MATH CONCEPTS CALCULUS
MATH CONCEPTS CALCULUS MATH 131
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Math 131 Exam HI quotSample Questionsquot The uver4174 55175 2 a 1 and a z The su1uuuns eanhe fnund atthe and 1me dueumem Gwenthe graph uf 7391 answer the ulluwmg z quemuns NOTE A15 smut aIVSEI B15 2mm 3173 572 C15 smut at 7211 1 1s 71 2 E15 amax at z a and F15 smut at 411 These shuuld be labeled unthe gzph ufthe denvauve beluw h 13de e nune ufthe abuve e nune ufthe abuve 3 In a lab experiment some mice are injected with a drug The concentration ofthe drug in the bloodstream ofthe mice can be modeled by 0 2 61 5 where t is in hours since the dru ms admmistered Find the bioaxmilability ofthe drug 3 hours a er injection d 117 e none ofthe above 4 IN 2 3391andfhas critical Wines 2x 0 and 1 use the Second Deriwtive Test to determine which critical Wlues give a local minimum of x a xl b x2x0 c x2 d xoxl e none ofthe above 5quot Assume that during the rst three minutes a er aforeign substance is I I39 I II ithemeR Mm 1391 391 I39 R39t produced in thousands ofantibodies per minute is given by where t is in minutes Find the total quantin ofnew antibodies in the blood at the end ofthree minutes a 115 antibodies b 300 antibodies c 1151 antibodies d 1501 antibodies e none ofthe above 6 Find the first derivative of W a 22 b 2w 1 c 2e 2x d m 22 e none ofthe above 7 Given below is the graph ow It has roots at 2 and 4 The o u onio above xaxis and 3pa11 belowxaxis1f lt gt 6 findZ and N and tell whether each is a local maximum local minimum or neither of f0 Note Thls is the gaph of WV 8 Find the derivative ofthe following functions You do NOT need to simplify your answers a mgtlt 2rgt5 b x 2cos x 35ln4x2 1 c x 2quot ln9x3 3x 1 x lsx lnoxz in x 9 Given x 7 4 4 10 answer pa1ts a g a Find the critical values off b Find the intervals where f is increasing and decreasing and give the x values of the relative local maxima and minima if they exist c Give the intervals where f is concave up and concave down and give the xvalues of the in ection points if they exist d Sketch a graph of x Label the x and y values of each relative max min and in ection point e Doesfhave a global max on mw l If so where f Doesfhave a global min on v an Ifso where g Doesfhave a global min on 0 2 If so where 10 A company s revenue from car sales S in thousands of dollars is a lnction of their advertising expenditure a in thousands of dollars Suppose 3a 5a3 300a a What is the revenue from sales if 6000 is spent on advertising include units with your answer b Find ST Interpret your answer c According to your answer for part b should the company spend more or less on advertising Why 11 Let fa 3 f U and fuel 8 Then atx 4fx must have an L a local minimum b local maximum c global minimum d global maximum e in ection point 12 For the function for 3 1342 2 find the xxalue that gives the absolute global maximum on the interval l 1 Let f K and EltXgtbe two functions values are given in the table below for these functions and their derivatives Use the information in the table to answer the 3 questions that follow x x g6 f x 13 IfhK x13 then nd W ml then find W 300 plx 15 If 10gt then find P W 14 Km 16 Given y quot02 1 a Use its derivative to find the absolute maximum and minimum on l 2 b Find any inflection points ofthe function y quot02 0 2x28 2x 2x2Ln 3 17 Given N x f X x2 ol x x3answerthefollowing a Find thex and y intercepts of x b Find the intervals where x is increasing and decreasing Identify any local maxima and minima c Find the intervals where x is concave up and concave down Identify any inflection points ifthey exist d sketch the graph of x 18 Find the constants a anolb in the function W W3 bx if a the point l 4 is an inflection point of x b the point 2 8 is alocalminimum of x o Let