MA 242 Sample Test 1 Model Answer Last Name First Name Student ID Seat Code 1 1 5 points For each of the followings show the definition for the concept and the theorem for computation For a b R3 a a b b a b Definition a b a b cos where is the angle between a and b Theorem a b a1b1 a2b2 a3b3 Definition a b cid 26 magnitude Theorem a b ha2b3 a3b2 a3b1 a1b3 a1b2 a2b1i a b sin RHR direction where is the angle between a and b For f R1 R3 a d f t t f t d dt f Definition d Theorem a f dt Definition R b Theorem R b t dt df3 dt f lim t 0 dt df2 dt f D df1 dtE t 0Pt a f dt DR b a f1dt R b a f dt lim b R b f t t a f2dt R b a f3dtE For f R3 R1 a f x2 Definition f Theorem f x2 x2 lim x2 0 df dx2 b Duf Definition Duf lim t 0 Theorem Duf f u f x t u f x t Theorem a b iff a b 0 Work a b h 5 3 7i h6 8 2i 40 6 0 Not orthogonal 1 f x1 x2 x2 x3 f x1 x2 x3 x2 where x2 is treated as a variable and x1 and x3 are treated as constants 2 1 points Let a h 5 3 7i and b h6 8 2i Determine whether a b or not i e a and b are orthogonal 3 1 points Find the area A of the parallelogram determined by a h 3 2 4i and b h 1 2 1i 4 1 points Find the volume V of the parallelepiped determined by a h6 3 1i b h0 1 2i and c h 4 2 5i Theorem A a b Work h 10 1 8i a b h 3 2 4i h 1 2 1i a b q 10 2 1 2 8 2 165 Theorem V a b c Work b c h0 1 2i h 4 2 5i h 9 8 4i a b c h6 3 1i h 9 8 4i 82 V 82 82 0 is given by r 0 h0 1 0i Theorem r t R v t dt Work 5 1 points Find the position r t of a rocket whose velocity is given by v t ht cos t eti and whose position at time 6 1 points Let f x1 x2 x3 1x4 2 Find the maximum slope of f at 1 1 Find the maximum direction 2 c1 sin t c2 et c3 cid 29 r t Z cid 10 t cos t et cid 11 dt cid 28 t2 r 0 cid 10 0 c1 0 c2 1 c3 cid 11 cid 10 c1 c2 1 c3 cid 11 h0 1 0i c 0 1 1 r t cid 28 t2 2 sin t 1 et 1 cid 29 and maximum direction f f Theorem maximum slope f Work f h3x2 1x4 f 1 1 h3 4i 2 4x3 1x3 2i maximum slope h3 4i q 3 2 4 2 5 5 4 5i maximum direction h3 4i h3 4i h3 4i5 h 3 2 7 1 points Let y x2 1x2 where x1 t1t2 and x2 t2 1 t2 2 Find y t2 Theorem y Work y t2 t2 y x1 x1 t2 y x2 x2 t2 2x1x2 t1 cid 0 x2 1 cid 1 2t2 8 1 points Let f 3x2 2 2x1x3 0 Find x3 x1 1 x3 f x1 f x3 Theorem x3 Work x3 x1 x1 6x1 2x3 2x1 9 2 points Find all real x1 x2 satisfying the following system of non linear equations Organize the solving process in terms of F factor split and S solve substitute 4x1 4 2x1 0 6 4 0 1 2x2 x2 2 4 0 4x1 4 3x1 0 3 2 1 2x2 2 4 0 x2 x1 4 3 2 16 2x2 2 4 0 S S S NONE 4x1 4 2x1 0 6x2 4x2 0 1 2x2 2 4 0 x2 4x1 4 2x1 0 6x2 4x2 0 1 2x2 2 4 0 x2 4x1 4 2x1 0 x2 0 1 2x2 x2 2 4 0 4x1 4 2x1 0 x2 0 1 4 0 x2 F 4x1 4 2x1 0 x2 0 x1 2 0 4 4 0 x2 0 x1 2 S F S S 1 x2 0 x1 2 2 0 1 1 2 1 2 4x1 4 2x1 0 x2 0 x1 2 0 S 0 0 cid 18 1 x1x2 S 12 4 0 x2 0 x1 2 3 x2 0 x1 2 2 0 3 2 5x2 3 0 cid 19 5 Classify them into local maximum local minimum or saddle Theorem Let D f11f22 f 2 12 Then a If D 0 and f11 0 then local min b If D 0 and f11 0 then local max c If D 0 d If D 0 then saddle then don t know 3 10 2 points Let f x1 x2 2x3 1 x2 2 It is known that it has the following critical points 11 2 5 points Let f x1 x2 2x3 1 x1x2 2 5x2 1 x2 2 and g x1 x2 x2 1 x2 2 4 a Write down the standard form equations for the critical points of f subject to g 0 Work f11 12x1 10 f22 2x1 2 f12 2x2 D 12x1 10 2x1 2 2x2 2 p f22 0 0 2 f11 1 2 2 1 2 2 10 0 0 3 0 cid 1 10 4 cid 0 5 3 D f12 16 4 4 16 0 20 0 40 3 type saddle saddle local min local max Theorem Every critical point x satisfies cid 26 5f 5 g Work g 0 2 10x1 2x1x2 2x2 cid 11 2 10x1 2x1 6x2 2x1x2 2x2 2x2 1 x2 x2 2 4 0 1 x2 5f cid 10 6x2 1 x2 5g h2x1 2x2i Standard form 2 0 2 0 1 x2 6x2 2x1x2 2x2 2x2 0 1 x2 x2 2 10x1 2x1 0 2 4 0 3 2 3 4 2 2 3 4 2 3 local type max don t know neither global type max x 2 0 2 0 3 cid 17 cid 16 2 3 4 2 3 cid 17 cid 16 2 3 4 2 f 36 4 27 2 8 min 76 27 2 8 min 76 min min 4 b Classify the critical points It is known that the critical points the solutions of the equations are the followings Theorem Assume that the solution set of g 0 is bounded and connected Let x be a critical point a If f x is minimum then x is both locally and globally minimum b If f x is maximum then x is both locally and globally maximum c Else Work Note that the solution set of g is bounded and connected Thus from …