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Unformatted text preview: MAC2311 MG? (4.34.7) Name: £0 lWlLlO t") ,5 
Fall 2011 Section: (circle yours} MWF 9.09 MWF 10:00 Directions: Use all available resources, including the math lab, Teague’s office hours, and fellow
students. To justify answers and earn partial credit, show work in the spaces provided on this document. MG7 is due in class on Tuesday 11/1/11. Late papers will be penalized on a casebycase
basis, but typically one letter grade (10%) penalty perday. “Geometry of Derivatives, Curve Sketching, L’HosPital’s Rule, Optimization” ‘1. The graphs of f , f ’ , and f” are shown. Fill in the
blanks with letters A, B, and C to identify which is which. f f’f”
Aﬁf: 2. Sketch a possible rational function f that meets all of the following conditions. Label asymptotes
with their equations, and local extrema with both coordinates. i) f(2)=1, f(1):2, f(2)=0, f(4)=1 iv) f’(1)=f"(1):0
ii) The domain of fis {xl xvi—i, andx¢3} V) f’ > 0 for x <—1and x > 3
f’<0 for —l<x<1 and 1<x<3
iii) 3311* f(x) = xlin‘lrfa) = +00 Vi) f" > O for x < ml and ”1 < x <1
_li_>1pf(x)=li+r¥f(x)=—oo f”<0 forl<x<3 and x>3
gym a 335; f(x) = 0
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'"I l 3. For each iimit, identify its indeterminate form and evaluate the limit using L’Hospital’s Rule.  [ 200x ) m
a) 11m TW:M
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XWWO ﬁllm “‘04,”; I 4. Sketch the graph of f (x) 2 31% — x. Show all ﬁrst derivative and second derivative work, and show their “sign lines”. At any point of 11011—differentiability, ﬁnd the one—sided limits of the derivative
and be sure your sketch portrays the graphic behavior suggested by these limits. Label local extrema
with exact coordinates. Note: Before you use DERIVE for this question, change the default Mode settings as follows to ensure that DERIVE
shows the complete graph: Click Options, select Mode Settings, click the dropdown arrow for the Branch ﬁeld, and
change the setting ﬁ‘om Branch to Real. 41cm 3;”th x ,3wa at some (.51va m). { wl/g
46x)»: m we) saw! it Y lnx—l 5. Consider f (x):'— over its natural domain (0,1)U(1,+00) with f ’(x)= 2 and
lnx . (1112:)
f ”(x): 26111 3: . Sketch f. Label each asymptote with its equation. Label local extreme points
x nx. and inflection points with exact 35— and ycoordinates. The limits of f as x —> 0+ , x —>1' , 2: —>1" ,
and x —> +00 may be'useful. x . ><  “25mm
iim —————~—: 0 in“ \n‘x '3 ”'00 ) \lm “x .5 +00
mm lwx )‘ wan“ 9‘4”?” Polew local) 5‘23“
alumna}, 0d” Xxl w + 44+ iceiii“ +1; w we.“ 6. Minimizing Materials A company must package 2000 cubic cm
 of its product in closed cylindrical containers made of sheet metai. The company wants to determine the dimensions that minimize the amount of sheet metal needed. (Use the back of this page as needed.) a) For a closed right circular cylinder with radius r and height h, surface area A = 27172 + 27rrh and volume V = arzh. Write the surface area of the cylinder described above as a single—variable
function of radius and state its open—interval domain. COW$+TC;\ Ni" IL] A E: 2:“, Fl'i" 211., f4“ Zoooe'lTr“ 2... Zone A(r)== Garrard,“ 4,000
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W Titre? domain: w A (r) at Zing—e zit:64. [Fact If function f is continuous on an interval, and f has exactly one local extreme value on this interval, then the local extreme value is an absolute extreme value 011 this interval. b) Use this fact to ﬁnd the radius and height dimensions that minimize surface area, and this minimum
surface area. Report answers first in exact form, and then in decimai form rounded to the nearest
hundredth. Use appropriate units. To justify that the critical radius value produces an absolute
minimum for surface area, sketch the surface area function over its open—interval domain. This is
acceptable evidence that the surface area function has exactly one local minimum value over its
domain, so the fact above guarantees that the local minimum is an absolute minimum. Tale.
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Answers: When 1‘: xiii} 2 $9533 5m ,and h: /\3}ﬁ w ‘3' (alarm, surface area is
minimized at A: 6:00 >3 [11“ a 3379.719 mt“ 7. Find the coordinates of the points on the eliipse 4x2 + y2 = 4 farthest away from point (4,0) . Note: This problem can be handled using either the “closed interval strategy” from section 4.1, or
using the fact stated in #5. It’s your choice, but either way you must state the interval in your work.
(See exercise #19 in section 4.7.) To Minimige (36.x); 1+2; ngﬁcimt “is Maximise...
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 Spring '11
 Teague
 Calculus

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