ML L r39 inquot A t on at elucit v in kmhr In other words v tells you how many liters ofgas the car uses to go one kilometer ifit is going at velocity v It is known that 75 e n rm and N75 n unn4 a Let gv be the distance the same car goes on one liter ofgas at velocity v Write gv in terms of v b Find E05 Interpret your answer c Find 75 Interpret your answer 20 TrueFalse a The 8th derivative off xis zero b Iffis always decreasing and concave down then fmust have at least one root c If NS then7 has an in ection value at x 5 d It is possible that f gt D everywhere f gt everywhere and f lt everywhere e It is possible that f gt Ueverywhere lt everywhere and f gt everywhere Solutions to Exam 111 quotSample Questionsquot 1B 2D 313 4A 5c 6D 7 f0 15 local maximum and f 12 local minimum 839 a x 5lt we 2 b f x 25in x Zoosatxi 1Ex 1 ma 27x2 3 of E 4h9x33x1 d fooixquot al 6 4 lnA e x xquot x 51 2x ln5x2 r 9 a critical wluesm 0 and 3 b decreasing on W N v 3 increasing on lt1 local minimum when 3 no local max c concave up on W N lt1 gtconcave down on v 2 In ection pointswhenx 0 and 2 d see graph below 3 points should be labeled 11 at 0 10 11 at 2 o and localmin at 3 17 e no Yesat317 g Yesat2o 10 a 1620000 b w When 40000 is spent on advertising each additional 1000 spent on advertising results in a 100 000 loss in revenue remember the units are in thousmds so Each additional 1 thousand dollars spent on ads results in a 100 thousand dollar log in revenue c Spend less rate is negative so revenue is decreasing Draw graph of 5a to see where max occurs 30 4500 97 57 0 9 absrnaxat2 16094andabsrninat00 IPs at 1 0693 andl 0693 no intercepts increasing on rm ltva and decreasing on HUN 1 localrnin at 28andlocalmaxat28 concave up on W hnd concave down on lt5 No inflection points zero not in the domain off graph L l a b a2andb6 a2andb 6 1 02 7 a N the units ofg are krnliters and the units of fare literskm 57 25 krnliter At avelocity of75 kmhr car goes 25 km on one liter ofgas 025 krnliter per kmhour or 025 hrliter At a velocity of75 krnhr the fuel ef ciency gv is decreasing at 025 krnliter for every additional 1 kmhr ou drive In other words as your velocity increases your fuel efficiency decreases aF bF cF dF eT E m thmmum Fall zoos Math 131 Weekin Review 4 Exam 1 Review Sections 1143 15 16 and 2126 F Note This collection of questions is intended to be a brief overview of the exam material When studying you should also look at your notes the suggested homework problems rm the textbook as well as the other weekin reviews for this material 1 Find the domain of the function f below A9 x2 3 x xz x 6 Ssxlt2 XL X793 W sxzn xlt 5 39 log7x9 BlJ 56 4 c 9 a X f 3 o a mm M w DWYL 513375 th o W 39090 QJ Qul SZrD3UO2WO 7 xm 7 smug M w x22 x 20 KWQ X222 2 Iffx 112g0c cos7cx and hx xx5 nd fogoh and state its domain chmm s C 3WED osrrm 3 mg xgt 6ampm cosxwmlgw I WC05TT 34475de kaz g 006 6x11 gta Low W t 0 4mm 9 ngm u 3 ourtes of oeKahli S Silve fo10wmgtx Iog7log32x1452 39 mm a I79r Loga gty a L 7 ags02xf lt 1 4 ow9H0 3 gawk egg4X gt 50 2K gym I may ammo i I J 3 I 105 9Tr Ms Z copyrigm Angie A m Fa 2003 4 Use the graph of the function f below to answer parts a c Graph courtesy of Joe Kahlig a Show fx does not exist MWquot Q N WJFOO 4 X34 Ya 7 xwa ampUwy shw w uwiiAaww xaq y5u yaq quth b Find the vem39ca asymptotes off if they exist Describe using limits the behavior near each vertical asymptote WA th5 wd mk ra mkf oo Xacf Y o V Owd WQ 60 Y 7d gt M 1 I do yaw JJJVW VX quotd3 Va cOnwhatintervalsisfcontinuous W U 010OLD U as 2 5 Fmdxgrgw x2x1x W powee 95 km 7 3W WLXMH XX 3 12m xamxm UXaHH X3 X940 Hawh X cm 16 Wiyd quot X X Mm WW 4 WW w Sum W1 x W3 i W X a 39 X y we W X a gt W 14 Xlt rm 15E Ji a w z x W k Xem lggt39LL W W24 Q i i XZ X 3907 E 6 A bacte al culture starts with 500 bacteria and quadruples in size every three days a Find a function Bd representing the amount of bacteria a er d days ECO 550 gm LMszm 143m MOD WWWSUD qgo 52m 7 st 500 32in 7 Determine the number of bacteria a er 135 days 930357 600 igwjooo I e r wwrighlAllg39e Allen Fan 2003 1 7 Write the mction hz which results from shifting the function gz E to the left by 3 units then re ecting about the yaxis then stretching horizontally by a factor of 3 and then shifting up 82 units L 39 hyger 23 M1C27 lg 330 W aimJquot O 1 L ta with w H 25 513 m hie 22 SDISHe rdhw Wblsa g 01 to Kgti5 3 W have Kg 12 f g a 4 l8 1 I 3 9 39ig 5 3 2 FOO 3x3 15x2 513x 8 Find the horizontal and vertical asymptotes of the curve y 2 vertical asymptotes describe the behavior near each vertical asymptote H 5 M that m 43 4345231 if they exist If there are xam has X X X q M 3 5347 3 Xe quzq 3 5 0 emHk PM wy tmg O b a V LA 3x x 5yto 5y x QMB 5x x Dbcm W I xx J pygx 3 a 3x 993131x 9 D 9axa33x3xb 76quot 7 X LDVJX Wm MR3 XH X 9 73 61M FZX3 3 7 W I l copyrighlAngieAllmJa mE 9 Courtesy of Joe Kahlig Find the values of A and B such that the function f x below is continuous for all real numbers x WW 839 W x2 xlt2 fx Ax2x1 2 x 2 Bx22 xgt2 b X Q 77 all x gm w msl 1cm MLMQAVH 4M3 no Wm m WWWE W 03 Xca 7 300 860 x va Hm c ROD 3H WM M a WWW A W mmth X 33 my Limsw irLAJLJ Tl M 19 mu m H nch E z 3 Mob 744 10 Find quot1335 39375 39 9 3 4 cw a XJX3q JAM wL gqz quot W0 kger 7 q P5 M X 7 x7 g munk 37 quotZ X 39 Xx W M Xa7 43 728 O I X s17 yam x T 1 W 39 quot capyri ltAngieAllm Fall 2008 The following problems are from section 26 11 Find an equation of the tangent line to the curve y 4 at x 6 using the de nition of the slope of the tangent line Mn m Hum 9m WW L MAD o k hem N jb Jn Jig o Jr w MO mum hao MWW W k c gL cm x vo put mm mo Wm NE 3ft 2 BQSFMOPYD out to W 47Wb 455057 gt L5 iSSos r g 9pr 12 The population of a small city in Texas from 1994 to 2002 is shown in the table below Midyear estimates are given mm P 29036 29672 32300 36205 38260 a Find the average rate of growth from 1996to 2000 m WSV T W W M t 960043010 5213 x10 ae WM 0N non can 39 mWM39i39SW 93x Sauna CLAJotLWWWB 39E 40 0903 x3 SS7x 755gt38 X 2 Q1070 Lsvl W I b Find an appropriate model to t the data for interpolation purposes M W b We oLodra 01m 39 13 The position of an object in meters is given by s t2 3t 15 where t is measured in seconds Find the velocity of the object a er four seconds using the de nition of instantaneous velocity ie de nition of the slope of the tangent line me sUHD SUD Mn wmi bmmw b lzhsl Mao k 0 h M EUW393H S1 h o W Ie 539O39 1 W ther n90 7 heo Mao n review 1 MATH 131 By H Greg Klein lf ht 716t2 100t 6 is the height of a ball thrown upwards at 100 gt then answer the following a Find the change in height from 2 sec onds to 4 seconds b Find the average rate of change of height with respect to time from 2 sec onds to 4 seconds c Find the average velocity from t1 to t4 d What is the initial height for the ball A a Between which two points is the aver age rate of change greatest b Between which two points is the aver age rate of change smallest 7 A car rental company offers cars at 40 per 2 The position of a car is given in the table day and 15 cents per mile Its competitors prices are 30 per day and 20 cents per mile tsec 0 5 10 15 20 25 30 st 0 30 55 105 180 260 410 a Write formulas for each company giv ing the cost of renting a car for one day a Find the average velocity of the car be b Write formulas for each company g1 tween t0 and t15 and between t17 ing the cost of renting a car that is and t27 driven 50 miles per day b Find a function that ts the informa C HOW would you determme Whlch com mon best pany you should rent from 8 Write a formula for this statement7 77The distance is inversely proportional to the 3 Find the equation 