Calculus III MATH 263 1L February 18, 2021 NAZARBAYEV UNIVERSITY Department of Mathematics Alejandro J. Castro Castilla Name: MIDTERM EXAM #1 20 points (1) If u = hu1 , u2 , u3 i, v = hv1 , v2 , v3 i and w = hw1 , w2 , w3 i are mutually perpendicular vectors, show that u × (v × w) = 0. 20 points (2) a) Find parametric equations for the intersection of the planes 2x+y−z = 3 and x+2y+z = 3. b) Find the acute angle between the two planes in part a). 20 points (3) Find an equation of the surface consisting of all points that are equidistant from the point (0, 0, 1) and the plane z = −1. Identify and sketch such surface. 20 points (4) a) Sketch the sphere x2 + y 2 + z 2 = 1 and the elliptic cylinder x2 + 2z 2 = 1. Try to guess the curves resulting of their intersection. b) Find parametrizations of the two intersection curves. Hint: take z = 2−1/2 sin t. c) Compute the total length of these intersection curves. 20 points (5) Find the curvature at t = 0 of the vector-valued function r(t) = 2t i + e2t j + e−2t k.