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# CAL MATH 1431

LSU

GPA 3.77

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This 277 page Class Notes was uploaded by Maverick Koelpin on Tuesday October 13, 2015. The Class Notes belongs to MATH 1431 at Louisiana State University taught by D. Vaughn in Fall. Since its upload, it has received 50 views. For similar materials see /class/222631/math-1431-louisiana-state-university in Mathematics (M) at Louisiana State University.

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Date Created: 10/13/15

Objectives 64 Fundamental Theorem of Calculus Find the area of a region under a graph on an interval using the fundamental theorem Evaluation of definite integrals Business Applications The Fundamental Theorem of Calculus Let f be continuous on ab Then aIb x ClX Fb Fa where F is any antiderivative of f that is PM fx Geometric Interpretation of a Definite Integral 39 aIb fx dx Fb Fa Represents the quotsigned area between the graph of fx and the x axis on the interval ab That is the area above the x axis minus the area below the x axis on the interval ab Recall from our previous class Find an approximation of the area of the region R under the graph of the functionfon theinterval24 Use n 5 subintervals Choose the representative points to be the midpoints of the subintervals fx x2 7 We found 3214 Let s now use the FTC To compute the exact area fx x2 7 24 36814 4 9 3 k e N X3 Fmd the area under the graph of Pi fx x2 6 On the interval 21 l 1 g Prde fbe 1 a Mod X gyww I 06 quot g 12 3 432 JV5 3 35 0 A Find the area under thelgraph of V x X g3e 382 fx 3eX 0 On the interval 02 2 o 9 3e quot39 D 3e2 1 L 39x3 WA E A 3 e331 Evaluate the definite integral 1i 51 2121 52 dx 82 07 51x 988 1 511k 5344 W5 477 f A 390 Find the area of the region under the graph of the function fx 5x on the interval 4 10 5in105ln4 new is 5X4 i0 4 ID 35X Obi 5 JMX 4 4 wkdb SLlrol 5 394 v 5039 5140 5 Evaluate the definite integral 4 42 i 39J 3VU du 53 4 3 3342 3 2 14 Find the area of the region under the graph of the functionfon the interval 8 1 i I V 5Ixy3 h fx 1 X13 9 b 3 Ix 3 4 l 392 4 i quot 3X irzro y fz es 8 I 9111 W42 3 75 zo 2413 Evaluate the definite integral Z 3 2X ly 122x3dx lg 4L1 7s 1 1 quotA z X 11 w L 9 a 139 7 I X 2 39 rvl The managemenT of DiTTon IndusTries has deTermined ThaT The daily marginal revenue funcTion associaTed wiTh selling xuniTs of Their deluxe ToasTer ovens is given by The following where R x O1gtlt 40 is measured in dollarsuniT Find The daily ToTal revenue realized from The sale of 160 uniTs of The ToasTer oven 3 g J map Note3 Noo Ra D x 4 934 Find The revenue realized when The 2Q40xc producTion an sales level IS increased from 160 339 To 260 uniTs 2300 R60 39 QofxzT HlDVw L 12490 gt ago MK quot 005 Llox 2 ROW 393 WHOM lDo0gt 1 mga Ring 1 00 539 lief viva ath 7am w 360 W 321be Nil 34 g3 A division of Ditton Industries manufactures a deluxe toaster oven Management has determined that the daily marginal cost function associated with producing these toaster ovens is given by C 39X 00003X2 012x 22 where C X is measured in dollarsunit and x denotes the number of units produced Management has also determined that the daily fixed cost incurred in the production is 800 do gao 406 619003 ifquot in JLRgt01X W 3 2 39MX39l f21xc 32 ax 0001x3201 xz mzx L MD 900 9 1mm yd is 990 r 1300 900 300gt3 D10300gtz113amp0gamp0 7 0 Find the total cost incurred by Ditton in producing the first 300 units of these toaster ovens per day 4700 Cquot C 1190 1 DDDI7I90Y DL7DD 11105 0 What is the total cost incurred by Ditton in 5 producing the 201t through 300th 47M unitsday 1100 2395 0 300 J at S60003xzenax21gt lt 300 100 3000 x5 06 x312 yl 30433 0 DD 0001300 06 so3325 12300 0 am fami dzod 2 22 The Algebra of Functions These pmbiems are aii en Wu r med i e Compcsition of o The Difference functions Quotient nrnent Previewer Window ntemet Explarer II It 3 x wxz10 4x 6 fg L5 Find the functions f g f g fg and fig I v y w xvquot i z m 3 14x 6 X 6 416 X 6 vwm5urquota 1IXH639X36 X 6 VH1quot G 35 i my xix m 393 33 i l 391 Let x x3 5 gpr x2 5 and hx 4x 5quot Find the rule for the function has x a Mi 4X44 Q 306 X gt 30 9 7 34 Q09 xgt 3 gt 38m Composition of Functions Assignment Prev3ewer Windows Internet Explorer Find the rules for the composite functions f o g and g o f Hitquot 5m am3 sitquot quotquot 99 3 J gt f 0 g 5172 5517 15 3 5045 b x 3 Mi 5Qquot w wag nzer 5 Find the rules for the compomte functions f o g and g o f 2 39 glix39inun W W 2 Ra6546 5 r Evaluate h15 where h g o f 910430 362123 54quot tha quotk 3quot 3 A srxzqa 36 MJQ leuu 39 h 30 bag 3xX 05 395 15 33115 513 1 maz mt 3 Ia Gag I3739H3 mLa39Q 355 THE DIFFERENCE QUOTIENT v I Hiquot quotgm 4 4 Jig E Find fa h d a for the functio 9 f fah ag Find and simplify the following for the function 37 5332 103 Sub l39r u 5 l quotw39k l Lied b New Hem 39 5quot q a oampn xxt v 4 3 5 vE 0319 u 251Ennivi RA n sr J o ae Bfquot Pufa r EWTNAQ Pg 57 vawasv Pym E r my W FVAV aroura ba pjr 39f Mfr scPQB F atmhtw av R A 9r rm ior 9t 7 8151 Ryanng r XIIIEle Ian 5AID 4quot Objectives 23 Business Applications Supply amp Demand Market Equilibrium Cost function Revenue Function Profit Function Business Functions 0 1 Cost Va 2 Revenue Number of items sold times the 1 Price of each xp 3 Profit Revenue Cost 29 Assignment Previewer Window Internet Explorer A division of Chapman Corporation manufactures a pager The weekly xed cost for the division is 30000 and the variable cost pagersweek is represented by the function Vx V i Ltiiilit101ar l m 0013 5m quot o The company realizes a revenue represented by the function Rx r k mm murmur 15m 1111105 a 5 75m 2 This is from the sale ofx pagersweek n u C a Find the total cost function g a Cm minnow u1u39i atmipiitiiinj 9 i X 395 0 X U mm b Find the total pro t function pix 0232 15UUIJUULU3 w Ulmz 5301 30000 at c What is the the company if 2300 saga t 134Q33 713 03gt 3quot DRQWv gags00 Comm u30033v 011309 500300 its are produced and sold each week a Three important business mathematical models 6 Suppiy functions 9 3 Six 0 Demand Fumtimns p Dix a Equiiibrium Mints SW 3 For the foll kolwyingprgblem x is in n its of a t h o u 525 akkde A gnment Previewer Windo lntemet Expl rer x 10 ng a For the following problem x is in unitshofa thousand W 34 p z 33 1 if a Sketch the supply curve SCI 40 3 5 Is V 1L251u 2 1 an 6 WV nment Previewer Window ntemet Explurer I p285 3X 2 and px2sx25 1 Fihd rhekediiilibriunjquantity D 3 x zx 5 thousand units 0 c Find therreduili hrium hrice o 439 r a i a s x V 7quot 9513quot W44 57 The weekly