thq_03 - Use T.C ble Calculus Qa-11 Q AOL/002/01 Q 1var1a...

Info icon This preview shows pages 1–8. Sign up to view the full content.

View Full Document Right Arrow Icon
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
Image of page 3

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 4
Image of page 5

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 6
Image of page 7

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 8
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Use T.C. ble Calculus Qa-11.% ., Q. AOL/002/01 Q 1var1a 16s MATH 2451 thquB Mult McCary Exam Course Due Date Instructor (Last Name) S’mgge E .,N Mm M ®@®®@® veeeeeeeweeeeeeseeessss fw. [email protected]@®w®®®®®®®©®®®©®®®®©®®®. .fi4©3U©®®®®®©®®©®®®©Q®®®®®®@®@ @®@®®®®@ W‘Peeeeeseeesseseeseeeemesseeee /[email protected]@@@@@@[email protected] cs®aeeeesasesen [email protected]fi@®ssessseeo®®sss©®s®aeas axesgéaé as. see es ésssseees see 0 @@@®®®®@ tw®s®efi ®@®@@®@¢@ . Net ID 2016-01-21 14:10 0. AOL/002/02 O 1. (10 points) (a) The inner product {i - I; on R” induces the norm on R”, as “63” = V62 - a , as well as the notion of angles betwi vectors, as a: - 5 = “an H6” 003(9). i. Find a unit vector in the direction of (‘1'. M: a s s a 7‘” 1F; vi“ ii. Find a vector b % U such that 63 J. b: '3’ 0‘1' 1 1 5" . __ __ ’51; /L’ ‘ Tile 7 _:_, 7‘ . 59 lum— ;- ' fie-3 Q (b) In later rnath courses, functions themselves will be regarded a vectors. Here is an inner product on functions: = 9(3)) dm. This inner product induces a norm on functions: = 1/ (fix), . D (log/€91 ,‘J’K '4 391:? T _.‘.‘..a«-.f:"“"'3r'~1 ‘ 1. Find a un1t vector in the direction of cos(3:). ii. Show that cos(m) J. sin[m]. 0y NI , .--_ ‘. . .J, x“ nmtlfflfié 7 ; l "—1: .Er‘ ._ 3 1 (a " M: 2”; I: w J / rm Ti Mirth: O — . flan" M 4-44 L 3 _-——r _—l— ! / {rat—u '_'r- f [‘1'— wig-"wt 0 , Ungraded: It is interesting to think about what it might mean for two functions to be orthogonal to each other. What would it mean for two functions to be at an angle to each other, say 7:/3'? .LGraderOnly . O® . O. AOL/OO2/OL Q ‘ \10 points) (a) Use the inner product {1' - 5 o and the induced norm to show the parallelogram law. "I: f fi/fli '4’" 2 _. 2 _. 2 ‘ 7 w :2 (“an + ) ‘ #2 .. Ha+fll+“a—b ;» _ , ,.1- “<7 g'zarr-g'f“; xii?” [QA’W‘x a; 211% M" 0? MI W M ‘“ V5- 7)? 11%? .55] gr" H Diszgorumé A _ f‘: ‘ 5 ‘ ‘ r! - "‘ b, .. ’ ' 3 ~/- ’. '1’ "in _- 7-? I -' j :— I'T f f_' LEE““”.::C'-~l" LL 3, U g-Vl ngi/{Aw 3‘ bit/“‘7, fl”; 0F £2 a .. AOL/OO2/O4 . (b) Now use the inner‘product and the parallelogram law to show the polarization identity. _. 1 _.2- -2 fi-bwlllwl ale-bl] f ’1 - P ' -r" a- x’TfiJUfiiflaS‘j :2; 77' =-— F’W‘b'fi‘r W7) lbs: L J 'v T: WI¢W g 1W1tear/tailleiflrrrlflrllrllhlifi W s creme) lN lrywaln..-‘ L; {NM/Q; VFEQUQLQfi’ 1; WEI" filing?" l‘vffiléifl-"i. Hews-giw 1; Wear: {R2 err: g. E "is [114";1/7 Hg {antigen '5? 31.73;. ;:, :lméé "We lI-‘=1‘i"f53*i~ EMT" NET F Jr” 1"! 'v’L-jr'; in: Q l5 INST-l 17.;1327if3.“ 97 AN INNM, PE?)- WET“. Ungraded: Why are these called parallelogram law and polarization identity? Draw a picture and try to figure it out! Note that one of the most common uses of these is actually the reverse of this question: if you have a norm which was not defined by an inner product then you can check if the norm can be induced by an inner product by checking the parallelogram law; if the parallelogram law holds then you can use the polarization identity to define/ compute the inner product which induces your norm. l Grader Only J, . O® 0 .. AOL/OO2/05 . \10 points) The indicated portion of the ellipse has the following parametrization fit). ~—* $13- 1254 g"- t: e w 3T: 35:: o 3 t 4 7» 9 pet): [2:13] on wan/2] 0 ’7” ‘99,; of}; m: k, i Additionally, the dashed ellipse can be understood as the zero level set of f = (93/82 + (ii/2)2 —'1. (a) Show that the image of 15' t) is art of the ellipse above by verifying (f o = 0. fr, N‘- “t f I CR?” ' 7- &0 WW W a, j / if p.