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**Unformatted text preview: **Bonus Total FINAL EXAM
Math 51, Spring 2004. You have 3 hours. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING
TO RECEIVE CREDIT Good luck! Name g6 IU‘I‘I W5 ID number (/30 points)
{/30 points)
{/30 points)
(/30 points)
(/40 ooints)
{/40 points) ( / 20 points) ( / 200 points) “On my honor, I have neither given nor
received any aid on this examination. I
have furthermore abided by all other
aspects of the honor code with respect to
this examination.” Signature: Circle your TA’s name:
Brett Parker (2 and 6)
Chad Groft (3 and 7) Joe Blitzstein (4 and 8) Ryan Vinroot (ACE) Circle your section meeting time: 11:00am 1: 15pm 7pm 1. Compute the following partial derivatives:
3x 2 - a? (a) 3% (x3y3e312—Z) : K3 e sthIad) Ln (sin (Ya—1)) 5$5(:lj2esiilllb'Z—l)111(sir1(1,,'z—1))) :- e (c) 8221,223er —. g3; ( ><3 exfx)
= (w)
z + (4X3) ex‘f : (qu -l' Chg) exy 2. The function f : R2 —> R1 is deﬁned to be zero at the origin; and for all other points, f is deﬁned with the formula
f = rsin(36) where 7" and 0 are deﬁned in the usual manner (7' is the length of the vector and 6
is the angle, going counterclockwise from the positive part of the :c—axis, to the vector .) (a) Compute (directly from the deﬁnition) an expression for Dnﬂﬁ) be“? = lzm Mﬁl v) _ HTS.) H
<1
8‘.
7
{I}
9/ (b) Compute the Speciﬁc vector derivatives D-e-1f(0) and D lipﬁﬁ ﬂﬁ’)
2 2 (c) Using these results, explain how you know that the function f cannot be diﬁeren—
tiable at the origin. SUWBSQ ‘9 “Wt gx‘gomiuue; yum “we "J'Wo thPu‘)m‘J-Im5
AW Wng in” u; #4 bin/‘6 (a) .— O ;W\ﬁ)ke§ I
‘yLere‘LWe bv $66 = 0 $01- an
emit to We am PM; (A. go ‘g: WM; 1): Angew‘lme‘e. (I . Consider the function 9 given by (3.) Find a general expression for the J acobian matrix of g in terms of 3:, y, and z. 3?. Am 32, y X+ 2— V
3‘1 be __
"' 2
3327 a 31 9 4x — 2 2/ a
3" 3‘7 a}
1
(b) Suppose we are at the point 2 in the domain, and that we are moving with velocity
1
—2
vector 1 . What is the velocity vector of our image by the function 9 above?
5
2 2 2 [1 I l 4. (3.) Find all of the critical points of the function =$+y—$2—y2+xy+100 W : r—2>< +v> : ('2‘7’ +>( 7:; K: 3; ’ﬂ = 0
=9 —3v + 3 :0 I {54L
=3 y:( :9 K=I :3 (6:97 “4+3. (b) Bob is hiking up a mountain whose shape is given by the graph of the function from part (a). At the moment when Bob is at the point on the mountain corresponding to the point in the domain, how steep is the slope of the mountain side there? AS Sanh in C(«sg/ Mm Sleefhzss O‘C Mae Smﬁi f: #2 5. Use Lagrange multipliers to ﬁnd the point(s) in the domain that achieve the absolute
maximum value of the function f([:l)=wz—6y subject to the restriction that 3:2 + y?‘ S 25 . Edi" NQQJZ 3'5"”
‘7'? 7‘ ca“ W“), LC ‘5: 0W7”er Since Q #0 ,
go ‘Hx-Erz are N inkn‘m— cri'S't‘Ca( pm‘Nl—S. 6. Suppose that QGEHEJ is the composition of the differentiable functions f, g, h, and k, composed as below:
with iEHii 9([ED=E§] 4H) [iii W) (a) Write an expression for 6623/63 in terms of other partial derivatives. jtoL : 1 TL 161
162
k3 3 I ‘ 7:“ 993 2 315; 91:; .33; i—
as a?“ h BS3 i (b) Suppose that at the point f (3') we know the following relationship between the
partial derivative vectors of g: 3.9_ 39 39
3u_ 6v+23w Show that the vectors
VQ1(E’). VQ2(E’), VQ3CE’) are linearly dependent. Hints:
What do the partial derivative vectors of 9 have to do with the matria~ JEN?) ? What does the given relationship between these partial derivative vectors of g say
about the determinant of Jgﬂ?) ? What does the matrix ngﬂ?) have to do with the matrix JQ’; .9
What does this then say about the determinant of JQ‘T; ? What do the gradients of the components of Q have to do with the matrix J93; .9 3Q : jkjkjsj; H
L
U I“. 11L030$(3) X'jogﬁﬁ TSFMR) F543,? TR “01‘” a» ray? , "3% «h an the Ou‘UMHV1c'LT‘S—
it . 36 it at 4U in {gm/7 atom 14Wj mm“) :o/ M midi 4wa 4L:
raw Vecgm-s OE 7Q}? ywusJ' La {imam-l7 ANPQ 0"; Course ‘Hw raw UNA—MS OL TQ/a. are Mt
jerEwlT 0‘: Wﬁnw"? 04: Q a} E? '— VQIK‘) / Pauli $7623 (3*).
gm / Malay vac—LG are ’ImowL/ I Bonus Question: Use the deﬁnition of differentiability to prove that for any linear transformation T : IR” —» R“, and any point 3’ e R", the derivative transformation D733; is the same as the transformation
T itself. In other words, for any vector v, DT,E’(?) = TU?) We (gar-{VALUE ‘ans OT]; her' 94659
HM wQeL‘m u an
: hm T(?> — (T( 7’? +(E—a‘ﬂ)”
3W" “TE—2:" 10 ...

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