demand and sumler functlons for Sportsman S X 7 tents are glva the following respective fum ms where p is mea is measured in units of a hundred 39 o I x x 39 x p L 1172 w r if qu p Huh2 2x 615 a 1 Find the equilibrium quantity r E D V t 2 x 3 x 7 W hundred units Find the equilibrium price all 756150 w r g Wu M ll 9 33 a we 719le Wt 53 l 391 x 13 I 3120 261 quot It X a 339 q 4352 10 I quotW W a Man Faoorlb 3X 3K 10 6 0 gxquot3ox 100 05 1x u X IOQgt 2 0 x lb x loo 0 x x 5 so 9 0 X1751 PMS 2514 X 9 0 P H929 x x 5 Help Patricia Wishes td have a rectangu With whichwto enclose he 9 8 r 3 4 i I v l at 39 X i quot39 Letting x denote the Width of the garden nd a function fin the variable x glvmg the area of the gar em if m fix t111 3 V What is the domain of this function W u I gtgtgtgtgtgtgtgtgt A 39 z x 3 L The owner of a luxury motor yacht that sails among the 4000 Greek islands charges 490fpersonjday if exactly 20 people sign u However if more than 20 people sign up up to the maximum capacity of 90 for n quot 1 used by 5 for passenger Assume at least 20 people sign up for the cruise and let x denot he number of passengers above 20 a Find a function R giving the revenueiday realized from the charter are 2i190 53 1in What is the revenueldav if 56 oeonle sion uo for the cruise I 7 A 0 07 th l il 74 5151 MAP Revamf39e z awe Petquot x ts 3919 quot Remlhlgr Pa 9n3 p5 24gt a Rev am x 4905x 1 55 W 40 W W mienaer s 450239 R39CV39CVHIC 339 g L304 74 JarJ70 x 7440 s kWenwe ab54 490 5654 Malffo 393 other dn 9DNE t State the degree of the polynomial mction If an answer does not exist enter DNE D g 2 The rise of digital music and the improvement to the DVD format are part of the reasons why the average selling price of standalone DVD recorders u drop in the coming years The function below gives the projected average selling price in dim oiqtandalone DVD recorders in year t where t c corresponds to the beginning of 2002T q 55 r I39 M 1 o 1S39J v q 0 e 0 At the begmf 2007 Round your answers to the A a 2484 76 2007 i iiww39 a 4 q A a s 323 What was the average selling price of standalone DVD recorders at the beginning of 2 Assignment Detaiis Objectives 725 5 W 66 Area Between Graphs 53 9 6quot U n Calculate Area Between Curves given functions and graphs 66 Let f and g be continuous functions suc L ZERO on the interval aab Then the area of the region bounde above by y x and below by y on ab is given by o w 50 quot t A X v aIb 99 x v a gtlt 6 Let f and g be continuous functions such 3 that 34 gt 199 on the interval ab Then the area of the region bounded above by y gx and below by y x on ab is given by 7 bo39H39DW39 a 800 fX dX 4i A b Assignment Previewer Windmvs Internet Explmer for st X Sketch the graph and nd the area of the region described beiow A Find the area of H19 renirm hounded balq hv the Oranh nf Had an ahnve hv 14 3qu m x 339 Tn r 1 V 39 Assignment Previewer Windmis Internet Aquot Find the area of the region bounded beiow by the graph of for and above by etchthe graph and nd the area of the raglan described below 0 fun a 9 uau o w 3 3 QMh i 9 961 3 gurfg d 27 ng3827 e xaxis from x 39 5 I27 mr yet9 11 signment Pre ewer Windnws Internet Expimer Qy n2x3 Find the area of the shaded region Assignment Previewer Winduws Intgmet Expierer Sketch the graphs of th ctions f and 9 Select the correct graph W a 2902 11 I Taulhea 15 1D 1039 15 15 331 It 154 15 J W Fail 4 area twoI95 18quot k5 0quotquot V P2gt0Ll his X a M x 0 Jib za 15 Me 75 Ab Viki 39 a Mr quot39 139 1 1 S a 3L x L x 4 d quot 4 f lt 4v j x Xa zx zquot V a a 1 4 WM 33 400 yzuz quot 43952 P M 69 v M at 2m 1 9 QQquot Mquot s L h 4065 square units I MEWI W Find the area of the region enclosed by these graphs and the vertical linesx 1 andx 339 Rnund your answer to two decimal l m as Imus 5V 39 ow 7quot Assignmem Previewer Windaws Internet Explorer St th graphs f 1 ntions ad 9 etect e crect Iph 1 x 2am 7 q 3 I e 11 39 1a 3 10 a 5 y 4 1n 5 5 1o 2 x 4 43 a 10 1 39 W 54 M M 3 n a w MK 1 xglwx Mx gzam pJm 9 314T 39449 quot a he ll M w M Jaw 4 K 7 quot law a w u ct t grep v Haj 12 la and at w 2amp4 w 3 3quot 5 9 2wquot 39 x3386 y u 2 C 086 246310 X V x I l b 3 numb M05 quotquot Hy g x pdx zlx 3 o x 2x 3 9 x 90 3 3 g xzwngm S exam 3301 quot9 quot o v l f D Wildquot is 154quotsz uh fwkz x at395 1w W I In w 3 a w I 3 r 16 a 23 u 3 1 3 b SAN3 mm3 s 3 3 T o x W 1N a 13 is x IlJ 3 My av X v IR ad m 0 a x E A quot Assignmem Editm Windaws Internet Explorer Sketch th raphs nickel functians f and g r on 0 a 0 1 x a mg 3 901 Ears x 1 L 1g4v3x 5 13 3 4 39 554 3x393 3 4 II 33 63 Gsogtgt 3 gt 3971 I in g 9 9 er gm m3 vrjw 5 all 9 29 1 Given the equation 3x 5y 1 answer the following questions a Is the slope of the line described by this equation positive or negative negative Vquot b As x increases in value does y increase or decrease decreases c If x decreases by 2 units what is the corresponding change in y 6 59 units 2Find an equation of the line that passes through the point and has the indicated slope m 25 21 m 10 y 10X27l 9quot 3 Find an equation of the line that passes through the points 3 6 and 3 2 x 3 v 4 The percent of homes with digital TV services which stood at 6 at the beginning of 1999 t 0 was projected to grow linearly so that at the beginning of 2003 I 4 the percent of such homes was projected to be 30 a Derive an equation of the line passing through the points A0 6 and B4 30 y 6t6 4 b Plot the line with the equation found in part a Submission Data c Using the equation found in part a find the percent of homes with digital TV services at the beginning of 2001 18 5 Let g be the function defined by the equation below gx 7x2 4x Find the following h gm V r Quit 1quot In HI gig 93 11 l g Vquot a 7 H 1Vquot aga win gr 7 7 l 7 l ga f Vm H l 6 Refer to the graph of the function f in the following gure A l l V y fh 4 f 7 quotJL a Find the value of f0 f0 2v b Find the value of x for the following i x 0 x 1V ii x fx 2 0V9 0 Find the domain off 3 0 6 E 2 6l E 2 6 E 0 6l w d Find the range off 2 6 C 0 6 V 7 Find the domain of the function IL fir it 3 4m 00 43 75 U 75 03 E 4 5 u 5 w r gt23 7 4 PK VUI V 8 Find the domain of the function H1 V Ur 11 U 0 11 11 co E 0 11 11 E 40 CO Vquot 9 An ef ciency study conducted for Elektra Electronics showed that the number of quotSpace Commanderquot walkietalkies assembled by the average worker thr after starting work at 800 am is given by NU x 5 131 for u r 1 How many walkietalkies can an average worker be expected to assemble between 800 and 900 am Between 900 and 1000 am 800 900am 20quot walkietalkies 900 1000am 26 walkietalkies 10 Match the