Name:Lab Section:MATH 215 – Fall 2004FINAL EXAMShow your work in this booklet.Do NOT submit loose sheets of paper–They won’t be gradedProblem Points Score1 152 103 254 105 156 157 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. (15 points) This problem is about the functionf(x, y, z) = 3zy + 4x cos(z).(a) What is the rate of change of the function of f at (1, 1, 0) in the direction from this point tothe origin?(b) Give an approximate value of f(0.9, 1.2, 0.11).CONTINUED ON THE NEXT PAGE2(c) Recall that f(x, y, z) = 3zy + 4x cos(z).The equation f(x, y, z) = 4 implicitly defines z as a function of (x, y), if we agree that z = 0 if(x, y) = (1, 1).Find the numerical values of the derivatives∂z∂x(1, 1) and∂z∂y(1, 1).3Problem 2. (10 points)(a) Find and classify the critical points of the function f(x, y) = −2x2+ 8xy − 9y2+ 4y − 4.(b) Find the equation of the tangent plane to the surface z + 2x2− 8xy + 9y2− 4y = 0 at thepoint (2, 0, −8).4Problem 3. (25 points) No partial credit. Evaluate each integral below. R > 0 and a > 0 areconstants. Note: If you correctly understand the domains and/or the symmetries of the functions,many of the integrations become immediate. CIRCLE your answers.(1) If D = [−1, 0] × [−1, 0] ∪ [0, 1] × [0, 1], thenZZDxy dA =(2)Za0Zx0dy dx =(3) If D is defined in polar coordinates by the inequalities: 0 ≤ r ≤ R, π/7 ≤ θ ≤ 8π/7,thenZZDpx2+ y2dA =(4)Zπ/20Zπ0ZR0ρ2sin(φ) dρ dθ dφ =(5)Zπ/40ZR0Za0r dz dr dθ =5Problem 4. (10 points) (a) Find the value of a such that the field on the plane~F (x, y) = haxy +1x, x2iis conservative. Find a potential for the resulting field.(b) Compute the line integral of the conservative field you found in part (a) over the curve whichis the image of het2, t cos(2πt)i, where 0 ≤ t ≤ 1.6Problem 5. (9+6=15 points) (a) By a direct calculation, evaluate I =ZCRdx + x2ydy, whereCRis the triangle with vertices (0, 0), (0, R), (R, 0) oriented counterclockwise.(b) Compute the value of I by evaluating a double integral, using Green’s theorem.7Problem 6. (15 points) Let S be the portion of the surface x = 5 − y2− z2in the half spacex ≥ 1, oriented so that the normal vector at (5, 0, 0) is equal to ~ı. Let~F (x, y, z) = h−1 , 1 , 0i (aconstant vector field).(a) Set up and evaluate an iterated double integral equal toRRS~F · d~S.8(b) It turns out that~F = ∇ ×~G, where~G = h0 , z , −xi. (You do not have to verify this.) Givean alternative calculation of the surface integral of part (a) by applying Stokes’ theorem.9Problem 7. (10 points) Consider the plot of a vector field~F = P (x, y)~ı + Q(x, y)~:11.522.533.5y1 1.5 2 2.5 3 3.5x(a) Mark a point A at which curl(~F ) > 0.(b) Mark a point B at which div(~F ) < 0.(c) Sketch a closed curve C such that the circulation of~F along C is positive.(d) Mark two points S and T and two curves C1and C2from S to T such that the work doneby~F in moving an object along those curves is positive for C1and negative for C2.(e) Can the field~F be a gradient field? On the space below explain why or why