0f the line through 275 square root of the force between two bod and 673 ies 9 Assuming that stopping distance is propor 4 Find the equation of the line With tional to the square of the velocity7 nd the horizontal intercept of 2 and vertical stopping distances for a car going 40 mph intercept of 4 and 150 mph if the stopping distance at 70 mph is 170 ft 5 Given the data in the table Find r as a 0 The table shows data for the average mile linear function of s and s as a linear function per gallon for US automobiles Graph the of r data and from the shape decide what type of function it should be Describe the shape r 5 of the graph in terms of concavity and slope s 950 900 850 800 Year 1940 1950 1960 1970 1980 1986 mpg 148 139 134 135 155 183 6 The graph of the function ft is given H H H to H 00 lfa population of clams is growing at 1 per year and there are 1 million tons of clams now how many tons of clams will there be in 6 years What is the doubling time for the tons of clams A substance evaporates exponentially in a closed container if there is 3 grams at the end of day 1 and 2 grams at the end of day 6 nd an equation that give the amount as a function of time How much material was originally placed in the container What is the half life for the material The red ant mound has 40 thousand ants and grows at 3 per year the black ant mound has 30 thousand ants and grows at 35 per year a When will the ants have the same pop ulation b When will the black ants have twice as many ants as the red ants c If each colony started with one ant when did each one start What annual rate will give the same amount as the continuous rate of 5 i 7 2 i i i Where is 5 05 increasing decreasing con cave up concave down Your parents have started a bank account for you with 6000 a If this account earns 8 compounded continuously how much is in it after 1 year b At the end of three years you make a onetime withdrawal of 1000 Find the times where the balance of the ac count was 7000 Give answers from the start of the account If a radioactive substance has a half life of 10 years how much is left after 5 years For the data below determine what type of function each could be and why 19 to OJ 3 q to a mo gltxgt W 1 94 96 1 3 85 88 74 5 76 80 64 7 68 73 59 9 61 65 55 11 55 57 52 13 5 5 5 The graph is Sketch the graphs of M 2 fm 521 m Let ft t2 gt 4255 Find a f9 0 908 0 9005 A polynomial crosses the horizontal axis 4 times what is the smallest degree it can have If a polynomial touches the horizontal axis twice but does not cross what is the small est degree it can have Find a polynomial that crosses the horizon tal axis at 1 2 and 3 To the previous problem add the point 04 What is the amplitude and period for sin2t 6 cos2t 4 The high and low tides differ by 5 feet at a certain pier The midtide mark is 12 feet from the bottom of the pier lf high tide is at midnight and low tide 12 hours later nd a function of depth of the water with respect to time in hours since hightide review 1 MATH 131 By Greg Klein 1 H QR HH Wm H q H U 16 DFi a 8 ft b 15242 4ftsec c 20ftsec d 6 ft 839 1105500 7ft55 37 324v3 8v79 1956 ftsec used the regression function to com pute position at t17 and t27 b Quadratic 5572 3975 2 yew mm 73 7253 97537 70047 39 a CD b BC a 71m 40 015m r2m 30 020m 1 40d 015 50 r2d 30d 020 50 c Answers vary 0 xF d40 555 feet d1507808 feet concave up everywhere decreasing to about 1965 increasing thereafter Tons of clams in 6 years 10615 tons 6966 years originally 32534 grams half life is 854 days a about 59 years b about 202 years c red ants 358 years a 0 black ants 304 g 7 years ago 5127 increasing foo 0 decreasing 000 con cave up 70071 100 concave down 11 a 649972 b The rst time the account reaches 7000 is 1927 years after the account was started The second time the ac count