statement with one of the graphs a f The slope of the line is positive and its y intercept is negative Va A I cgp 4 4 l 4 E a E b E c y J 4 t 3 L 3 o o o t 4L 4 39 E e U f 11 Find the slope of the line in the gure If the slope of the line is unde ned enter UNDEFINED m 2V 2 1 l 1 gt 1 2 2 1 7 12 Find the slope ofthe line that passes through the pair ofpoints Ifthe slope ofthe line is unde ned enter UNDEFINED Do not use mixed numbers in your answer 4 1 and 6 5 m 2V 13 Find an equation of the vertical line that passes through 19 1 x 19V 14 Let f be the function de ned by the equation below fx 6x 2 Find the following f4 22V f125 55Vquot f0 as a V 1 fa f 15Let f be the function defined by the equation below H will r 1139 r39 1 1 1 Mr W I if r 3 1 Find the following f0 10V f1 9V f3 05v r 16Find the domain of the function V j r Jll1r 1 2 10 1 U 100 E 1 12 u 12 w E 710 12 u 12 on C 12 CO 3 100 v 52 53 39 Logarithmic Compound Functions Interest Properties of Logarithms P ert Graphs of Logarithmic Functions met Expiurer The percentage of fam res that were married housemlds between 1970 and 2000 is given apprwtimately W PW 8716405 0 5 t 5 3 Ms M admin V Wham t is measwred in decades with l 1 c msponding to 1970 C 1 l a what percentage of fammea were married houae hoids in 1970 In 1986 PM 1970 P 39 e UW ngeazmm A cf l m Va 198a 39 L 1990 l g 939 W e 31me 3 2000 quot w w mmwm 1 Sketch the graph of P Select the act grep 7 l e u n 115 1 1 5 3 r M l w 39 The cancentra on of a nug 7m an crgan at gh fl time t in 5a ds is given by Q m e 0m 04tme 241 Wequot 0 Jq39u 6 39 24 m quot166 75351 r w here 60 is measured in grams per cubic Centimate r glcm3 o a 4 D quot a what is the initial cancentra on of drug in the oSan 39 b cu4c 2uue b What is the concentration of tha drug in the organ after 6 sec 291 mama 31 c What is the concentration of the drug in the rgan after ea 60 slams c3 g 239 c e d What wilt be the ammunition of the drug in the long run I 3 3quot wgo p 52 Logarithms of x to the base b Ylggbxifand only ifxbV Li J 232 3 Expand quot 9 IQ u 3 L93 rH mxu Wt 5 Logarithmic Notation 1 3908 e X In x W 2 i0g10Xlogx 5 Logb 2 Properties of Logarithms for positive numbers m and n Logb mn LogIo m Logb n LogIo mn Loglo m Logb n 0 Logb n Loglg 7m 4X1 Logarithmic Function The function defined bd fx log b Xq bgt0 b 1 is called the loga Ithmic funct on with base b oat L x The domain of f is the set of all positive numbers Properties of Logarithmic Functions The function is onetoone Its range is infinity infinity Its graph passes through the point 10 It is continuous throughout its domain It is increasing on Oinfinity if bgt1 ancl decreasingigcmsif blt71 Mb i bum Properties of Logarithms for positive numbers m and n Logb mn Logb m Logb 391 LogIo m n LogIo m Logb 391 03b mquot lll39ogb m 4 Logb 1 0 5 LogIo b 1 Q 0 Use the properties of logarithms to expand and simplify the expression log gt562 58 WI 7r 0p r0p 3 l Expreas the tier in fogari unic form 3 l u y 1 quot 3 81 q 3 g l 1 logggw W V Write the ressio a Iogari m of a sin leuani v In 2111y 3 111 z 2 ma v Rah 2 ix a Use the taws o loitho expand and simpli the expression i av ago To 1 1 103431 05 Lx w 1 I 2 Prop 3 203 Q 5 293 as Sketch the graph of the equation Seiect the correct grap y loguacvl a I 5 gab 4 2 6 d 2 4 l5 ngm s39vznt Previewer Win aws Intem t Ex mer Sketch the graphs of the equations on the same cooyngte axes Select the correct graph 9quot y 33 and y In 31 W A property relating ex and In x m X 1amp9 1 In e x wm because Kat K Le XI Use Euga hms to solve the equation for t w 42quot a VW Ti Stake I Use ozga as to soive the equation for t g 904 5 O rithms to ve the equation for t t 40 0 u x H m a l 1 gt A no 30 7 702 q 397 19 9quot The height in feet 0f a certain kind of tree is approximated by z 130 W 1 195e0w eDVVV 9quot years The temperature of a cup of coffee 1 min aftea39 it is poured is given by T 75 SOs0W where T is measured in jag quot a What was ih htemperatu 155 Wop Interest Formula A Peu P 2 Principal Dr camal r Annual interest rate compounded I t Tie is years A Accumulated amount at the end oft years quot Find the interest rate needed for an investment of 8000 ta grow to 9000 in 3 years if age compounded con 39 uously answer to the r1555 Imam afa ercenta e dint 3 000 g R p w DD 3 1449 3b 0 mg s i i Find the interest rate needed for an investment of 4000 to triple in 8 years if int tit is cumpu 5222 How long will it take for an investment of 8000 to double if the investment er t at the rate 0 ear compounded uni 39 I 5 Find the accumulated amount af 4 4 yr if a 5 invested at 10year compo nded comely will 644Dz 56903 Find the present value of 3239661 due in 6 years at an interest rate of 5year compoundedVcontinuouslx 4000 t I a quot 1g 38W W A condominium complex was purchased by a group of private investors for 12 million and sold 5 years later for 67 million Find the annual rate of return compounded continuously on their investment Round your answer to e decimal place 344 How long will it take for an investment of 6000 to double if the investment earns interest at the rate of 7year compounded continuously Round to one decimal place 99 I of W wquot w Objectives Change 301w Browse 4 Web Ass in 065 MW 443 problems Maniu Jer 63 Riemann Sums bow Larva Y Jkga Alah film Approximate the area un er a curve by usmg graph Approximate the area under a curve by using an equation of the function a given number of rectangles and an interval value to use in calculating the heights of the rectangles How does one approximate the area between a graph and the x axis 63 Riemann Sums FOR EXAMPLE to approximate Use a R with two subintervals of equal Iengt 39 of the the H R Cose the be th subintervals For fX 9U it ntervai 13 Repeat with f of equal length n 5 square units For fx x2 Calculate the area be een the graph an the x axis on the interval Show graph HOW A Riemann Sum can be used to approximate area under a graph Rn z 39 on ab i 1 T for Z O 16ng Wadivb Where Ax p a a quot For a given numbe partitions For fx x2 Calculate the area between the grap s x axis on the interval 13 Using 5 endpoints and 3 n 2 x n 5 gig n 10 9 w W 39 quot quot I Can you guess the area an i 3 4 2 3 a go 5 gquot n 0 ll 39 3 3 n9 Hag H5 60 Lef39k Endpoln39rs MPJxA Ab i X 3 393 1 x31 1 3quot 9 7quot Am 2 9 M0Jlt s M Ia4 3 Lef lquot Em 10133 quot HEMHTS L yam Ax 12 Jazq Xat HI 1 b 5 339 39 xl4 cm H MWa 5th 0 L6 3 Xq532 1430 x5 quot7quot L 9LL 29 Am 29 4 01 LVN91 4 342 712 Amt 20 111 9 3Lz939zfz 1 14 I 26 1 MAM gt 779 For fx x2 Calculate thearea between the graph and the x axis on the interval 13 Using the right endpoints and n z 2 513 n 5 n 10 Can you guess the area l 3 4 39 33 w saws 4 my X L 3 xx 339 W x 39 4q s w 7919 103419 L3 3 a g I Mm 522 r 2 if M 37 pl 448 12 1 X3 19a 1L x4334 3 I0o31 For X2 Calculate the area between the graph and the x axis on the interval 13 Using the midgoints and n 2 n 5 if 3quotquot 39 n W E 39g Can you guess the area M M 358ng 3mm M J39PJ quot A 3 a 2113 m WW x Q 1 L Amx 441247 X 33 25 2 3 K7 7L f AX 3 31 39 7 1A 1 8 um TH 11 22 2613 Rf 02 X M IJ x kquot x xquot 5 x 1 1quot x 24 24 x 18 1 1 1 a 1 Am 461M 224 29 SCL w g l z39t39i 39z39t 6 2394 rt l 4 v42 7 1 5 14 m t i bw m l r 1 tx 4 32 C 7 c 39r W 3 Pt Pig X 391 b 6 Lo 3quot 515K Wit I39M39X 1 W W H 1quot quotW quotquot39W aux bx 4 lg 139 6 x 3 a 17 Find an approximation of the area of the region R under the graph of f by computing the Riemann sum of f curresponding to t of the interval into the eubintervale Shawn in the accompanying gures In each case use the midproints of the sublntervals representative paints v m q1 l 70 39 2 a x vumaw ei Figure 3 shows a vacant lot wid a L 120 frontage in a development Road a To estimate its area we introduce a coordinate system so that the x axie coincides with the edge of the straight road formil helium property as shown in Figure 4W Ax WW 0 23 wt 3quot 9 14m 35 VWIDOHl 392 ID f 6 ft Then thinking of the upper boundary of the property as the graph of a continuous function 3 over the intervai 0 120 we 5 problem is mathematicaliy equivalent tn that 0f nding the area of the lot using a Riemann sum we divide the WI 12 equal subinterval s of length 24 ft Then using surveydr39s equipment we measure the distance from the midpoint of each of 1 gnment Previewer W dew Internet Explorer Figure 3 shows a vacant lot with a L 110 fl frontage in a development AX 00quot0 5 a To estimate its area we introduce a coordinate system so that the Xaxis coincides with the edge of the straight mad forming t of the property as shown in Figure b 39 A ovum A 216m HOD m 3 40 X ft L Then thinking of the upper boundary of the property as the graph of a continuous function f over the interval 0 110 we see mathematically equivalent to that of nding the area of the lot using a Riemann sum we divide the interval 0 110 into ve eql ID 3 13 Road V a 39 k quotD quot To estimate itS area we intraduce a coordinate system so that the x axis coincides with the edge of the straight road formii boundary of the property as shown in Figure 3 m 2000 0 witw 5 Heights 7 L 4 7973 h Then thinking of the upper boundary of the property as the graph of a continuous function f aver the inte m LL i J n t J39l u wgw 1 run i z nLx 25J an 1 the Choose representative 5the subintervgls g M n 9 a 4 hi IM L xke 3 H 3 N M u N 5w a g Find an approximation of the area of the region R under the graph of the functionfon the interval 0 2 Use ubintervals Choose the representative points to bethe V mipint ofthe subintervalsfx f x I 1 X4 33 D Vlvg AX 7 l 0 l s z u 1 L 362 1quot 3 Iquot 39 397quot quot X2510 619 5 3600 X x3 10 Dec 3 1 X451 3664 Lj Xo 16 30 695 3 3 3 3 3 3 3 4Lo H L4 7 WW P a Find an approximation of the area of the region R under the graph of the functionfon the interval 1 3 Use n 4L ubintervals Cse the representative points to be the ibendpoints of the subintervals 285 fx 3x 4 AX 12 7M Ww x 5 3605 7735 V I K2 0L lzgt i X33920 5116 333 xqa 93 3 42a 2 quotI v 07 93 Nts39f i he Chain Rule 1 For Fx m F60 3 F X 3 x 439X 2 For FxfgX PU 9quot3 PM f gX g ix n words the derivative of the outside function run on the inside function times the derivative of the inside function run on X ggfiwgfrsignment Prg glgfr Mozilla Farefnxm F39 Ed ew Higtory u okmarks Xahno my w M M M R NA 1 I gt quot lmt 1 L Fund the derivative of the functlon Mobwa wJ 5mg fwiunv 2 m 3 1w m2 my 3 Stdwga j gtaw b b 4 egg f t 4L Find the derivative of the function 1072033 10 3 o and over 3 changing at the beginning of 2018 Round your an h quot per veer 5 ID 2 83 t The popuiation of Americans age 55 yr and over as a percent of the total population is years with t 0 carresponding to the year 2000 ogt zo per year flog What will be the percent 0 th oulation of Americans age 55 yr and we opeK 0 3 l 1071 969 Io 3 334 39r 43 6 1 at2 13 O A 41 5183 301 4 i39aP three decimal place Wnill grcent 0 961 HO 7 410 i 3 Ithfvrfcmvti we 294 c 394 quot 4a3 3 AQme o Anwsm A quothaw 4 h 3amp quotw 157 2 54 r her 34 Marginal Analysis Derivatives of Business functions Objectives 34 Marginal Analysis with Business Applications R x C x P x Average Cost Function Marginal Average Cost Function Elasticity of Demand 35 Higher Order Derivatives Today we shall see what the derivatives of Business Functions tell us Marginal analysis is the study of the rate of change in economic quantities Marginal Revenue means the derivative of the Revenue Function ie R x Marginal Cost means the derivative of the Cost Function ie C x Marginal Profit means the derivative of the Profit Function ie P x At any quantity these functions are approximating the effect of the Next unit Assignment Freviewer Windows Internet Explorer oebsw mm quot a I7 Pulsar also manufactures a series of 19in color television sets The quantity x of these sets demanded each week 15 elat rice b the followin function P P Y 9 R t P x h 39 c 39 p oosx 300 o g x s 120 quot I a Find the revenue function R The weekly total cost incurred by Pulsar for producing x era is repr nted time following fu t39 L0 Cx 0300903 009 S4Ux seooo E o R a 4 v hvw39 Cl 1 05 a Find the pro t function P t D x t D D o Pix bags 240x 000003273 88000 K a 0 in Find the marginal cost function D 1 c v I Find the marginal revenue JZEt Ion D amp1 R 39x 12L 300 Find the marginal pro t function a P 39X 051 240 000009182 g 39 o D 08 q 1 1 obxaw 0 c Compute the following values Round yo r 3 er to 2 decimal places C lBOO 432121 V I t R 39E1300l144 W 03005 DDDODQ I300 v 03060 I 9 g r39mgd39G S M3 gt win 3012 M Esme revenue P 113 l12 ll Illquot Williams Commuter Air Service realizes a monthly revenue of Rx 16000 x 200 x2 ollars when the price charged per quot quot quot l isgtIltdollars Find the Marginal Revenue R21 4090 max Compute R 39 R 40 R 41 Enter as three numbzrs separated by commas in the order R230 quotDpo39qaoa listed above 3939 4 00 R240 3 lbDDD lwo In 40004oo 0 Based on these calculations wiice should the airline R39 u Moog ch a a 400 73 x 579425 p fg 2 Val t wow m 141 Lynbrook West an apartment complex has Egg two bedroom units The monthly profit in dollars realized from renting X apartments is represented by thequot following function Px as 7639 SEN H 930cm 350m a What is the actual profit realized from renting the Slst 3 Dinr unit assuming that 50 units have already been rented s 1425 Wm 393 flra Mauls 3quot w quot 4 two b Compute the marginal profit when X 50 and compare your results with that obtainedin part a e51quot ma w 51430 39x m kl I930 p T ruCW From 1430 P250 39chfD lq3l mow memgtlt g6 cmsEcmo Avequ ecost Function Suppose Cx is the total cost function Then the average cost function enoted by quot I Cx cm i X Th e 39V39 a rain 339 function measure the rate of change of the average cost function with respect to the number of units produced Custom Office makes a line of executive desks It is estimated that the total cost for making x units of their Senior Executive model is Cx 300 x 200000 dollarsyear Find IimX H00 of the AVEragQCost Function 3 Find the unction 390x awaaog 3 in 3 3w a m ca 3 3 Ac xivwvw 3w zww W303 Ciquot swag x X 9 K 12 I 0 DH Avg 395 Jpgpgaiz 300F ammwux Elasticity of Demand re ntia ble demand function defined b then the Elasticity of Demand at EP f I M quot 39LJ39E Me m a r mew fp Elasticity of Demand values The demand is said to be if Epgt1 The demand is said to be if Ep1 The demand is said to be inelasti if 3 Assi nment Previewer deows Internet Ex inner a 0 A E Q 3 quot P 29 For the following demand equation campute the elasticity of demand and determine wheths the a is etastic unitary or In indicated price Round your answer to three decimal places a q a a X P11f 54 rm Em elem M 6 4 Need Help XP aasbmw v 39W 7 x 5233 7 399 9 9 For the following demand equation compute the elasticity of demand and determine whur the deman astic unitary or in indicated price D8Xp220 p11 39 Etieet Q a prce can 5 revenue Pr I If the demand is elastic at a price p then W3 W An increase in price will cause the revenue to decrease A decrease in price will cause the revenue to increase Etiect pi a prce change revenue 39 I6 Tr39u fw39o If the demand is inelastic at a price p then An increase in price will cause the revenue to increase A decrease in price will cause the revenue to decrease E 3 Wm mmgg YQVQHUQ If the m WW VMWW m dwrmmi m ymmmiwmu 45 Guidelines for Solving Optimization Problems 1 Assign a letter to each variable mentioned in the problem If appropriate draw andlabel a figure 2 Find an expression for the quantity to b waged m 905 3 Use the conditions given in the problem to write the quantity to be optimized as a function ofgng k variable Note any restrictions to be placed on the domain of f from the physical considerations of the problem alaui Optimize the function f over its pertinent domain D agem W 3 The owner of the Rancho Los Feliz has 3000 yd of fencing material with which to enclose a rectangular piece of grazing land along a straight portion of a river lmngb not reauired alepg the river what are the dimensions of the largest area that she can enclose 150 yd shorter side 1500 yd longer side What is this area 1l1250005q yds l Jena 2 905 E hkfza M X f w ifs res quotw 39 300a 3615 055 cg x x 31mg M 342x 3 ww 3 MwJx 4 r x becomesxj v ALA X500 3934 439zo 3 30cm X ax 394 Am 3WQ IIx 0 30w 4x 4 3mm X 7f 50 3 A a m gz3 00750 u m MAX39MLM Area 3 750205a I ll oaj The owner of the Rancho Los Feliz has 3540 yd of fencing to enclose a rectangular piece of grazing and along the straight pom owner would also like to subdivide this area b means f a fence running parallei to the sides 0390 x 1 Wyaxzwm Eveer men13 3 L 715a Redm39mw 352mg 5amp x44 3 1 30 girl 44 46x X 3331 391 la Am 3540 xw 3x amp V CAI WLVUS W0 35 x Miima39wo 6 0 gm W quotm i X33 3977U 5 95 Z q 33354A H Assignment Previewer Windaws Internet Explnrer The management of the UNICO department store has decided to enclose an 854 zparea outside the building for displaying pot owers One side wilt be formed by the external wall of the store two sides nstructed of pine boards and the fourth 5i gatvanized steel fencing material If the pine board he 39 quot V running foot and the stee n u ive each answer corre w an g hrunning 1 to one decimal place t 3 5 55 I0x39 ggt aquot Wax 554 r12 KJ r0121 10 x df a x4 xiii 0 2 Mg 3129 625 zDx v 2319 s x Mfg x I 6be39 t V004 Sale M quot 0 5 5 a 3 a 5335 b 3123 534 x quot5 quotJQJ answer j w er ems uU 39 Assignment Previewer Windaws Intern t Explnrer see and a volume of 104 in3 and is constructed from a tin sheet nd the dimensions of the box 2 used in its constr reb 5 in largest value A rectangula ox is to have a s mare base and a volume of 20 If the material for the base costs 33 per square feet the m sides costs 10 per square foo a e material for the p costs 17 per square foot determine the dimensions 0f the box th constructed at at LMWJMIZB Madeth v xquot uwwww ew 96 Hag My 3 2amp5 ea RESTquot 39 f39ww V39Q Lwag x x M i ht LEI x35 33 k 1 h v gar R k m 39 X 561 xfiw39 xx 67 Mfr 392 j 5 quotquotz Ma a x w 37W 5 nquot 4 0quot 39 g 723 23 MWsz Cm aw a V Md has 13 hW t g L031 5035 1 x3 39R 39 uon W v szz VOLUME 320 Xxx 920 50 f xl k I g H htw Each side of a square piece of metal is 42 cm A small square is to be cut from each corner and the metal folded totorm a box with no top 9lf x represents the length of the sid removed find the 391 H i e rqur which represents Vx 42 x422xx Find the side length of the square which should be cut to maximize the volume 7 What are the dimensions of the largest box that can be made Round your answers to 2 decimal places Height 7 XV M Length Width X2 x 7 r Xl gnjmw y an g ux 342 x 27 WM 0 EX14m 37 WU TWA Nix5 xpsa xhzvh l TOOw WW z39w39ww quotM lt9 0 X E wZv xv27 WA w q l fzhhx m Tm 8 x 07 IVE F1717 jg 4 xf I Y I 7x 3007 The owner of a luxury yacht that sails among the 4000 Greek Islands charges 210personday if exactly 20 people sign up for the cruise However if more than 20 people sign up for the cruise up to a maximum capacity of 90 then each fare is reduced by 3 for each additional passenger Assuming that at least 20 people sign up for the cruise determine how many passengers will result in the maximum revenue 45 What is the maximum revenue 6075 What would be the fare passenger in this case 135 5quot Kg 05 2X0 z X7 0 79 X7 X7 L quotquot WW 5 00X x05vz39gtlt39gquotw x L x0e oazfvmxgmxgm WH 9 XE OXPJ 03 ghuaAaX w M 4g 0027 Maw St Emua ZX 2 C X l 7 a Haw Mam PQSenjer539 i I J Maxr39Muwkexenue 5 now 6219 615 0 quot395 95 75 3 4M Q Fart As 70 3525 57935 54 Derivatives of Exponential Functions rt A162 Find the Bresent alue of 3239661 due in 6 years at an interest rate of gigyear compounded continuouslxl 4000 3 x a g a L I my g A g mi V w 33 5 u fquot 5quot 19 2 a k h A A condominium complex was purchased by a group of private investors for 12 million and sold 5 years