— . . ‘l _. - ;\ «ff-“r ~ : Jr a *1 I ‘ ’s Misti/l (b) Find a parametrization 97(15), including domain, for the indicated portion of the ellipse below. Note the orientation (the arrow)! 76’ 7+ 7’ 5X" '7' E W X7 (K Cir-“£69 ‘ -- I" .1123 ‘ 6103:}91“ mfg—('9 7.5“? 0x5“?- 2/ l0?!“ :— J, Grader Only ( .. AOL/OO2/06 . 4. (10 points) Let f n [m y] _;] That is, fie) fiTAi,for£=[$]andA=[2 1]. y 7 —3 y (a) Carry—out and simplify the matrix multiplication to obtain an expression for f as a polynomial in ac and y. 'jxi’, 3:: .‘7 .r be? _ (b) The level sets of f Will be conic sections (which you need to review). What type of conic sections will they be, and why? (This is an algebraic question, you’re not being asked to draw something.) 2r><s+€8zy~e7e 41:0; 'X$4’--’ii‘=-x’ 7 ’94 xfi _ ? ‘F’ a» r" ‘7? ux/t 5:». INT’é/ggééfGi—lwe’y x JFAHYJrig/o i’ffi’r’ ' v -- - / Liwéé 3: PX} ~— "“ 1—33 ._ flagella”; (C) Observe that-El A 75 ALT/Find a difierent matrix Q which can also fine f such that Q =- QT. That 151 find a matrix such that 1°55): “TQi‘ and Q = qT_ !/ t J i. I”. __ _, ‘ _ :— 0:1/ ‘ ," g ,t r 7 ,w -7- ,H k a if; a :7 :4 3"”: T ‘ e " — - i’“ ,’ 51 L} 9"“ __ ii) ‘I 1 1 , f _/ J, I fl 1: {'2‘ ' )f/‘f f, l l , F) 1 [Ki-'7‘ 7 , _ it“ i016 r In 1 wk i 5/" \_.—‘) _._./ r/ [thi- ' % J 1 l f" ‘ 1‘2"", -‘ J I f“ '\ _ _/_ 'z x QIX/EU’ iv A; I; n 1' A r! ”‘ f V e, .. r i 7» o [7/ * ~‘ “ 3“ I Ob< «i 2 15:4” to. "H The function f is known as a quadratic form, a kind of generalization of quadratics from grade school. i Grader Only l o o .. AOL/002/07 . ‘10 points) Ungraded questions Which you should be familiar with. CH/ZCLé/CLLIWf’y’é 1’? 1:59.114 .- “\tfli’fii’vb‘vz’i / .— fi— '=— // ' ‘ 1 In ‘1:- m _A)=+(7_ b)—>:rr¢ if? I @142: F: _ ‘:__ ,1 2. Given f( ) = 4 — 33: + 2y, plot level sets c = —2, —1,0, 1,2. J 19' 1. Able to plot any conic sgction. a 3. Given f = 3:2 + my. plot level sets c = -2, “1,0, 1, 2. 7 4. Given f = m2 + yg, plot level sets c = —1,0,1,2. 5. Given f = W, plot level sets c= —1,0, 1, 2. 6. Given f = + lyl, plot level sets c = —2,—1,0,1,2. 7. Given f = maniac], lyl), plot level sets c = —2, —1,0,1,2. 8. Describe the surface in R3: 4x2 + y? = 16. 9. Describe the surface in R3: 22 = y2 + 4. 10. Describe the surface in R3: 4w? — 33/2 + 2 22 = O. 11. Describe the surface in spherical coordinates: p = 6. 12. Describe the surface in spherical coordinates: p = 6 and = 0. 13. Describe the surface in spherical coordinates: p = 6 and q‘; = 1?/6. 14. Describe the surface in spherical coordinates: p = 6 and 6 = 0. 15. Describe the surface in spherical coordinates: p = 6 and 6 = 7r/t3. 16. Describe the surface in spherical coordinates: 9 = 77/3 and qt 2 7T/6. 17. Describe the surface in cylindrical coordinates: r : 5 and 9 = D. 18. Describe the-surface in cylindrical coordinates: r = 5 and 9 = 7r/'.3. 19. Describe the surface in cylindrical coordinates: 'r = 5. 20. Describe the surface in cylindrical coordinates: 6 : TF/s. And once you are comfortable will all of these equalities. begin thinking about What regions / surfaces would result from inequalities, e.g. SSpSGandUfiHSfiT/BandW/sgquW/z O. AOL/OO2/08 . This page won’t be graded. ...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern