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**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
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(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
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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.
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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
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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). ~—* $13- 1254 g"-
t: e w 3T: 35:: o 3 t 4 7» 9 pet): [2:13] on wan/2] 0 ’7” ‘99,; of}; m: k, i Additionally, the dashed ellipse can be understood as the zero level set of f = (93/82 + (ii/2)2 —'1. (a) Show that the image of 15' t) is art of the ellipse above by verifying (f o = 0.
fr, N‘- “t f I CR?” ' 7-
&0 WW W a,
j / if p.—
. . ‘l _. - ;\ «ff-“r ~ : Jr a *1 I ‘ ’s Misti/l (b) Find a parametrization 97(15), including domain, for the indicated portion of the ellipse below. Note the orientation
(the arrow)! 76’ 7+ 7’
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0x5“?- 2/ l0?!“ :— J, Grader Only
( .. AOL/OO2/06 . 4. (10 points) Let f n [m y] _;] That is, ﬁe) ﬁTAi,for£=[$]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"“
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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. ...

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