Name:Recitation:Multivariable CalculusMath 215 Fall 2003First ExamOctober 9, 2003Please do all your work in this booklet and show all the steps.Write your final answer in the corresponding box.Calculators and note-cards are not allowed.Problem Possible points Score1 152 103 204 105 156 107 108 10Total 1002Problem 1. (15 pts.)(a – 5 pts.) Find the area of the triangle with vertices A(1, 0, 0), B(0, 1, 0), and C(0, 0, 1) and compare it to the area of its“shadow” on the xy-plane.Answer:(a)(b – 5 pts.) Find the distance from the point P (14, 4, 8) to the plane 6x − 2y + 3z = 2.Answer:(b)(c – 5 pts.) Find an equation of the sphere centered at the point P (14, 4, 8) and touching the plane 6x − 2y + 3z = 2.Answer:(c)˛˛˛˛˛153Problem 2. (10 pts.)(a – 5 pts.) Find an equation of the plane containing the line L : r(t) = (2 + 3t)i + (1 − 2t)j + (−1 + t)k and the pointP (1, 3, 1).Answer:(a)(b – 5 pts.) Find a parametric equation of the line of intersections of the planes x − y + 2z = 2 and 3x + y − z = 4.Answer:(b)˛˛˛˛˛104Problem 3. (20 pts.) Find or estimate, depending o n the type of data provided, partial derivatives fxand fyat the pointP (3, 2) for the following four functions:(a) Function f1is given by the formula:f1(x, y) = x2cos(xy)Answer:(a)(b) Function f2is given by the table:f2(x, y) x = 2.8 x = 3.0 x = 3.2 x = 3.4y = 2.1 1.20 1.35 1.50 1.64y = 2.0 1.26 1.41 1.55 1.70y = 1.9 1.31 1.46 1.61 1.75y = 1.8 1.37 1.52 1.66 1.81Answer:(b)(c) Function f3is given by its level curves:54332211–2–1–100012345y12 345xAnswer:(c)(d) The only thing we know about the function f4is its gradient vector at P (3, 2):Answer:(d)˛˛˛˛˛205Problem 4. (10 pts.) A gnat with a keen grasp of multivariable calculus notes that the relative humidity in the greenhousein which it is flying is given by H(x, y, z) =12sin(xy) +1z + 1.(a – 5 pts.) Which direction should the gnat fly from its current position of (π, 2, 1) to decrease the humidity of the air aroundit the fastest (it is not necessary to give your answer as a unit vector)?Answer:(a)(b – 5 pts.) How does the humidity change if it instead flies in the direction of the point (π + 3, 2, 5)? (Give your answer asa rate of change in this direction).Answer:(b)˛˛˛˛˛106Problem 5. (15 pts.) Suppose that three quantities x, y, and z, are constrained by the equation 2x2+ 3y2+ z2= 20. Thegraph of this equation is a surface S in space.(a – 5 pts.) Verify that the point P (2, 1, 3) is a point on S and find the equation of the tangent plane to S at this point.Answer:(a)(b – 5 pts.) Near P (2, 1, 3) we can think of z as a function of x and y, z = f(x, y). Without finding f(x, y) explicitly,determine its linear approximation Lfnear x = 2, y = 1.Answer:(b)(c – 5 pts.) Approximate the value of z corresponding to x = 1.97 and y = 1.12.Answer:(c)˛˛˛˛˛157Problem 6. (10 pts.) Corn production, C, is a function of rainfall, R, and temperature, T ; C = C(R, T ). Of course, Rand T are functions of time t. If at t he present moment CR= 3.3, CT= −5, and the current climate model predicts thatRt= −0.2 and Tt= 0.1, do you expect the corn production to increase or to decrease?Answer:˛˛˛˛˛108Problem 7. (10 pts.) Multiple Choice.(1) The gradient vector ∇f of the function f (x, y) at the point P(a) Must be parallel to the level curve of f through P .(b) Must be perpendicular to the level curve of f through P .(c) None of the above.(2) If r00(t) is parallel to r0(t),(a) The trajectory r(t) must be a straight line.(b) The trajectory r(t) must be a circle.(c) None of the above.(3) If r00(t) is perpendicular to r0(t),(a) The trajectory r(t) must be a straight line.(b) The trajectory r(t) must be a circle.(c) None of the above.(4) If for two non-zero vectors v and w we have v × w = 0(a) v and w must be parallel.(b) v and w must be perpendicular.(c) None of the above.(5) If for two non-zero vectors v and w we have v • w = 0(a) v and w must be parallel.(b) v and w must be perpendicular.(c) None of the above.˛˛˛˛˛109Problem 8. (10 pts.)Match the following functions, graphs, level curves(a) f(x, y) = cos(x2+ y2) (I)–3–2–10123x–3–2–10123y24(A)–2–1012y–2–1012x(b) f(x, y) = 10xye−x2−y2(II)–3–2–10123x–3–2–10123y–101(B)–3–2–10123y–2–1012 3x(c) f(x, y) = cos(3x) (III)–3–2–10123x–3–2–10123y–4–20246(C)–3–2–10123y–3 –2–1012 3x(d) f(x, y) =5x2+ 2y2+ 1(IV)–3–2–10123x–3–2–10123y–101(D)–3–2–10123y–3 –2–1012 3x(e) f(x, y) = x − y + 1 (V)–3–2–10123x–3–2–10123y–101(E)–3–2–10123y–3 –2–1012 3xby filling up the table below:Function Graph Level