Name:Lab Section:MATH 215 – Winter 2005FINAL EXAMShow your work in this booklet.Do NOT submit loose sheets of paper–They won’t be gradedProblem Points Score1 102 153 204 105 106 157 108 10TOTAL 100Some useful trigonometric identities:sin2θ + cos2θ = 1 cos 2θ = cos2θ − sin2θ sin 2θ = 2 sin θ cos θsin2θ =1 − cos 2θ2cos2θ =1 + cos 2θ2Spherical coordinates:x = ρ cos(θ) sin(φ) y = ρ sin(θ) sin(φ) z = ρ cos(φ)1Problem 1. (5+5=10 points) In this problem f(x, y) = y2x − x2+ 2xy and P is the pointP = (2, 1).(a) In what direction is the rate of change of f greatest at P ? Express your answer in terms of aunit vector.(b) Suppose ~r(t) = hx(t) , y(t)i is a parametric curve such that ~r(0) = h2 , 1i, andddt~r(0) = h3, 5i.Find the value ofddtf(x(t) , y(t))|t=0.2Problem 2. (8+7=15 points) Suppose that, in an experiment, the temperature of a sample (indegrees Celcius) is given by the function T (x, y, z) = 2y2+ ze−x+ 16, where x, y and z are variablesone can control in the experiment.(a) Using the linear approximation of the function T at the point (x0, y0, z0) = (0, 1, 2), fin d anapproximate value of T (0.2, 0.9, 2.3). Note that T (0, 1, 2) = 20◦,(b) Suppose one wants t o change (x0, y0, z0) a little and yet maintain the temperature at 20◦.Using the linear app roximation, find an equation between ∆x, ∆y and ∆z so thatT (∆x, 1 + ∆y, 2 + ∆z)∼=20.3Problem 3. (4×5 = 20 points) For each item, circle the correct answer or indicate if the statementis true or false. Assume that the functions, fields and curves below are smooth.Think carefully before you answer –no partial credit on this one, -take your time!(a) Let C be an arc from (0, 0) to (2, 1). According to the fundamental theorem for line integrals,RC(y − 1) dx + (x + 2y) dy is equal to:(1) 2(2) 1(3) It depends on what C is.(b) For every smooth function f, the integralZ10Z2y2+10f(x, y) dx dy is equal to(1)R30R12√x−10f(x, y) dy dx(2)R31R12√x+10f(x, y) dy dx(3) None of the above.(c) If (a, b) is a critical point of a function f , and iffxx(a, b) = −2 and fyy(a, b) = 3,then what can one say about (a, b)?(1) Nothing can be concluded from the given information.(2) (a, b) is a local minimum of f.(3) (a, b) is a local maximum of f.(4) (a, b) is a saddle point of f.4(d) If~F is a field such thatHC~F · d~r = 0 where C is the unit circle, then~F must be conservative.(1) True.(2) False.(e) If C is the boundary of a domain D and C is oriented as in the statement of Green’s theorem,thenICx2y dx − y dy equals(1)RRD(2xy − 1) dA.(2)RRD(1 − x2) dA.(3)RRD(−x2) dA.(4) None of the above.5Problem 4. (10 points) Let a > 0 denote a fixed constant. A uniform plate with mass density2 gr/cm2occupies the region bounded by the curves:y =√x with 0 ≤ x ≤ a, y =√−x with − a ≤ x ≤ 0, and y =√aFind the coordinates of the center of mass of of the plate.6Problem 5. (5+5=10 points) Let C denote the oriented closed curve consisting of the linesegment from (0, 0) to (√2, 0), followed by the arc of the circle x2+ y2= 2 from (√2, 0) to (1, 1),followed by the line segment from (1, 1) to (0, 0).(a) By an explicit direct calculation, compute I =HCy dx. (You have to break the calculation intothree line integrals.)(b) Verify your answer to part (a) by computing I using Green’s theorem.7Problem 6. (7+8=15 points) Let S be the portion of the cylinder given in cylindrical cooridi-nates by0 ≤ z ≤ 3, r = 1, 0 ≤ θ ≤ π/2.Orient S by normal vectors pointing away from the z axis.(a) Compu te the flux (surface integral) of~F = h2x , y , −3zi across S.8(b) Let C denote the boundary of S, oriented counter-clockwise if one looks at S from the point(5, 5, 1). Consider the line integral I =HCyzdx − 2xzdy.Without computin g the numerical value of I, determine whether I equals the surface integral ofpart (a). Justify your conclusion carefully.9Problem 7. (10 points) Let E denote the portion of the solid sphere of radius R in the firstoctant, and let~F = (2x + y)~ı + y2~ + cos(xy)~kApplying the Divergence Theorem, compute the net flux of the field (surface integral)~F across theboundary of E, oriented by the outward-pointing normal vectors.10xy1 2201.510.5 1.500.5Problem 8. (5+5=10 points) The figure above is a contour plot of a function f and of itsgradient. The values of f on two adjacent level curves differ by 5 units.The plot also includ es an oriented curve, C.(a) Wh at is a pretty good estimate of the value of the line integralRC∇f · d~r?(1) 30(2) −29(3) The integral cannot be estimated with the given data.(b) According to the plot, which of the following appears to hold?(1) div(∇f)(1.7, 1) > 0.(2) div(∇f)(1.7, 1) < 0 .(3) div(∇f)(1.7, 1) = 0 because ∇ · ∇f = 0 for all smooth functions