Applications of Differentiations

Module8 Applications of Differentiation 8.1 optimization problems Finding the min and Max of a quantity 1 Redd Problem identify quantities and Sketch 2 Writeprimary equation that is to be max or minimized 3 Write Secondary equation equation that relates variables if primary has more than one variable 4 Write primary equation 5 find absolute Max min Example 1 findinterval of intrest or local maxi min of primary equation wi Single variable maximizing Area A rancher has 300kt of fence for a rectangle yard One Side of the yard has a barn so nofence is required 3 exterior fences b 7 interior Separate the yard into 2 parts Tha dimensions shouldbetused sothe yard will be maximized who Is the maximum Dreaof the yard IF 1 It 4 Y 300 3x Solve for y 5 Maximize A 1 Critical point Evaluate at each C P Evaluate at Endpoint FIEndpoints To Y 300 3 20 x I 100 domain sopossible Os x 100 A is Max when lard 2 ADD 3 31 A X 300 14 300 3 3 300 3 2 Subintoprimaryequate A 300 3 2 6x X O 7 A 300 6150 50 CP 2 A503 300150 315072 7500 0 3 A 100 3001100 321005 0 A 300 o 3107 so when x 50 4 300 36503 150 deminsion 50 150 Max area is 7,500ft X X Minimizing perimeter A farmer plans to fence a rectangle field across from a lake w 3 sides of fence He wants 180 000 sa meter Of grass area What is the least ammount offending used 3 P 2x y Xy 1801000 y 4 9 Y 18090L p Domain O P 184 00 2 180,000 2 2x x 2 180 000 12 x 90,000 2 X 180,0005 2 0 180 00 1300 X 30 Second derivative test P 2 1800007 3 x P 300 When x 300 36 P 00 13070 600 y 183 800 P 213007 600 360 1200 Therefore 1200 meters will close 180,000 meters 2 C Optimizing a postal package A man wants to send apackage via ups all packages must be boxed shape w sum in Lxwxn I 108 inches find the dimensions of the package of Max Ime can be Sent win Iten 4 4 108 5 Domain 3 2 wth V 2W OE W W 108 2W 54 V 108 108W 2W 108Wh 2W 216W 6W X 6W 136 O W V 367 10813672 213633 0 108105 2105 46651 O V 34 1081347215473 0 V IS Max when X 36 therefore dimensions I 46 656 in P 36 Y 108 36 36 2136 36 Max volume A rectangular tank wi square base an opentop and a volume of 23,328ft is to be constructed of Sheet Steel find the deminsions of the tank that has the minimum Surface area Minamata h s't 4h5 4 23,328 h 54 A S 4h 5 45 23,3328 A 52 If 5 5 93312 I 356656 say 25 1 23,328 93312 9331 23 8 D X A flower garden A 190m is surrounded by a grass boarder I m wide on 2 Sides and a 2m wide on the Other 2 Sides What dimensions of the garden minimize the combined Area of the garden and boarders w 2 3 4 4 A A A W 4 BE ht 2 E A Wh 190mi 4h 4 Chtz wt a h 3 388 95 76g 194 19g W WZf5mA Liff E 8 2 Linear Approximation and differentials linear Approximation L feast F'case as F x feast F a x as x Error in Approximation E X the approximation to f 1 find linear approximation to f 3 Inemata A L IT p P x FIX Fx at 9 1 a F x a LF X FAIR When a L a stemsIniater.z F a f underestimate Fx Example Fix of 0 f call atthepoint a is an overestimate if F is l linear approximation LIX 1g FLI L X LIX I t F x i x 1 i E I 42 42 42 42 LLX Yzx 2 P x at Ex 2 f X f't 1 3 LIX 42 Tx 42 1 1 d x 1 4 20 Overestima 2 F 2 112112 2 42 1 I T.az Calc says 1.09 50 Its overestimate Linear approximation of a function Consider FIX ex underestimate o A fix When a O 1 over F e Flo x f mation to FCO X I hcxt fcastficascx.ae e x b I e at a o f'D 0 1 F O X O I X O I X L xS 2 3 F x LYE When X f e 0.4 0.4 x f O e e I 70 Overestima 1.4 4 e Cale Says 1.49 so