MATH 201 Exam 2, Fall 2001

MATHEMATICS 201 SUMMER SEMESTER, 2000-01 Makeup Quiz II Time: 60 Minutes. Date: August 17, 2001. Instructor: Prof. A. Lyzzaik Answer the following questions: 1. (a) Express in polar form the rectangular equation x3 + y3 − 3axy = 0. (10 points) (b) Express in rectangular form the polar equation r2 cos(2θ) = 1. (10 points) 2. Consider the polar equation r2 = 8 cos(3θ). (a) Indicate the possible symmetries of the graph. (10 points) (b) Sketch the graph. (10 points) (c) Find the polar equation of the tangent line to the graph at the polar point whose θ = π/4. (10 points) 3. Consider the polar equations r = −6 cosθ and r = 2 − 2 cos θ. (a) Find the points of intersection of the two graphs. (10 points) (b) find the area of the region lying outside the graph of r = −6 cosθ and inside the graph of r = 2 − 2 cosθ. (10 points) 4. Consider in rectangular form the equation 9x2 + 4y2 − 36z2 = 1. (a) Write the equation in cylindrical and spherical coordinates. (10 points) (b) Sketch the graph of the equation by showing its traces with the rectangular planes. (10 points) 5. Identify the surfaces given in spherical and cylindrical coordinates, respectively, by ρ = 1 − cos φ and r = 1 − cos θ. (10 points) 1 www.amal-aub.org