exam1_sols - M ath 2 50 Y our N ame E xam 1 Your S ignature...

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Math 250 Exam 1 Spring 2007 Your Name Your Signature I 1< t Y [ ~ Student ID • Give your answers in exact form. Do not give decimal approximations. Graphing calculators are not allowed. • In order to receive credit, you must show your work. Do not do computations in your head. Instead, write them out on the exam paper. Place a box around I YOUR FINAL ANSWER I to each question. If you need more room, use the backs of the pages and indicate to the reader that you have done so. Problem Total Points Score 1 11 2 5 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 7 11 7 Total 100
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Math 250, Spring 2007 Exam 1 Page 1 of 4 1. [11 points total] Answer each of the following as true (T) or false (F). Do not justify your answers. [£] The speed of the curve r =< cos t, sin t, 3t > is < - sin t, cos t, 3 >. OJ The curvature of a circle of radius 2007 is 1/2007. OJ The gradient of f(x, y) is normal to the level curves of f. m If (1,1). is a critical point for .the fu?ction !(:J;,'j/) then (1, 1) is also a <;:.it~c~ P?i?t +lK forthefunctlOng(x,Y)=f(x~y2). (~t( I}I' _ (t~(\ll) i.fll~J l), - l ; VI ~)) tV 3 t' , \ \ : ( 1-)( (I I I ) 2. X I (1 l' ,,) 21 ) ~ l VI ..:>, [1] The function u(x, t) = ~2 + t satisfies the equation Ut = U xx (heat equation). [1] If u is a unit vector tangent at (x, y) to a, level curve of f(x, y) then t~ directional derivative satisfies Duf(x,y) = o. 0 V\ - .(<:'tJ t. I l>A{.te- I ), \J- -e- -A.!\ l t 'v7 d ~ e) ,,-.;J \ (. h .:".-01\ S e { \,. (' ... - JI' [ i J (~1H .... ., ~ b 1 (,II (' f"1 ) [£] ::. fx(x, y) = Jy(x, y) for ~l x, y, ~hen i(x: y) is_a constant. I .- V- ( [, f ,~ .;): '\-t f ~ ~ \. 'f' "1> ._ ..,. t 1 m If in spherical coordinates a point is given by (p, (), ¢) (2, 7r /2, 7r /2), then its rect- angular coordinates are (x,y,z) = (0,2,0).
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