reaches 7 000 is 3683 years after the account was started 707 gx is linear fx is quadratic or exponen tial hx is a power function fx2 is a shift up 2 fx2 shifts 2 left 2fx stretches vertically by 2 fiw does strange things I ll go over it in week in review degree 4 x m 1m 7 2m 7 3 105 35 1z e 2z e 3 sin2t amplitude 1 period 7139 6cos2t 4 amplitude 6 period 7139 25sin t 6 12 25 cos t 12 WIR Math l lrcopyright Joe Kahlig 07a Page 1 Exam 3 Sample Review Sections 5155 and 7174 This collection of questions is intended to be a brief overview of the material on the exam This is not intended to represent an actual exam When studying you should also look at the homework problems in the book as well as the other week in reviews for this material 1 F 9 7 A bat starts out traveling towards the exit of a tunnel 90 feet away The graph describes the bats velocity HQ in ftsec vs time 4 a What is the interpretation of fvd give the units of the de nit integral 1 b c How far from the starting point is the bat after 6 seconds When does the bat change direction d How far has the bat traveled in the 6 seconds e Does the bat make it out of the tunnel in 10 seconds 20 20 The following is the graph of f Use the fact that f0 40 along with the given areas for the regions to answer the following Region A 8 Region B 48 and Region C 34 a Find the coordinates xy for all of the critical values Also classify the critical values b Find the x values for the in ection points c Sketch the graph of If x is the number of years from 1990 then the population growth of a city in millions per year can be modeled by the formula 156025 a Write an expression that will give the growth of the city A years after 1990 b If the city had a population of 5 million in 1990 nd a formula that will give the poplation A years after 1990 If f 5cos3x2 and f71 2 then nd f1 WIR Math l lrcopyright Joe Kahlig 07a Page 2 5 2 5 Compute 7Cos3xdx x J A z 6 a Com uteidz p 0 12 z 1z2 dx converge or diverge b Does 7 7 Set up the integrals that represent this shaded area 8 Find the value of B so that the area between the curves f 26m and g 3x2 from x 0 to z B will be 110 9 Compute these integrals 1 a 7 sin6 55 dx 71 b 2 dz 5651 7 86781 C 651678m3d 10 Compute the area between the X axis and y em 7 1 from x 72 to z 2 9 1 11 a Estimate 7dx using a left hand sum with 4 rectangles s 1 b ls the estimate an over estiamte or an under estimate Justify your answer with a graph 10 12 Estimate f fxd using a right sum and the information in the table 1 1 a 0 0 1111I I 11lt9M119 I lt1 2 I P5 1110 sou2A E111m001 o1p 1110112 popuu0 oq 112 121pr 3991 329 32 1o 399 39111111111111111 210 2 oA21 1111500p 1111 111111111X2u1 210 2 s21 121p 11011911111 2 101 2u111101 o1p Q1110 39 31350 31 101 111111111111111 210 olp p112 X2111 210 o1p pug 39 lt9 391012u2 E1111d21 o1p 53111511 10 su0112u2 X112 S11111110119d 1110111AA u011soub s11p 1omsuv 1111111 20 2 10 X2111 20 2 u21p ssos1121p ou2A21111 01p 51 V 11101101d 103 39 15 390 5339 gm 101 1111011 11011o11u1 112 51 3V 1 1111011 01p 121p 1us q p112 7 5111215110 o1p pug V 83 Mg 32 p112 519111111111 201 2 51 f 10 11121110p o1p 11 f 101 51111011 11gt11o11u1 o1p 10 sou2A x o1p pug 39 CO 91mm 3905 11 it o1p u0 x2111 210 2 oA21 11011911111 01p 5001 a 91mm 390511 quot111111 210 2 oA21 11011uu1 91p 590 p 91mm 3905 11 32111 210 2 oA21 11011911111 01p 5001 1 39SQIHOCI 11gt11o11u1 o1p 10 sou2A x o1p pug 39uAA0p oA2u0 s1 11 o1o1m p112 in oA2u0 51 f o1o1m SPAIO 1II1 o1p pug q 39sou2A 21111 o1p X11882Q 395111s2o1op s1 11 91mm p112 5311152919111 51 f o1oIAA S2A1o1u1 o1p pug 39 01 gI01 fIz I m 39z 9 0 2A1o1u1 o1p 110 f 10 111111111111111 210 pu2 111111111X2u1 210 01p pug 39iu 0A211gt p112 5311152919111 51 f 91mm s2A1o1u1 91p pug lt0 lt2 39uAA0p oA2u0 p112 E111s2o1op 51 f 91mm s2A1o1u1 91p pug 39SOHPA 11gt11o11u1 