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Spring 2007 - Yeap's Class - Practice Exam 1

# Spring 2007 - Yeap's Class - Practice Exam 1 - Math 20E...

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Unformatted text preview: Math 20E Practice Midterm Exam April 17, 2007 Name: ID: Section Hour: —— TA: Guidelines for the test: 0 No books, notes, or calculators are allowed. 0 You may leave answers in symbolic form, like m, unless they simplify further7 like \/§ 2 3, 60 = 1, or cos(37r/4) = —\/§/2. 0 Use the space provided. If necessary, write “see other side” and continue working on the back of the same sheet. 0 Circle your ﬁnal answers when relevant. 0 Show all steps in your solutions and make your reasoning clear. Answers with no explanation will receive no credit, even if they are correct. 0 You have 50 minutes. Question # Perfect Score (1) Suppose that m: y) = {‘Hi if (m) aé (0,0) 0 if (ray) = (0,0) (a) Compute ( gin?0 0) f ( y) if it exists. If not, justify Why it a: ,y —> does not. (ma—90> 0) f (yawn; 0) X3 : ﬂ; Dowax 9 Ag L/%ﬂﬂw [yy)—)(wo) JXL 3 ‘ —— (9x (>{%—9(00) 4°C“ % f HvMJ ,i I 3 ‘ » : Vivid» QL/a~e.. mar; ‘9)(2Z7L~ (b) Is f(3:,y) continuous at (0,0)? Explain. WM onwww £70) M going; 74% gnaw/5,) m ””360? \ 790]) ‘ 7170/50 ﬂV Zrk/ /w m\/ 4v“ 94%“! /-){0 17) (C) Is f (513,31) differentiable at (0,0)? Explain. wt); Oéfqﬁ’MKiﬂﬁ/M 619% “>907; CO‘VT/y/‘MW y (94).) a \ . > x [W ; jﬂ ,P /, \$71194ng m 23/7: J z: mAW/ Page 1 (2) Let f(:E, y) = exy sin(:c + y) (a) In what direction, starting at (0,7r/2), is f increasing the fastest? V71): <§chj §ﬁ£> (ye Gﬁfx+5l>+ éVjWL/Xﬁgy)) Xcz’xi/aLc’X—ég) + ex7m(x~eyj > a Vf (of/A) = <31}, <9>. (b) In What directions, starting at (0, 7r/ 2), is f changing at 50% of its maximum rate? . ff? Wax/‘mum W A! /V:Q (02%)] :— g (ﬁn: {<54 ,mtrs/ => )Let 005 )be a ﬂow line of F: vf with 6(0) )Calculate | | A O :1 \ [\3 v c/ le Gag/j DfC 6(a)) ”EA/[0) = ﬁﬁézEJZJQ) 0:69 26% ;2e 7&0“ng r2? 2%): NW2 be. Z/(0)= Fde}: 7560943)): (§,0> 75km (A). .71 - IL JinQQEm/EO :(ﬂ,0>ﬂ(>lo7v 7* Page 2 (3) Let f : R3 —> R3 be a give mapping and write f(\$, ya 2) = (“(3): ya Z): ”(:37 y) 2)) 111(1), y? 2)) Suppose that g : R3 ——> R3 is deﬁned by g(u,v,w)= (u—U,u+w,w+’u), andh:gof. (a) Write a formula for the derivative matrix 5h. (b) If u(x,y,z) = my, v(a:,y, z) = xz and w(a:,y, z) = yz, ﬁnd 8 \$9 0 f(17071) yXMJ of! #2 m 36% if) H: mm 3/ WW 6% X 23L: : I 0 I ’5 - l O l 1 9X 0 312 0 i j 0 t O i i W _ »~ I : O Page 3 ’35 1402’ (do 0 (4) (a) Suppose that ﬂuid is ﬂowing through space and the velocity of the ﬂuid is represented by the vector ﬁeld F(a:,y,z) = (avg/,2). Show that the ﬂuid is expanding and irrotational. ELF: /'/L/1L/ :3 70 if SVW (b) Suppose now that the ﬂow is disrupted in such a way that the velocity vector ﬁeld is now stable and equals —. _ (m,y,z) m / F(””’y’z)‘ll<x,y,z>|t3' ‘ (ﬁy’vz‘f/Z 0%?» Is the ﬂuid expanding or compressing? 2 ’/ E (Xz+g2+2)%_ X "z“ﬂixéxf‘ﬁzﬁz Z— ” F * “Ejzgzﬁzﬁ j. r [/2 Q ’2 3 (xiii—+21) gég ‘7 #2 —3xZ——3d"l—32 (go+v?+£33 Page 4 ...
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Spring 2007 - Yeap's Class - Practice Exam 1 - Math 20E...

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