later for 67 million Find the annual rate of return compounded continuously on their investment RoundMLanSwer to ki onech 344 535 39 aw 6 How long will it take for an investment of 6000 to double if the investment earns interest at the rate of 7year compounded continUously Round to one decimal place 73 54 Derivatives of Exponential Functions 1 9 Rule number 1 a be Iff x X e 09339 f x equot x 01 10 I h 139 Wow that s pretty easy ms is f9ke Find the derivative fx 46 WW ya ef vii Fin the derivative 39Pro ucl fltxgt 6xgt an mquot w 6 nex 9 99quot Find the derivative of the function ew X 5 pm a u Find the derivative of the funct mn X fm quot X K I V f 39x Find the derivatss the nation39 2 quot1quot Mrs e Ruiz i w x Rule 2 j Jil 0 5 O y I ag h39ow p hi 4am y efXf x 3 In words if y e to a function of x the derivative is e to the function of x times the derivative of the function of x As the problems become more complicated You once again will have a choice about 1 whether you want to use the Quotient Rule or I Proguct Rule after bringing the denominator up and changing the sign of its exponent W Find therivatzve of the functio 7 t Se gt 3e e 39 M 2 V 7 39 ub v 1amp1 39 7e 3e 43921 quot7 I V M33 quot 39 W a ma The unit selling price p in bre and the equation p 1226430005quot o g x 3 200 WOW 3 I x 39e one 3 Find the revenue function R uwgawos 500m b Find the mar inai r v ue function R39 mane AWE 0005 e Wm V 3 9quot 135fmva 60005 mac 95 be me puffth I c What is the marginal rave B when x 1000 I calm MW 000 e 3qu v 4 3 0 dat39 33 53905 aei bcoaa tae 0 518 393 173 351 3 4 9k fly 127 L633 t 6quot w a 84 m m yw e Find the derivative of eX3 2 etoxcubed 3 mmus 2 Jet 524quot M r flan Ffz 3 1 if ignmant Previewer Windows Intemet Exp The percentage of alcohol in a person s bloodstream t hr after drinking 8 uid 02 of whiskey is given by At 021te 9393t o s t s 12 a What is the pmgeoelcohol in a person39s clstream a zer 12 hr After 10 hr gym answers to 4 decimal pisses l V V V WM after 12 hr t 9 I after 10 hr c 1 a 1 39 v M Jar Noted b How fast is the percentage of alcohol in a person39s bloodstr er 12 hr After 10 hr Round your answers to 4 decimal I 2 39lquot R5 ozo hrfaii ii A t a 7 we Hula 111 quotSt2 I Assignment Details The percentage of alcohol in a person39s bloodstream t hr after drinking 8 fluid oz of whiskey is given by the function below A39 t quot3quot quot3quot At 017te03toctc12 quotl e m e 6397 At what time after drinking the alcohol is the w percentage of alcohol in the person39s MaXIMizL bloodstream at its highest leve A ampn t 333 hrs g V Wha is e lev gag 0 v 11 l L H 3 43 A gz g eggDZZ H tatl5 5333 hrs Find the equation of the tangent line tcl graph of QROEQ I fx ex 2 at the mini133 DP 39 2 1e4 I cb 3x 39 39 529 3960 e fax amt2m I f a v a 639quot i v AD 19 mxxj39 I If x2f qgt3z EidxaDlJ Given fx x2 ex stalemm hwkjobs Determine the intervals where the fur Etion39 39 A Increase Ira X18 quot l e 41x B Decreife s Zx xz ax 13 Mus vfad g xx39zampgt ax 5 21 A 05m l 02 x6444 690 023 19 mm 35 lt 39 l 2 D 2 j s faluas c393 l uniW 52 lt o altquotgt1e 396quot 40 Me auboL P amp5 4 03 quotM e 0 103 6 m 5 o 2 Tamas 3 J 3 3 m ab e 3 quot3 quot3 39 dwhrvals 395 ea 40 Ne a39hvcg tie1 beneath c M 0 U a 3M I 39aj Given fx x equot Z devwa39t JLGk Jo Determine the intervals whereathe function A Is Concave U p x 393 B Is Concave Down 52 X x gt 7 l 3251039 5quot xl VG ej Y t Ha 393 I63 7 I uampor39ex 444414 a iim fu c 3 xfxquot Mama wmogt z muzy f zg 25 4 90d 04 CT 7 ag gr a f WG9 V S Ca I 393 0771 7 Z l 53 0 awn 571 1 5 r 2 ax Qt V CTN mm m an WW 2 m L The development of AstroWorld quotThe Amusement Park of the Futurequot on the outskirts of a 39crease the city39s population at th j below in peopleyear t yr the start of I construction P3 3 3400 in 7 000 cage quot th t ulation 16 yr after construction of the park has 53 Dbegun 5 Nail Emui3 Emul awake n 714 S ww quot 7000 it 35 t lquot 7000 t ya x 28 9 3 45 away I iv 10w 6 7Q Mpg 3quotquot 4 7mm 6 Q p02 53000 33 ll 00 D Mag 0 J50 3 43 Sigma So we Amt moatm j ues aw is b 3 5 Wm M00 0quot 4 790Q kquot 53awa a Z 4w Given f 39X 6quot 3X 0 4 Find the functionfgiven that the slope of the tangent line to the graph offat any point xfx isf 39X and that the graph of fpasses through the given point 29 I350 S 3amp x a 0 amp 3 LL new e32 32 m I Schg 4 ha a 43 903 6 340139 4 I Inc 5 ls wan It 3 J mixzw nenquot3 Objectives 9 4a m 2 62 9396 413744 Integration b ubstitution How do we undo the Generalized Power Rule to put the original function back together We will use the technique called integration by Substitution Noatomquot Integration by SubstitutiBn 39 1 v a 1 Leta ggtltwhere gx is a part of the integrand usually the quotinsidequot function 2 Find du which equals 3 Convert the entire integral into one containing only u and du by substitution 4 Evaluate the resulting integral 5 Replace u by gx to find the final solution as a fu nction of x y 3 Myquot x w 3x o w lt mm w f Find the indefinite integral Description This assignment covers section 62 computing inde nite iniegrais using substi u on Find the ind y integral H s 1 f1 3915quot 1328f 1 Emmi E3 289 m I 44quot 1 Need Help Find the inde nite integrai 399 a 1 33 13 36 L1 w gt m3 rz2 37 i w E 33 y H Need Help 5 hag1 0 AK XS x x I m 93 u 01w 5 3 2 4 5 cw ltQabxiz gte Mad 3 z 7 5 294 b I Ltd M W WN 4 Find the IndEfan t 39in39Ee ia lNb uxg M Aw 1 dx 4 39 f L f w 2L gnmem Praviewer Winduws Mamet Explmer Find the Inde nite integraf 8m 4 x3 4x 213 411 I 1 in H 69fo 171 Mend Help Find the inde nite integrai f 33 31 2 la 1 108 3w4 29 3 27quot 2quot C Need Help Find the indefinite integral I 5x2 dX 9 4x3 391 Rim 8 a A 01 974315 304 u VTijsWP Am A 7754 mam Mk 939 Assignment Previewer Windaws Internet Expla 3 Sm my 31 quot3quot IxeM A 4 Suddw lt5 l n zbs 05 V Need elp Find the inde nite integraf f L 4 1939 13 12m 1 quot 1 y v 1 E111 abs 33 121 1 Eln z 127 If 0 Need Heap Assignment Previrawer Madam Internet Bcp39lnrer Mead Heip Find the inde nite integral 3 9276quot 2 d3 Wad Hem Find the inde nite IntegraL 841 f 6 34 at W 1 1116 if 1 In 6 643 C Mead Help S GALA tuxI A 3 3 x14 3 I llll yo X dx A 3 9va 5 A a f m a A v a PO Find the inde nite integr w 1 L 1 11133 if L VC The average student emolted in the ZO Wk Court Reporting I course at the American Institute of Court Repurting progresses the rule below where 0 S t S 20 and N39t measures e rate of change in the number of we 39 machine shorthand after 1 wk in the course e dictation the studen N 1 49 035 iigsezthe diefetio speed of he student after t weeks i 5e N 3 4 5 I m l I 5 U Vd39ll Objectives 32 