underestimate A F F In 1.04 I x 1 9 fax x 1 thx 1.04 Lex Fca F'case a d 161.04 I 4 0.04 1 fX 2 2 3 F F F A at a Gx x f 3 F 35 a a X 25 F 2 9 3 25 find 25 18 1876 3 54 18 18 21 B 18 LC 2 9 find 29 x 1812.93 29 2 231 C FIX 2 3 2 I 2 9 error 23.23 100 Approximate find Exact b f 2.93 2 362.912 Error Lex Approymffittact 100 1 23.2 C 23.2331 1 2 3.231 0.129 9 FIX Sinx II Ala B X Flo X 0 02 a at O 0.02 F 0.02 O 4 3 0 C A 0.02 1 x X o 51910.027 100 101 20.02 0.02 Use linear approximation to evaluate 1 V69 Lex Fla f'casa Fg Fg F X 2 Aca ate of f 564 8 a d 4 F x I Aca V27 3 e 007 8 5 16 a II I t 24 a 46 69 64 57 I so P I 8 8 a I LIX 130 3 27 3 111 3 FIX ex flex ex 9 0 x 0.07 X F Fca a Lex eoteo Ceo 07 x X O I X I 0.07 1.07 a Differential of a functionusing differentiation Rules If f We Rewrite DX dx AY dy x Fca F a F f x Fca then AYE F'calx as x a a LX C a dy F'G dy AY FIX DX FG Consider y f x x2 Approximate the change in y AY when X 1.00 Calculate by exactly p compare wi approximation 1.02 I 1 X Changes from f x x2 dy F x dx 1.00 to 1.02 F 2 the exact AY x dy 2x 2xdx fix DX f 1.02 4 1.0 x Dx 1.02 AX dx 0.02 2 1.011002 fax FLI O 94 0.04040 0.04FAY 1.02 1O eat Fox e4 f 2 x F FIX F x use 9ea ke 3 2x 6 2 2 Cotax x dy 9csC ax 66.2L F x dx 83 L Hospitals Rule Theforms 010 and Alo Indeterminate form are called indeterminate forms Limit problem If If nm Tech Cancellation technique timex 13 2 3ft L Algebraic I F Leadingterm 3 behavior Hospitals Rule If the limit Fox goes as x approaches a produces 8 or then we take the derivative of the numb den to find the lim Ha Only use rule YE if indeterminate forms 8 F F or Evaluate All.mx 3 B 2 Y 2 2x 1511 2 34546 Im Guys end i 8 D 21 2 8 C limo CEE Ziff 1 Igor e 8 8 HE 8 1 3135 3 3 EYE T.IE 8 0 4 It 81 tagger 7sec 7sec G lim 1 30 É E F 0 1 csc x E csc 8 x E Is 7 8 7 491 2 28 101 0 4 EXTRA Evaluate a I 3 1 13 2x X fine 2 2 341 g day 31 1176,1 f v if den 261 2 III II B limo CNET.IE isinxs7 ftp 2 gz 2qIox cosx nxtsinx eoiici isnt It II E 1 Imo cost Y's 5431 Iho 3 1 0013 at 81 3 oF D Infinity II I'I Is 41 Infinity over 3 3 45 00 3 Er 1 É 1mg f Ecosx I 1mg 2cosxC sinxJ_2cos zjiGTzI AImx se.x I'm t Er time o Imo 8 E D L Hopitais Rule for Evaluate A 5 life KE th 3 Simplify B Ing XT 22 I 3 E 8 Eg time csa six 2sinyc_I DYsEII.s III 3 lextilextis 5 3 8 idenxy Hmo to to hot III Y Imo C x xthot 2,5 1 AIM lety 1 D 1 thy x1 en.IE eq 9 E II buy 2 E Ing s In 4 Ex I L en É t.im I 4th 2 In C I 21 1 lay Itsy II LD X D 41 I Y t 2 Other indeterminate form O Evaluate A dim x'e x Rewrite 2 to E E B I p x'tnx 3 Simplify Rewrite É EY 3 00 and Xo 1 Evaluating With these indeterminate forms L Hospitals rule cant be applieddirectlyFirst apply natural logarithmic and use the properties of natural logarithmic to Convert Evaluate A IM Itt Apply Natural Ing af Logarithmic en any 4 Itt fq.tt L Ttm thything limo Any limo IF 1 I simplify en y Elmo X lux x tnx 1 8 tha buy 1 enthfxen its ily B limo X thy lng In enfE by the 5 4 116 I xx e therefore e I If It I C kagginx Imo C Cny x o Any enftp.go.lsinxstan buy ftp.sflncsinxstanx lay 11 tanxlulsinx Initio MEI X xmo 411 lay Emo EEE Timo L cosxsinx a lay o Evaluating form A ftp.ccotx x X cscx Imo sent Imo GEE BY't II I et EenÉÉ III xent Etta 8 Imo If ta tnx x 1 xt en 8 eax dy F x dx f Ffs F s 1 8 x 5 9 O 0th I f 5th 200 810th T 815th t5 T 200 210th T I 2h2 80kt 8h F 2h h 8078h 2h 8048h PISO Y Y 200 200 10 74 T h 710th 801 80 5 400 7 7 10 10 10 Y Y 80 10 200 x 111 4 4 4 2 2 0 O 7 7 0 10