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MATH 137 Multivariable Calculus Solutions to Midterm 1.

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Multivariable Calculus Fall 2018 Solutions to Midterm 1 1. Find an equation for the plane containing the points (a; 0; 0), (0; b; 0), and (0; 0; c), where a; b; c 2 R. Assume that at least two of a; b; c are nonzero. Here you are asked to give a single equation that describes the plane, not a set of parametric equations of the plane. Two vectors that lie in the plane are Hence a vector normal to the plane is = hbc; ac; abi Using the point (a; 0; 0), we construct the equation of the plane: Argue that lim (x;y)!(0;0) f(x; y) does not exist. Along the path y = 0 to the origin, we have lim Along the path x = 0 to the origin, we have Since the limits along these paths differ, lim f(x; y) does not exist. 3. Our hero, the Dude, Jeffrey Lebowski, is traveling through space when he hits a wormhole and is hurtled into R5. His spaceship’s clock resets itself to 0 and his navigational computer says that he is at the point (x1; x2; x3; x4; x5) = (2; 0; 1; -3; 5). One hour later the Dude is at the point (-1; 4; 1; 2; 0). The computer confirms that he travels a straight-line path. (a) Using time t (measured in hours with t = 0 corresponding to the Dude’s entry into R5) as the parameter, give a set of parametric equations for the Dude’s path. The appropriate direction vector for the line is Working with the point (2; 0; 1; -3; 5), we consequently obtain the parametric equations (b) The Nihilists have set up an energy field along the hyperplane x2 - x1 = 19. When will the Dude reach it? At what point of R5 does he pierce the energy field? Since x1 = 2 - 3t and x2 = 4t, we have It follows that So the point at which the Dude pierces the energy field is (-7; 12; 1; 12; -10). 4. (a) Find an equation of the tangent plane to the paraboloid z = x2 + y2 at the point (-3; 4; 25). An equation is given by z = f(-3; 4) + fx(-3; 4)(x - (-3)) + fy(-3; 4)(y - 4) where f(x; y) = x2 + y2. We have So the desired equation is (b) The equation xy2 + yz2 + zx2 = 1 defines z implicitly as a function of x and y. Deterine @z Taking the partial derivative of both sides of the equation with respect to x, we get 5. (a) Express 2i + 5j as a scalar multiple of i + 2j plus a vector perpendicular to the latter vector. The scalar multiple of h1; 2i that we seek is Thus the vector perpendicular to this must be (b) Find the area of the parallelogram that has the vectors u = j and v = j + k as adjacent sides. The area is 6. If a; b; c; d are four nonzero coplanar vectors in R3 (meaning that the four vectors may be represented by arrows that all lie in the same plane), show that (a × b) × (c × d) = 0. (Hint: Use geometry and nothing messy.) Both a × b and c × d are normal vectors to the same plane. Hence they must be parallel vectors. The cross product of parallel vectors is 0.

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