two different ways:a) by verifying that F is conservative and replacing the path with a different path connecting(0, 0) with (4, 1),b) by finding a potential U satisfying ∇U = F.Problem 11) (10 points)a) Find the line integralRCF · dr of the vector field F(x, y) = (xy, x) along the unit circleC : t 7→ r(t) = (cos(t), sin(t)), t ∈ [0, 2π] by doing the actual line integral.b) Find the value of the line integral obtained in a) by evaluating a double integral.Problem 12) (10 points)Consider the surface given by the graph of the functionz = f (x, y) =1001+x2+y2sinπ8(x2+ y2)in the region x2+y2≤ 16. The surface is pictured to the right.A magnetic field B is given by the curl of a vector potential A. That is, B = ∇×A = curl(A)and A is a vector field too. SupposeA =z sin(x3), x1 − z2, log(1 + ex+y+z).Compute the flux of the magnetic field through this surface. (The surface has an upwardpointing normal vector.)Problem 13) (10 points)Let S be the surface given by the equations z = x2−y2, x2+ y2≤ 4, with the upward pointingnormal. If the vector field F is given by the formula F(x, y, z) = h−x, y,√x2+ y2i, find theflux of F through


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HARVARD MATH 21A - p-1

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