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**Unformatted text preview: **Math 41, Autumn 2007 Second Exam — November 13, 2007 Page 2 of 11 1. (15 points) Differentiate, using the method of your choice. X~arcsrmx “(I W) Since I" (MO) 2 In ((i+x7f)&rc4am><) :1 ONC’I'QMX- In (Hy?)
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569 ’ in (X) * ﬁ{ar$"x‘ﬂﬂ+xz)): i+XZIM [/H'xz) "f‘ mrﬁnx _——- .ng Math 41, Autumn 2007 Second Exam —— November 13, 2007 Page 3 of 11 2. (10 points) The equation 332 — mg + y2 = 1 describes an ellipse. d
(a) Find an expression for all in terms of m and y.
a; (b) Find the equation of the tangent line at the point (1, 1). A" {X)7)‘(I2’), 3&2 E: AI‘ t "a in W m (c) Find the coordinates (23,31) of all points on the curve Where the tangent to the curve is
horizontal. Fr (ﬂy-x - 0/. we 146600 y“09<:0 gm? 97-x¢o 15w) . 501:3th 0+ 2&7 yrs/62X ,7me #4 pfcmL/oaq ‘é‘r‘ﬂe 9/7/2951: Mike! KZ’“ X‘(3X)+(Qx)22 ‘ => )Z-“QxZ-rl/XZ: I => 3x2: 1‘
This is Sdliggeo’ wk» X=igj 50 'ﬁt ‘ﬁvo {1054‘1‘5 are m: (gag) m: {tyres/)0; Math 41, Autumn 2007 Second Exam — November 137 2007 Page 4 of 11 3. (9 points) (a) Find the linearization of the function f (as) = 302/3 at the point a = 8; that is, ﬁnd the
linear function L(m) that best approximates f (x) for values of a: near 8. 4‘6<1=xz/3 a) «0(8):??2/3: If (b) Use the linearization to estimate (803)2/3. Is your approximation an overestimate or
underestimate of the actual value? Explain fully. (80th is N303); m! M 8:03 ,3 W 3/ J we can say mm H303) ,9- Lil y: L/x) ~ IS comm/re,
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W [193 aha/{Me CUFW) 50 ‘f‘éaf :13 + '3‘? 8‘33“?) ‘Wbﬂvﬂﬁ/j‘ Math 41, Autumn 2007 Second Exam — November 13, 2007 Page 5 of 11 4. (10 points) Dennis, a 5—foot—tall man, notices a small spaceship on the ground, located 40 feet
from where he stands in a ﬂat ﬁeld. The spaceship suddenly begins a rapid vertical ascent, at
a rate of 10 feet per second. Throughout the ascent, a brightlight on the ship illuminates the
entire ﬁeld below, casting a shadow of Dennis onto the ground. What is the rate of change of
the length of Dennis’s shadow exactly three seconds after the spaceship has taken off? (Hint:
at any moment, the head of Dennis’s shadow is always located on the ground, and on the line
determined by the ship’s light and Dennis’s head.) é; mu diagram) x is 4% Izhjﬂ 501% SM“,
mam mama We y is m Mjlnl 0!?
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By similar ”frim’ijleg) LXJKN HOW—J
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Drggmtmélj wh‘A respeai ”to ’flimﬂ) we 1%? When 359(0nJg have @5560?) we [40149 7530) So 724d!“ 30x15€i0hd ”I Mi 075x :QOOJ 50 FE,
Aiso) W9. (view 33%;“), 7503) 52+ our WOW/4+ 0‘? Hat/eff") We tat/e 30 at. 5:: 1%an 255%: Math 41, Autumn 2007 Second Exam — November 13, 2007 Page 6 of 11 5. (10 points) Consider the function f (t) = g + v3 5 — t. (a) Find all critical numbers of f. Show all reasoning. w, W) =2 %+ amt) ° “91%) is vmﬂe'CmJ Wm ‘Hle expression (5’19’7’320) £9, (MAM 1‘65
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59 (5—02: I) (b) Find the absolute maximum and minimum values of f on the interval [0, 13]. We, Mud +61!“ ilk). WW 010 ‘P at 10m painfs: ) So ’H/S (Sher/Ila!” Math 41, Autumn 2007 Second Exam — November 13, 2007 Page 7 of 11 6. (17 points) Consider the function Mm) = 932 In 9:. (a) State the domain of h £16013 MM when I“ ‘5 J€GMQQ 50 ”(or X70 (b) Determine if h has any asymptotes; your reasoning must utilize limits to determine the
behavior of h at each end of the domain you found in (a). (E) ”Miler/rials? Nome “ jusfiﬁm’tiom belew
“M L} .._. {I‘M 2W 0Q [39mm 507% ‘43:!le X2 Kg“ 30 {’51 as K400 )(“Pﬁ In Joes (deli (9)0le é’QwUSe A ”015%!ij ‘fgr ﬂeﬂa‘ﬁw X (3'0 Vermr '7 Mia : New édow.
ﬁrh +9 Ll‘WR'CM Mgr; hLe limi‘fj only posslhiiity is X‘) 01: $5168; A I151 COWI‘Woug om {7‘s JOWIQI'M‘ XL -m ' (rm .. If LIE .——- if ﬁ/{f .. 1' x? _.
