1. (a) Consider the points A = (0,1,2), B = (2,2,3), and C = (−1,3,4). Compute the vectors v =−→AB and w =−→AC .(2 points)v =�,,�w =�,,�(b) Consider the points D = (0,2,1), E = (1,4,1), and F = (3,8,2) andthe vectors a =〈1,2,0〉, b =〈3,6,1〉, and c =〈2,4,1〉as shown atright. Find a normal vector n to the plane containing the pointsD, E, and F . (3 points)DFEbacn =�,,�(c) Let P = (2,−1,1) and u =〈3,2,4〉. Find a linear equation for the plane that contains P and has normalvector u. (2 points)Equation: x+ y+ z =(d) For two vectors v and w in R3, which of the following does|v × w|measure? Circle your answer. (1 point)• The length of v − w.• The area of the parallelogram determined by v and w.• The volume of the parallelepiped deter mined by v, w, and v × w.2. (a) Find the midpoint M of the straight-line segment between the points P = (1,3,−2) and Q = (3, −3,4).(2 points)M =�,,�(b) Let L be the line parametrized by r(t ) = a + tb for a =〈1,0,2〉and b =〈2,−1,1〉. Find the point Q ofintersection of the line L with the plane whose equation is 3x − 2y + z = 14. (3 points)Q =�,,�3. Consider the function f (x, y) =xy− x2yx2+ y2for (x, y) �= (0,0). Evaluate lim(x,y)→(0,0)f (x, y) or explain why i t doesnot exist. (5 points)4. For each function(a) −x2− y2(b) cos(x + y) (c) y2e−x2label its graph and its level set diagram from among the options below. Here each level set diagram consistsof level sets�f (x, y) = ci�drawn for evenly spaced ci. (9 points)5. (a) Compute the partial derivatives of f (x, y) = x3+ 2xy+ y. (2 points)∂ f∂x=∂ f∂y=(b) Consider the graph of f : R2→ R shownat right. If P is the point�a, b, f (a, b)�,find the sign of each of the quantities be-low. Circle your answers. (1 point each)fx(a,b): positive negative 0fy(a,b): positive negative 0fxx(a,b): positive negative 0PxzyGraph of f (x, y)(c) Use the contour plot of f (x, y) shown atright to estimate fx(2,1) and fy(2,1). Foreach, circle the number below that is clos-est to your estimate. (4 points)123012f = 3f = 2f = 1f = 0f =−1f =−2f =−3fx(2,1): −3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3fy(2,1): −3 −2.5 −2 −1.5 −1 −0. 5 0 0.5 1 1.5 2 2.5 36. Suppose a function f : R2→ R has f (1,2) = 5,∂ f∂x(1,2) =−2, and∂ f∂y(1,2) = 3. Use linear approxima-tion to estimate f (1.1,1.9). (3 points)f (1.1,1.9) ≈7. Supposef (x, y): R2→ R, x(s, t): R2→ R, and y(s,t ): R2→ Rare functions. Let F (s, t) = f�x(s, t), y(s, t)�be their composition.(a) Write the formula for∂F∂susing the Chain Rule. (1 points)∂F∂s=(b) Suppose x(s, t) = 2s + t and y(s, t) = s2t − 1 and that f (x, y)has the table of values and partial derivatives shown atright. Compute∂F∂s(2,1). (4 points)(x, y) f (x, y)∂ f∂x∂ f∂y(2,3) 0 3 6(2,1) 2 -2 -1(2,4) 3 4 6(5,3) 1 3 5∂F∂s(2,1) =Extra credit problem: Let E : R2→ R be given by E(x, y) = 2x + y2.Findaδ > 0 so that |E(h)|<0.01 for allh = (x, y) with |h|<δ. Carefully justify why the δ you provide is good enough. (2 points)Scratch space below and on
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