Rules Shortcuts for finding Derivatives Basic Rules Product and Quotient rules Real World applications 32 Rule 5 Product Rule F0r px fX39 9x the derivative p x Kis given by 9 x Bf x 444 2 p x 2 4st B Find the Derivative of the function J N M fx x12x2x Q50 Judi 39Z39J Jan Iquot 4quot 397 L844 u D 1 zx vx 6 50 X3 3 Xax Find the Derivative of the function r1 5 x r 9 le3X j2tw s 4 5quot My 330067 f 6 wx CX gt tVQJ Qx itlxxl39uq 50 P 32 Rule 6 Quotient Rule For qx tbs p gx the derivative 10 3733 q x is given by 0W gm f x fx 9 SIM 2 m mm to 982mg Bo om Find the Derivative of the function fw 3w2 14 5w2 22w fit w aVltIoW quot 3w 14 ID W 7 L5 wquotw 1 0k Find an equation of the line tangent to the wgraph offat 1 1 quot o x3 1x2 jquot 39 3 539 2st LL Ia3 op Ej 3 Q fxd 3920 xscj39 in 33x1x Jag Ql Cx aquot 0 b I a quot 0 3HS I 5 S39Q 39 3 6 a vb Q3431 a 2390au 9 5 7 393 gquot 4quot3392 113 xXX2 81 4 9W 6 chi39 W ww aw or the function f find the points on thgz a graph where the tangent line is horizontal If there are any unused answer boxes enter NONE in the last boxes o x y fj sma m value of x 9 X y wargecvalue of X 521 6 903 39 xc x wn 415 amquot a w 7 939 1 1y N a w o 13 it0 a osqvxmw oqx2xa Assignment Previewer Windows Internet Expls The demand function for the Sicard wristwatch is given by the following function where 2 measured in units of a thousand is the quantity demanded er week and dx is the unit price in dollars a F 7P6 391 US320 d a 6039 Dlta quotLOY 71 0011 1 Sigma L v 0 x H L b Find the following values Round your answers to 3 decimal places z NW I 0100 1 Al 2 lt 7 m 9 6 on an I 11 d 3914 I The total worldwide boxof ce receipts for a long running movie are approxi ted by the following functio what 700 is m red millio of dollars and x is the number of years since the movie s release t 1 TM v r 390 H 5WD x How fast are the total receip yr 4 yr and 7 yr after the movie39s release Rn d x ane992 decimal places s v I 90 H V millionlyr after J V millionyr after 7 yr V V minionyrrarter39ur TI 30 33 Chain Rule amp Generalized Power Rule Objectives 33 33 Generalized Power Rule lffx gum then f X n gX 391 8 X 39 W 9 1 For FX g fx 3 PM 839fX 39 f39X 2 For FX f gx Jquot F39X f39gX 39 839X The Generalized Extended Power Rule 39 If fx gx then 3 nah n M n goo ltn1g39x rm c 3 Powc39 L unb 15hgt er waquotquotquot I0 b 50 lt3x44x3397x HZ a J 7 3 4 3 J a ID 3K 41 7 44 if Max 7417 X 5fvlgt 12 Find the derivative of the function 39 fx 3x 16 fix a 3x4 3 5pp Find the derivative of the function aim Kw n7 39 fx 6x2 6x 712 I an 5quot chz gw o laxe Find the derivative of the function Mug f 5X3 6 x X2 x 1 mxwm NV 331 q 39 7 quot 4 w titmi 33900 390 f0 9 15 122 2 a 4 w a 4 u t 3 3 I 4 w it 3 4 35ng gt Need Help 51 1 V quot39139 397 Find the derivative of the function V u v39311012 9 3 u a 5 f M L x 0 lav2 3224 2011 96 3V 3 390 v a 13v v Need He 3 p Mid39 1 30 m5 735T53 35i33 We tquot3ae5 a 4 Him a x 3 5quot seo Need Help LDQMJ 7 f 39 g ssigr gqi Mozilla Firefox Eile Edit yiew Higtory aookmarksxa eip Find the derivative of the function 5 H mi 391 r39 f U 1 1 4 a if Em RQDW WT 3 5mm fwmu 3 3mm et quot39 ut4 91 1 39 New Find the derivative of the function 1 g t an 832 8f2 2 1 2 w is i w W9 192 15 31 2 Suppose h 039 5 f 39BII 4 90 3 and 9 Ta 7 h m quot Evaluation of a Definite Integral Using Substitution 65 Definite Integrals by Substitution 1 Let u gx where gx is a part of the integrand usually the inside function 2 Find du which equals g x dX 5k 3 Change the x bounds to u bounds 6 4 Convert thgentire integral into one containing only u and du by substitution 5 Evaluate the resulting definite integral using i u values Evaluate the given definite integral W x Mr N at Mei5r 2 quot WM means 9 aluf 24w 4 if 5 M 01088 d m 240 a H8 at quotElitequot 9 9 wk e gnment Previewer Windows Internet Explnrer 3 New 50 lt1 szop 3C7 quot39 739 u 4 l l 13965 4 Evaluate the de nite integral 601721 10 m 7 j 2H3 Xa3X39 le3 0 cit 4 x w a 3x L W 23 XM 2 d x c x 4 AliEgg 1L Wagofi tf 340 quot 2mm 6 4 4 3 4 4 4 3W0 3 3039b39 39L gualwzy i Lj2 w 290 I If 4 4 391 391 a 7 Assignment Editur Windnws Internet B39plnrer m Evaluate the de nite integral 1 1 la 1 339 db Evaluate the de nite integral 3 1 2 rf m i 1 3 S lt2 agtozx amp w 4aquotu39 I 39 20m WATKBI J39XJ m M 02X A x 6 k cwu aw vquot 395quotquot quot393 3 6 Mg 9 a 0W Jgt LL e a M e JM 2315 Assignment iditor Windaws lntemet Explurer Evaluate the de nite integral f Nita 1amp9 Evaluate the de nite integral gaff 303 3 w L M g m tww uhd u pw 2 6 e Dobi zxz A 707 Evaluatethe defmntemtegral dvdx 53 If sz xe f x 3 Zap 24 V 39 4 w w 1 Wardquot u 1 0 Se Axux 400 MA 3970 D 9 quot Average Value Problems Suppose f is integrable on ab Then the Average Value of f over ab is 1ba aJ39b fx dx 5 1 N M 0 a a L M H t 8 fax bu I i S X n 00 Find the average valu of the 7quot function ub gnme i Previewar W dnws Internet Ev Find the average value of x Se X 55 Objectives 1 will 14PM msz rada i 0m391h3f39 mini yam kt 30 itsf jauru Md 6 i39vv Mantle WC be 639 Z V 9o 41 30 quota Makeup class or hurricane AmpW are on schedule with pao erla 55 Derivatives of Natural Logarithmic Functions Business Application rate of growth of online households Writing the equation of a tangent line Natural logarithms have made it to the 21St century 39 The number of online buyers in Western Europe was expected to grow steadily in the early years of the 21St century The function below for 1 th gives the estimated buyers as a percent of the total population where t is measured in years with t 1 corresponding to 2001 Pt 282 143 nt 282 603 UV How fast was the number of households in with online o the beginning 2 Derivative rule 1 for Natural Logarithmic Functions 39 If y 1I1X Derivative rule 2 for Natural Logarithmic Functions If y 1nX where fX gt O u 9 w L Ndkvi x w gutted ch 39 theny f XfX a 33 W 5R g 5quot Aszignment Previewer Windaws Internet Explorer Find the derivative of the mctinn x I V x x In5x1 Find the derivative of the function gx ln7x 3 9 39x Find the derivative of tha function Mt 5 was i6 t Assignmeni Previewer W draws Intemet Eprorer Find the derivative of the functian quot I 39 l a f 3 11103 22 1 a f39ix 1443 2 v 14 2 aX39F39 Tw223r1 739 2er1 m4 md r on in Find the derivative of the function Be sure to enter arguments to the quotInquot function in parentheses quot x 4 max WP M Find tlle derixative of the function 5411 PM fx 5 3W9 m 5 gtquot gt mygg A wxra 964 My 1 MW 154 auo hin39l39 nmen Previewer Windew s Internet Exp lmer The number of con ne buyers in Western Europe is expected to grow steadity in the coming years The function below for 1 S