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Unformatted text preview: Exam : 165 MATH 2451 thq_05 TC Use
Course : Multivariable Calculus Due Date : 20990204 (Wed) Instructor : McCary KEZ (First Name) (Last Name) 20160212 16:21 (10 points} Find all values of a and b which make the function differentiable at. all points. You must. use left. and right. 1.
limits, the limit. deﬁnition of continuity at. a point, and the limit. deﬁnition of derivative at. a point. a332+bx+4 3:32
J60”):
(133+?) I>2 1H5; Fuumonl :5 9L; (lair: Even/WHERE EXCEPT foééwb/ 96:3, 5/0 PIECES A212
POLY§. F012 SE Ar 99:52,) NEED: (m)= (M an
ﬁéx‘tmey’) = EQEGME) = ﬂaw we 7la+ll+¢ = «CZ/Mb
Qa+b+tf : o Foe {Iii AT ¢6=.2) a! 1%) = xx»:
Jfg”; (QMHLJ) : {66:54 (0‘)
fa+b = a 3a+b = O Cox/191mm» mesa mthrronc, 2. (10 points} {a} Compute using the limit deﬁnition of partial derivative. (b) Compute using the limit deﬁnition of partial derivative. {c} Show f is not continuous at 3: _ I my (.3 0 otherwise _ [430 ‘77—
., mm (s)
4:) ‘ “Lg? Lb ll
0 @ jzmx PAW F0 (g) .3 LIMI’r 02m” 0N P/‘r’rH ‘ LIMIT ,. DNE
NM g2! The purpose of this exercise is to illustrate )ne of the weaknesses of partial derivatives: even if all partials exist at
a point, the function may still be discontinuous! This is in contrast to scalar calculus where the existence of 9’01}
implies continuity at a. See the comment about symbolic calculators at the end of the next exercise, which also holds
for this exercise. 3. (10 points} Find 82f and 82f using the limit deﬁnition of partial derivative. a a?! a 593N933 (93:69};
2 .2 . _. '3 PARTI/HJ
I _ lava$2341?) 7&0 > Foe THE 9/ MIXED I _ n I t: 93(0 =u<3—i((o)+m)g§(%)
. O . erw l‘ne 9x 0) hr)" kl __.—
AT (9‘); 5’ 1» sawo
j 0K)  l’WO k
'0 \~ 0 u
Jam ¥((°ll(ad ’ " >  < kgi_o 143:0 l!»  M90 h
J» L 3’0) M L.
= w v» (7)
— M r) 4 L1
. o Noﬂm J» awnaw)
9 M M 99693 ‘7 Pm k
i M l’ ,u sumo
952(6) = o W k )
' k“ In.
W 9:? ‘ 9L +4941 9'
’9'” W 2 (a
2i : (ml4x1 91 K
9” (maul : 1
40 (um: 9. m 9: 6o
0—19 _ —”;§—#) 9 09
axill WM gjgxm 7; ago)
0 orMERwICIE
) WH)’ DOES ms WT WNW/(Um cum/1W3 macaw?
(with; 9._ or i _ iﬁzér) a
7M ' (K m
j 0 otMEtheg The purpose of this exercise is to show that mixed partials in different orders are not always equal. Don’t get lost in
the limit formalism try plotting f and its ﬁrst and second partials on W'olfram Alpha (or, even better, download
SAGE Math and start using it, MATH majors}. Can you see why the mixed partials are not equal? Feel free to use a
symbolic calculator to compute the partials of f away from 6, but remember that you must use the limit deﬁnition
to compute the partials at [5 because [5 is not in the domain of that rational expression. 4. (10 points} Compute the Jacobian. You must. Show your work and not. use a symbolic calculator. (a)
I cos(3:}
f = 332?} + :92
‘ 5111032 — M (04) 0
[if = M5 06””‘5
(MW9) m m4 (“’45)
(b)
I /I2 + 92
9(a) = ' (5111(I39D2
% ’21— .24—
[TW] = W
'x.
3 w (W)? (MW) 9 WW “W” °‘ 5. (10 points} Let = Show, using limits, that for any number m, lim (f([}+ h} — m} —m h) = n (1) 31—30 but lim 100a} + h} — fan — m a) = f} (93 31—30 is never true: there is no m such that m h is a good approximation to the increment Af at f}. Fm m NR (90
J/W‘;<Iul~ioIW‘ ‘>0
AM“ MDT?
A>o 
M(iﬂm\=1—wx "” 1)”
Imo“ V”
Am:
Jlm =4m —>
1 =1 IS Fifi/‘35 \I m
a" ‘LIMTT (9.} D.NE. WV» Tug; exeamg 6mm; WW THERE «:5 No LINEAR FuNLTIo'J or 1m; maewsm wurcn MppoxtMMES lad AT 0) Em: a
mow." tr Is US NEW. m,uo
WI
(75
MW)\i WI.) N0 DOB; LIMIT (:0 weak? “0) 3/0 THEQE 139 ND WI who” 246.22% V/ LE MEIR 0 M»
1/55 ruwe ME 0mm MOT£OM5 as Danni/“IVE WHERE N“)
Is DIN—,7 4V”: ,4; A WEN DearujTIUE, BUT [46) :5 N07— Vwa‘W FR’ELHET DIFF Does LIMIT (4) weak? ﬁg 7 Vin) THE/I MIQBEAT 0.
The purpose of this exercise is to wean you off of the scalar calculus deﬁnition of derivative and to get you accustomed
to the Fréchet deﬁnition. ...
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 Summer '09
 EYDELZON

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