ﬂare X234 [1 ’— K—aat X210” ‘- x‘_:3+ X—z .__. x’eoi‘ :ixﬁg " x134“ 1556!: 0
[30:2] [gﬁm‘] [L’YLfg‘gfkq m éf‘m/‘ll'r? (c ) On What interval(s) is h increasing? decreasing? Explain completely. Neel W6C) : 12111,)?“ 1‘» xiii-z RXAHX = X(&‘/nx+_/)n ‘ k" never Uﬁale'lziner? 0M (0,”)
“ l«’(x)=o For Xfélﬂmxﬂ) =0 11> #0 0” Slime—I
a) )(‘O or )(r: 8‘sz [30+ X¢O [3y O/émam wmsldwa'ltmm. {119 intervals to :1th: O<X<ey : 313.4,? 1' = (Sigurd/517M; 30mg 1. @e 1 e)
X > 84/2 : 35?“ ”f ("i = ($53,4pr {5’59“ opﬂax-rﬂ :@@ =6 \ 75123 in Mater/251173 ow (76”: as) J (2140’ olecrxmg,‘ 0M (0) 6%), Math 41, Autumn 2007 Second Exam — November 13, 2007 Page 8 of 11 (d) On What interval(s) is h concave up? concave down? Explain completely. Need A”{x]=j%('2xﬂnx+x): lﬂnxrax-KLr I = (RAWB.
v A” newer UKIC’HMQJ 0n (0)0“) "l”x)=o a apmg=o «=9 Pig/2‘
190 intervals 1‘0 duck: o<x<g3zz : 8:3” 90 h” = Sigzno‘pélﬂmr’B)‘: e)
A 3w A” an Harms) = o. X>é3
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Thus h Cewcave UP 0.4 (e A?) 0%) ) anal (Man/9 (£91091 on (OJ 6 3/2‘) (e) Using the information you’ve found, sketch the graph y = h(a:). Label and provide the
(:c, y) coordinates of all local extrema and inﬂection points. LOCK/fin 0+ 56:3“? 5y Nelle);
So y:A(e“‘é)= gig); -1, $1 Medllom Fowl: al‘ Xc‘eﬂ's/z: 5/ Parli- (2’)) 5”” F New: = 53%): 333‘" Nﬁlié hm Mil—<0 cask/El
X~90+ I l ‘ Math 41, Autumn 2007 Second Exam ~— November 13, 2007 Page 9 of 11 7. (10 points) Compute the following limits, showing all reasoning. , (2m — 7r)2 0
1 ~—————— 6-— .....
(a) $3752 cos 2x + 1 O WQN’M ~ ~ 5109*? ‘1
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x~a0+ I+3x ’ 3 J "”103 IMngj 30 +14%" (L:Qg(_ Math 41, Autumn 2007 Second Exam — November 13, 2007 Page 10 of 11 8. (10 points) A landscape architect plans to enclose a 4000—square—foot rectangular region in
a botanical garden. She will use shrubs costing $20 per foot along three sides and fencing
costing $5 per foot along the fourth side. Find and justify the minimum total cost. We have 4000 ‘X/ J am! ‘th ’l’ofal (031‘ is; “”22 $0544 a?
“lei/13% ‘h‘mes dollar cost/1 for +115 ’Four sides ; ;z«(,fe¢e)~{ao¢/aa) +(xawsﬂag magma ﬂier) =- Wy+25x i Moi/m) 000 ~
Since yﬁ ix“ 1‘51 7mm}; Q We wisé 7‘6 Mini/44126 lS ) Q(x):515x+40°(i€<Q-Q) = 35;” Kim) for >29» Crif’)’ . 41/504“ 1 I c “ [60000 ‘ .
cal Q Q60 625 x1 ) ”SIC ”he”? Q/ is 0mg: ' ”PO/(ﬁr Zero.
9 Q50 “Meet/W? 0146/14; X10) Ail—f. M51 Jomaﬁy,‘ '2 G/x)=o => agr’ﬁgggooro 12> I€OOOO=525X2 :1: K2“ reoooo ._ d -’1 c‘ - gm
:59 X: 80 [Ammo X: -30 El” I'n (imam).
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04‘ {ALSO/Ute m 47 41:1 semi 09mm: 753+ a Ala/we Emma. ’llws) #44 mm @511 is @{30} = a5oxo+ 40. 1%) 3 2000 + 40 .59 = (“momma alt/000 l Math 41, Autumn 2007 Second Exam — November 13, 2007 Page 11 of 11 9. (9 points)‘The picture below depicts the graph of the function f (3:) = —m3 — 22:2 + so + 3. xﬁré)
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l / XX?or +M3M+ “l. X: I
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(fa-#519) (a) Suppose that Newton’s method is used to approximate the value of the positive zero—
crossing (root) of f with initial approximation x1 = a. Use the expression above for f to
ﬁnd a formula that gives the value of the subsequent estimate 952. Your answer should be given as a rational expression involving a alone. Xfxliini mt Wx=-3x2—~fo+/) was (b) Suppose the initial guess is —1; on the graph above, draw the tangent line that you
would use to ﬁnd the subsequent estimate $2 for the root. Then use your formula to ﬁnd
the value of $2; simplify your answer as much as you can. What do your results suggest
about the usefulness of Newton’s method for this initial guess? x 14'," ran? I ﬁr {,i 12" m Sew/tr! affmxlmziﬁom 55‘ Mr ‘R‘OMA #2? root ‘Mom ﬁe +3316 aP/OYUK) 50 7%)5
was mi a very useful 608$ {it gives (Mm/fable new/7‘s) (0) Repeat all aspects of part (b) if the initial guess is 1. (Use the same diagram above, and
give your calculations and conclusions below.) ~I~BZH+3 j " ’
X a: (. = ﬂ" :: .1... Z
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so New/Ms Mid Shot/lo? give good approximate”; 015 We Npem‘ MC
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