o1p pug 1 P 391o1p1ou 10 39u1111 20 39X2111 20 s2 111o1p 1 11ss2 p112 f 10 sou2A 21111 2 pug 0 I 5338 gIg I m 391 39R1IO 1P11 SUH 10 81411011101 11 OOAA IOI 1gt 0W1 SP OAA SP OOq mm 11 SIUOqgtId gt10mo11101 mm 12 gt00 052 pu01s 110K 39Eu1Xpu1s uo11A 391112xo 21112 112 1uoso1do1 01 popuo1u1 1011 51 s11L 391112xo 01p u0 211012111 o1p 10 mo1A1oA0 1o111 2 oq 01 popuo1u1 s1 su011soub 10 11011o0 s11L 17quot pm 9 8 1 3 VZ I Z 5110398 1219111911 spinneg z 7 ung 1 aged mo ullrx 001 n 09121 11mm 111M 39IIOEQJIIH sum 10 oAnszpop pug exp go qdm exp IpQOBIS 39 JCT2169 3910 pul g I 1 7119 L Sr q 5 on making exp go uonmlbo exp pug 39M 5111 ammo s 91qu go 5 qdm exp H p Asqugod uopwgu oAIzq soop 91qu go 5 IiPI exp H Lunuugupu Hgt P OARq 800 010qu go 5 qdm exp H q P5 u ammo pm Emmaqu s 91qu go 5 qdm exp H 39suogsonb again Iomsmz gt9r qdm exp osn 39 39Hnsm exp Q Oldl d l pm A 0erqu 39smnop go puesnoxp u omle exp 5 A pm 399003 U101 519 It may 5 I OIOIM 39Hgg390gg IL1 Hopping exp Xq poqmugxolddxz oq mm 9003 u posmpmd mm P go mum mu 390 V8 0 0 lt 33 00 mm a 0 rm 0 00 K100 gt lt22 00 3 PH 8 U 0 0 lt 33 0 60 HO 0 gt IL L a two 3 moqumu 901 HP 10 g gt f s1oqumu Hal 9 10 snonupuo s f 39sonladmd osolp seq amp Hopping P go IiPI P Ipn oxg 390 z aged 1720 ullrx 001 ul zx iorlal mm mm 3 I Hm 1mg 39aIfW m IN 11 1 my 11111 31001 gr 331111 lt1 OM17 11111 9801 1111 P5 ZLN VHS 39 N H P Xgt1 39EIIpmonog 01p o39mdum 039 11011211110911 51111 osn 391ip1 mp 111 111111 11011911111 mp oq 9911 0mm 91p Xq 119111 911 sonPA oA1911A11p p111 59mm osoqm p111 f 11911 39g g aged 1720 111117 001 111114lt109111 11mm 111M Math 131 Applied Calculus I FALL 2007 Solutions to Extra practice with em and ln 1 y x4 3x 4 is of the form x to the power of a number7 so we use the rule for x with n 4 to get its derivative 4x 1 4x 3w is of the form number to the power of 7 so we use the rule for b with b 3 to get its derivative ln33w Then we add their derivatives to get the derivative of y 3 4x3 ln33w 2 y ln6x2 Solution A Rewrite ln6z2 ln6 lnx2 ln6 2 ln Then take the derivative to get 5 0 2 Solution B Use the chain rule with y lnu as the outer function and u 6x2 as the inner function Then i3 i and ii 62x1 12m d We combine these to get 3734 i12z Then we substitute in u 6x2 to get le 3 y 6315 Solution A Rewrite y 63m5 as y 63m65 e5 is a number as is 637 so we get 2 ln6363m65 363 Solution B Use the chain rule with y e and u 3x 5 Then 73 e and ii 3 We combine these to get 5737 6M3 Then we substitute in u 3x 5 to get 3 63m53 363 4 y 1232H9 Solution A Rewrite y 1232 9 as y 1232w1239 1232 is a number as is 12397 so we get jg ln12321232 1239 1n12321232H9 21n1231232 9 Solution B Use the chain rule with y 123 and u 2x 9 Then 73 lnl23l23 and ii 2 We combine these to get 3737 ln123123 2 Then we substitute in u 2x 9 to get 2 ln1231232w92 21n1231232m9 5 y more Solution A Rewrite y ln3m2 as y x 2 ln3 ln3z ln32 This is the equation of a line 3 ln3 Solution B Use the chain rule with y lnu and u 3m 3932 3w9 Then 73 i and ii ln33m9 We combine these to get 3734 iln3 3w9 Then we substitute in u 3w9 to get 5 39 ln3 3w9 ln3 6 y lnz 3 Use the Chain rule with y lnu and u z 3 Then landd7 1 du u 11 We combine these to get 2737 Then we substitute in u z 3 to get 7 y ln Solution A Rewrite y ln as y ln17 7 ln Then 2707i7 Solution B Use the Chain rule with y lnu and u g 17x4 Then 7 i and Z7 717 We combine these to get 3737 i717z 2 Then we substitute in u 17x 1 to get 7 171717x 2 77 8 y ln2x 7 ln3x Solution A Rewrite y ln2x 7 ln3x as y ln ln This means y is a constant7 so Solution B Rewrite y ln2x 7 ln3x as y ln2 ln 7 ln3 lnx ln2 ln 7 ln3 7 lnz ln2 7 ln3 This means y is a constant7 so a Solution C Use the Chain rule on each term For ln2x we get a derivative of l For ln3x we get a derivative of E The derivative of their difference is E 7 E 0