234-oh-07sp01

234-oh-07sp01 - Math 234 Midterm I Instructor: Yong~Geun Oh...

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Unformatted text preview: Math 234 Midterm I Instructor: Yong~Geun Oh No calculator allowed. Write detailed work to obtain full credits. Your Name: Please circle your TA’S name : Weiyong He Seth Meyer J ie Ling Score 1 P roblem / pt _ P1/20 . P2 / 25 P3/15 - - P4/20 P5/20 Total / 100 1. (5 points each) There is no connectiOn between sub-problems in this problem. No partial credits for these problems! (a) Let Qbe the point (1, 2, 0). Find the point P such that 315462 = (3, —3, 3). (b) Change (—\/§, fiflfi) from Cartesian to spherical coordinates. (We require that p20,0§9<27r,0-S¢£7T.) (c) Find the unit tangent vector of the parameterized curve 7705) = (t,et,sin t) at t = O. - ((1) Find the numbers 0 such that the two vectors 40'?— 8} and 375+ (c + are perpendicular. 2. There is no connection between sub-problems in this problem. No partial credits for these problems! (a) (5 pts) Let F(x,y) = 33211. Find the values of PAL-2), Fy(2, ~3). (b) (10 pts) Consider the function f (x,y,z) = 3:23; — xyz + acyz. Find the linear approximation of f near the point (2,1,1). (c) (5 pts) Find the directional derivative of f at the point 13’ in the direction of ('1’ : f(a:,y) = (172 —- 2503/ + 2312, 13': (1,2),6 = (1, ~2). (d) (5 pts) In What direction 21’ does f (3:, y) = 1 -— 2332 + 3563/ increase most rapidly at 13’ = (1, 2)? ' O'l 3. (15 points) Find the equation of the plane through .(—1, —2, 3) and perpendicular to both the planes x—3y+2z=100, 2x—2y—z=—49. 4. (10 points each) You have to provide the reasons and justify your conclusion. (a) Check if the following function is continuous at (0,0) : _ 1 I for($,y)=( ftp: _ {tan($2+y2) for 7L4 x2+y2 (b) Check if the limit lim(z,y)_,(0,0) gig: exists. \1 5. Consider the points ‘A = (2,1,0), 0 = (1,7,2), E = (2, 10, 3), F = (3,5,0) .4 .4 and the two vectors 17 = A0, (U = E F, (a) (5 pts) Find 25 x 17. ’(5 pts) Find the area of the parallelogram with the above two vectors 17 and if) as the adjacent sides. (c) (10 pts) Find the equation of the line through (1, +1, 1) that is perpendicular to both of the above vectors. ...
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234-oh-07sp01 - Math 234 Midterm I Instructor: Yong~Geun Oh...

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