t S estimated buyers as a Bertth of the fatal population where Us measured in years with t 1 can ndiing to 2001 pm 29314031nt1 P l 0 V In a What was the percent of unline buyers in 20111 t 1 0 waded 3 I J 4 T a 03 1 3 Wal b What is the percent of on ne buyers expected m be in 2006 t 6 a to PM 34 3 3403 40ng w y WM W 5 2331 a Determine the intervals where the function fx In x Increases Oe Decreases eoo 47 x mac 94321 Wkimxvu 1amp7va x x r a ciem v 002quot v mwa a w 1 msme Cwa REE me wand V 7 Decreasmj am 933 Determine the intervals where the function fx In x X x is Concave Up e32 00 is Concave Down 0 e3 gy 0231 beriva l39 o 48 F ON 3 Io ILL 0 J a 8 00 6 mx at quotGNP k u amp f 3quotquot U xquot me mad i393IMej7x f oH x x 391 X TaMNW quot R I x4 31quot24 qu quoth e x 41 XI Wquot W M I 3x MgtDgt 3 V 393 giggw faw V mz 2 Laban XO at quot3Mx 0 3aM0 xii 4M ki Wxgy ay 139 46quotquot x a gt 0 8 0 25 if hfng a w in I t w W 4quot 4 3 5 m 3 Ngja39hvlb 3lt0 5quot5 3 quot 35 4 20 Comme DM g 5 fuse 57 520 I0 w f 021751 gt0 D5 u p x Continuity at a point A function f is continuous at 925 the point x a if the following conditions are satisfied 39 a fa IS defined 133 40 339 lim fx exists quot4 Xe a 1 39 399 ligufuq 0 lIm fx fal x gta Assignment Previewer ows Internet Explorer a nd at xa Determine the values of X if any at which the function 15 discontinuous At each number where f is discontinucy e the co continuity that are viotated Select all that appiy w 1 ifs 0 f quot 101 7g 39 D a 0 WM 9 40 0 Y 3 it 0 32quot xvw39 4 NETI S Pem L pgt Herc U L graph A lms liquo x is not equaf to fijo The function is not de ned at the paint 0 The limit does not exist at the point x D a g 5 Lgfm Asgignment Previewer WHK IDWE Internet Explnrer Determine al vaiues ofx at which the function is disc 39 ou a v fx 1 XQ D x goaD 12 9v 10 x q x H N smelter value 39v X X i1 JV tamer vaf a limu f 11inr m 152W 9 Defining a value of k to make a function con nuous nd 2 Find the value of k that will make 1 39 continuous on 0 0 if m a 3 at quot ng it 4 4 m 73 3EME39 a 515 X 393 73 v o Jud trmmn M Definition of the Slope of a Tangent Line Finding the slope of a tangent line using the fourstep process Writing the equation of a tangent line Definition of the Derivative f x Finding the derivative of a function by use of the definition Real World application Q 599va I I I I I I h Mab iqv 15 l 50 r 3403 The ope of the tanqent line to the graph of fat the point P x fx is given by lim fxh fx h gt 0 h if it exists 27 the slo the tangent line to the graph of the function fx at any point M a 1fXh g IPHCKPW I 39 2 fXh39fX quot quot WEJL Qyj 4k 45 394 k I MWK 3 fXh fx h 4 Iim fxh fx M kgtu the 1 tangent Inn to the graph of the function fx 7x2 9x 3 at any point 39 7 7xh q4xh 3 X f 7x zxwh 5 qiqhes 3 r3 u VK39Hka Th Qxqt5 o X X r W 7 77 7 a 1k qh t1 3 4 rh e derivative of a uncti f wit respect to x is the function f read f prime f x lim fxh fx h gt 0 h The domain of f is the set of all X Where the limit exists 30 V7amp Bible Derivative Rules 39 ie V Short Cuts for the fourstep process 7 Rule 1 The Derivative of a Constant If fx c 39 Then f x O If a function is constant its instantaneous rate of change is zero ie it doesn t change Rule 2 The Power Rule If fx xn for any real number n Then f x nx39iil a 51a X3 51 ix Rule 3 The Derivative of a constant times a function Is the constant times the derivative of the function flag 5x1 47x v 2 Rule 4 The Derivative of a of functions Is the sum of the derivatives of the functions 3 0 yb li REPS f BF 30 xx H N 3 20 Jot the derivative of fx 7x2 9x 3 WANquot is Hm aler39waf equot 7 JF U quot39 H wa Na W39 wej39s d0 Find the tangent line to fx 1x at the point 3 13 Sh w E lay 431 is g 5 393 hawk 5 mm H A 33 Assignment previewer Windows intemet Explurer Find the slope and he equatio nvor anaeg gine to the graph of the function fat the speci ed pomtDo not use 39 SH 4 fwv 1Eat 4139th V 4W Irer 7 9 m D609 ux x quotMush e minus on the graph 01 f where the tangent line is horEzontal If there are any W 4 uised aiswer boxes enter NONE in tl39 amquot J 00quot 39 II 3 39 ma quotw iv zg I 28 a 4439 I smaller x vaiue X 2 3 53963 X m 1 39 X9 g d ope 1 3 xa w 41x 0 2 QM39 g 4 0 09 k 394 X 394 aka M 394 3 yaw C1456 h q P73 39 S m M 3 Assignment Previewer Windows Internet Explorer As use of the Internet grows so does the number of consumers who shop online The number of online buyers as a percent c j represer ed by the following function where tie measured in years with t 1 corresponding to the beginning of 2002 I I I quotL39M mq 2 eat 912 where 1 Q t E 7 J a How many online buyers as a percent of net users are re expected to be at the beginning of 2005 Give your to 1 decimal place m 525 Moe odtjdpp r 139 V t 3 Q V 139Lb How fast ili nlbe 30nineuyere as a percent of ne users expected to bat the beginning of wdf answer correct to 1 decimal place g K WEEYr 1 r wk n A 1th Y Maxiuse vwa ve u v V gt I 39W Ks iat w M The demand function for the Luminar desk lamp is given by the following function where x is the quantity demanded measure andp is the unit Price in dollars 4 4 g y u 2 pfixo1x2 04x32 0 I 6 3 Find f39x b What is the rate of change of the unit price when the quantity demanded is 8000 units 5 8 7 Enter INCREASINI rate DECREASING for a negative rate or NO CHANGE for no rate change Enter the absolute va tie for the rate itselfy A W nrrwunninn decrees H 2 g 3 a 33 As 39gnmem Previewer Windsw Internet Explmer Perform the speci ed tasks on the following function fvm912m a Find the derivative 1 39 of f we v in nd the point on the graph 21 f where the tangent line to the curve is horizantal Hint Find the value ofx for whict m u D a 2 I h x g L graph offend the tangent line to e curve a e point found in part b M L1 3 03393 9 a m Assignment Previewer Windnm Internet Expiarer b 5 2 12 3u5 39 J a i 39 A ball is thrown straight up with an initiai veiecity of 123 at its height in feet after t sec is given by 503 128t 1 a What is the average vetocity of the bali over the following time intervals 2 6 t 39 II II 6 av sec so 23 0 Values 3401 gt L Lgr9 2 Manta agov1 r N v v instantaneuus elect at t 1 2 237 220 quot i Lagi 543M c what is the Instantaneous veiocity at time t S 1quot quotmil44L Is m falling or neither at this time I72 r39 24 w as 7quot 3 4 27mm SQ 12 mt s 6 air 316 25154 50 I gg n 5 gat d when wilt the ball will hit the ground t who 03 27 C ILD 5162 7 seconds later a Q Q L Efb Xmi m x 3 m x mm m x33 m m f a 35 33 mad w W 3 WE m 3 x3 x nd We mam mm m ahangg M y rewem m x a3 x mm 3 3 X333 m x34

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