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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 subproblems 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,0S¢£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 subproblems in this problem. No partial credits
for these problems! (a) (5 pts) Let F(x,y) = 33211. Find the values of PAL2), 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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 Spring '08
 DICKEY
 Multivariable Calculus

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