_t l A correct ans_wert unsupported by calculatl_ons_t e_pl_a_____nat_llont or algebral_c worh l _ _lll)(_��������Math 237_ Name (Print): _: _�Fall 2013 Student ID;�Midterm 1 5ection Number:�October 2, 2013 (Morning) Teaching Assistant:�Time Limit: 50 minutes Signature:��This exam contains 7 pages (including this cover page) and 5 problems_ Check to see if any pages�are missing. Enter all _e_uested information on the top of this page, and put your initials on the ,�top of every page, in case the pages become separated_ You are allowed to tahe one doubled-sided�8.5 inch x 11 inch sheet of notes ìnto the exam.�Do no' t give numerical approximations to quantities such as sin 5, _, or _. However, you should ,�simplify cos -_4 = _/2, e0 = 1, and so on.�The following rules apply ; .�l Show your worh, in a reasonably neat and coherent way, in the space provided. All an-�swers must be _usti_ed by valid mathematical reasonjng, including the evaluation�of de_nite and inde_nite integrals. To receive full credit on a problem, you must show�enough worh so that your solution can be _ollowed by someone without a calculator. ,,�l Mysterious or unsupported answers will not receive _ull credit. Your worh should�be mathematically correct and carefully and legibly written.��will receive no credit; an incorrect answer supported by substantially correct calculations�and explanations will receive partial credit.�l Full credit will be given only for worh that is presented neatly and logically; worh scattered�all over the page wîthout a clear ordering will receive from little to no credit.�������1 20 pts�\ 2 20 pts ,�3 25 pts�_ 20 pts�5 35 pts�TOTAL 1 2O pts�

______________t _zt _________) _ _ _ __ _ _ __ _ g_ _ _ _ _ )))__t�����Math 2374 Fall 2013 Midterm 1 - Page 2 of 7 October 2, 2013 (Morning)�' 1. (20 points) (a-10pts) Find the equation of the plane that contains the line e(t) = (O, 2, _1) +�' '' t(2, 1, -3) and is orthogonal to the plane _ + y _ _ + 4 = O. Your answer should be of the form�__+By+C5+D = O.�, __ nri ___ _ _ __ _�_ x _�5 __ _�� J _ :_ _ _ ( _ 0 __ /e___) _ _ '�_n _ _ __ _ _ ' l��__ _ ___ __ e + _ __ 0 __���_ __ _ __ + 3 _ _����(b-_0pts) Find the constant(s) h such that the volume of the parallelepiped spanned by the�vectors a = (1, O, O), b _ (O, k, 4) and c _ (-1, O, h) ìs equal to 4.�_ _ _ __�_ _ __ _u_ b_ _ c _ _0 _ _ _ _ 0�_ _ _ _��S 0 ___ __ _ r_, _ __, _ _' Z ,�S�e _ _____ __ _ __ g __ ___ _ _ _ _�0 _ _.�

___g_tt ___:__te___0qqD0_____g_0 ___g__ _ ) _ ____ _tc_______+___ ___00___00c0________ (___(_ _)t _ __ 3 f _ __ltl_l_)_lll_t���Math 237_ Fall 2013 Midterm 1 - Page 3 of 7 October 2, 2013 (Morning) ,�2. (2o points) In this question, consider the function J(_, y) _ __4 + 2y_ _ _ , _ : !'!�(a-15pt,s) Find an equation of the tangent plane to the graph z = J(_, y) at _ = _1 and y _ 2. ,�Write your answer in the form z _ A_ + By + C. ,�/__ _ J 0 (4._ _ ) __ __ _ _ ,0 (_ 1,. __ _ _ ___ _�_ _ ' _ __4+._ _ _ _ _��_ __ 7 c ) _4 __ _ _ i,�' g '0 '_/ _ __ : _ _ ' _ _ ^ '' '_T ''�_ ___ +n _ '��_ __ (_,_,_,_ _ _ ,���S'�e __ 3 _ _(____ _ __ ____�_ _��__ _ __ _ _ _ _ _e,'�_ _ _ _J _;�(_-5pts) Estimate J(_O.99, 1.99) using a linear approxiamtion.���_ gg D�_ _ ' _ / _ _ _ 0 _�S "�__ 3 _ __ _ _ o 0_) _ _ ( _ o_ '�3 3 "���_ g _ _, __ __ _ s_ _�_ _ _ ' ,�

__/__tt____0_\___g__: (g____ _ __ ) : _A 1 __ ) _?g__ _ _____ ____xt___________ _ / _ __ r __ __���Math 2374 Fall 2013 Midterm 1 - Page 4 of 7 October 2, 2013 (Morning)�is a functjon whose matrix of partial derivatives at ( 1, _ , 1) is given by��Find the matrix of partial derivatives 0f the function J o g at the point ( 1, 1 , 1 , O).�_ _ _ _ _�' '__ _ __, / _/ _ _ _ e _ ,�_ 0 _ __e��, C _, _, __ / _ _ __ __/ _/ _�S�_ '__ ( __0 __g _ _ _) _ __D(_ _ _) _ _' (g _ _�i / / _ _> /' / / / / '�_ _ _ _�_ _ _ _�_ 0 _ _ _ _''_�_ _ 3 , __�_ _ __ _ ' _ __ _��, _ _ _ _ _ _�_ _ _ _ _ __ _�m_ , / / _�_ _ __ 3�

________c t _) __ _(__ _: ) ) __(____)_ Jc___ ____v))_:)__ )___ __ _g l)__llllttl����Math 2374 Fall 2013 Midter_ 1 - Page 5 of 7 October 2, 2013 (Morning)��_nts _n th_.s uest__on cons__der the ara_etr__2e_ curve c t _ 4t2 g sin t cos t _"�(a-_5pts) Give a para_etri2ation of the tangent line a_t the point ,t = _/2. _,,4.,,_ ,,,,_,M,,,,_�_ _ _ "�, N _ '�_ _ _ _ + _ _ _ __g _�_�_ _ __ ,�__ _/ _, _e + _ /_ " '_���_ ___ __ __ + __(___ _ g _�_ "_ / '_ _ ;����__ ) _'_, _ __ c_) _ _n�__ __ / _ _ '�(b-5pts) Suppose that a particle follows the path curve c(t) until it _ies off on a tangent at ,�t _ _/æ with no _urther _orces acting on it. Where is the particle at time t = _? '_���_ _ D _ __ , _-0 _ _0 : _ __ _ _ _�a ^ ^" _ '���J ___ _ _ _ __ _ _ ' "'�t /����_ _ _ a ,_ _? _e __. _00 _ _�_ ^' _��_ _v_N ___ _ _ _ 0 , _ __ __ ,�_ / _ _�

_: & __gg________ _ __/ __ (_ )_: __t��Math 2374 Fall 2013 Midterm 1 - Page 6 of 7 October 2, 2013 (Morning)�(a-10pts) Find the directional derivative of J at the point (2, 1, 3) along the vector v = (1, 1, 1).�_,_,_ (_ _/ _ ,g) _ C _ _/ m_ _ _ , _ _�____ C _/ _/ __ __ ( ; , _ _, _�__:,,___,_____ _.,a ____0__ , _�Dv J c_) __ ____ C__, _,_ __j , ( __, _, _ _ __ _ ' _ _ + _ _, _�_�(b-10pts) Find the unit vector n along which J is decreasing the fastest at (2, 1, 3).�_ __D_ C&/_ 3�__ _ _C _, _, 3�_ __D_C. _/ _/,'__ _ _ 4, _ _ / b�,, _ \i __D_ (_, _/ g_ __ __ _ _ __ 4 + 3 _ __ 5 _�_ _ _ _ _ _ _�_ /�5G 5b�

___tgs_____t __l __()_( __00 ?J ?_?\ ; _ l ___a&_t ___ _ ___D0_ 0_______ _ _0_____\ ____l I_ll_t_)���Math 2374 Fall 20l3 Midterm 1 - Page 7 of 7 October 2, 2013 (Morning)�(c-15pts) Shetch the level surfaces _, _ ((_,y,z) e I_3 l J(_,y,z) = c) for c _ -1, c = O and _�c = l on three different _gures. Next to each _gure, precise the name of the corresponding�surface.��___0 ., ___ e_ __ , _ ___, __ A_ _ ,__a a��_ D __ o_��������� __ _ _ _, __D,: _�c _ _ _, _ m _' __ ' _ _ ' _ _ __0�_ _ J_ _ _�' ,_ o00 _ _ 0 ^ _���������_ ___0,__oD,, ,_�_ _ _ __ Ca 0�_ c __ __ ; _, ^m _ _'_ _� ,,, D a_ :_, _ _ , OD 0�, _ _ 0_ _ __�\ ' _