2311_MG7_GeomDeriv_Sketching_LHopRule_Optimization_solutions_F11

2311_MG7_GeomDeriv_Sketching_LHopRule_Optimization_solutions_F11

Info icon This preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 4
Image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 6
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MAC2311 MG? (4.3-4.7) 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 case-by-case basis, but typically one letter grade (10%) penalty per-day. “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 WWW ~F mmmwetwgtjjiwm H e“ I I ,3 . .F sixth!;._:.i;;‘,'t;fi::?i”Shit-3%:giintiqnzxmngghmmsmlfig '"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 1’: 19+” .. +x+1 [ 50352 J 0 [fax [00% .. z: b Iim : m ) 35—)0 1—0081? C) XWQ S‘VNX [rm 10C) g: )Lué‘go (”cigfi X hm LAO-fa}; .2) C) Hm[1+3]:§ exw #3; g x-—)+oo x / \lm “(My 3)x -::: [tram xthI‘f’wjug .3990 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 drop-down 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 y-coordinates. 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+ ice-iii“- +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 ._ 2°99 . AU”) i: 2719?“ “i- 211‘“ W(?;§~ WY“ 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. Lfnrbm- itOWeQ Pie 4000 2:: long an "T It) at: in: 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... 433m 2:. s t— 2s w 2»? 7c M "e a... 6x ...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern