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MATH 251
Simon Fraser University Department of Mathematics Burnaby Campus Midterm Exam No. 1 Feb. 9, 2005 Instructor: Dr. Akeel Shah
DO NOT LIFT UP THE COVER PAGE UNTIL INSTRUCTED. Once the test begins, please check that all pages are intact. Answer all five questions, clearly showing your working. Each question is worth 8 marks. If you run out of space use the extra space on the back of each page, CLEARLY INDICATING WHERE THE SOLUTION CONTINUES AND THE NUMBER OF THE QUESTION THAT YOU ARE ANSWERING. Ordinary scientific calculators are allowed. The duration of the test is 40 minutes.
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1a) (4 marks) Consider the plane surface x − 2 y + z = 3. Find equations for the intersection of this surface with each of the coordinate planes.
1b) (4 marks) Consider the surface of revolution generated by revolving the curve y = x^2 around the y-axis. By examining the evolution of a point P (x 0 , y 0 ) on the curve, determine the equation for the surface of revolution.
2c) (3 marks) Evaluate the principle unit normal to C at the point (x, y) = (1, 2).
2d) (2 marks) Evaluate the curvature of C at the point (x, y) = (1, 2).
3a) Consider the vectors ~u = 2i − j + 4k, ~v = i − 3 j − 2 k and w~ = i + 6j + k.
(i) (1 mark) Are ~u and w~ perpendicular or parallel? Explain your answer.
(ii) (3 marks) Find two vectors which are perpendicular to both ~u and ~v.
4a) (4 marks) Find both the cylindrical and spherical coordinates of the point (0, 0 , −1), given in rectangular Cartesian coordinates. Note: Rectangular coordinates are (x, y, z), cylindrical are (r, θ, z) and spherical are (ρ, φ, θ).
4b) (4 marks) Convert the equation 2x + y − z = 4 into both cylindrical and spherical coordinates.
5b) Let f (x, y) = x^2 + 2y^2.
(i) (3 marks) Describe and sketch some typical level curves of f (x, y).
(ii) (2 marks) Using your previous answer sketch the graph of f (x, y), indicating some typical contour curves.
