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Date Created: 09/07/15
MATH 121 Fall 2008 Additional Review Problems for the Final Examination Covers 427 66 in Stewart Note These problems are in addition to the Review Problems for the Midterm Examination which covers 117 41 in Stewart A portion of the Final Examination will include material from 117 41 in Stewart 1 Find the intervals on which the function is increasing Find the intervals on which the function concave upward a 2mg 7 3m2 712 71392 b 9m m3 7 3m2 3m 17 C ht 3 g 32 d 81 Us 7121 2 e 9t T f hu Que 2 2 Let x 2x 7 7 7 4 Find the domain of 1 Determine the zeros of 1 Find the intervals on 1 which 1 is increasing Find the intervals on which 1 is concave downward t t 3 Let g be the function de ned by gt 100 20 sin 1 10 cos 7 For 0 g t g 8 the function g is decreasing most rapidly when t Hint Use a calculator A 0949 B 2017 C 3106 D 5965 8000 4 Given f m cos2z 7 sinz7 0 lt z lt 27139 On which open intervals is the function 1 increasing 5 Find the absolute maximum and minimum values of x and the corresponding m Values on the given interval a fzm373x227 723 b megj 03 c fmm728inm 077139 d fmze m2 70000 6 The absolute maximum and minimum values of y 3 7 9x 8 on the interval 737 1 are A 8 M7 8 B 8 M3 0 c 8 0 D 8 7 M3 0 E None of these 7 At what value of z does the function x 3x 7 z13 change from increasing to decreasing 8 On what intervals is the function gt decreasing i t 7 t2 1 9 How many points of in ection does hz mge m have 10 Let m4 7 4x2 a Find the critical numbers of f the intervals on which 1 is increasing or decreasing Find the z y coordinates of any local extrema b Find the in ection points of f and the intervals on which 1 is concave upward or concave downward c Sketch the graph of f by using the information obtained in a and 4 7 3 11 If mg 3547 as then fm 2 3 3m 4 7 m a Find the critical numbers of the function 1 What is the domain of f b Determine the intervals on which 1 is increasing or decreasing c Sketch a rough graph of 1 below 12 Evaluate the limit 1nlt3gt em7m 27mm2 a 1320 A 32 W 11320 f 1 m C 3520 2 1 g 6x2 sinz as 7 d lim 51 e lim 5 sin x f lim gt 170 z ma m7xi 1 1 i 7139 i i 71 7 7 7 7 7 17 g 131117 ltln m m 7 1 h 11332 lt 2 3 tan a 1 1311 3 am 7 bx 13 If a7 1 gt 07 nd the value of linj m7 m 14 A three sided fence is to be built next to a straight section of river7 which forms the fourth side of a rectangular region The enclosed area is to equal 1800 ft2 Find the dimensions of the enclosure to minimize the fence material 15 A threesided fence is to be built next to a straight section of river7 which forms the fourth side of a rectangular region There are 96 ft of fencing available Find the maximum enclosed area and the dimensions of the corresponding enclosure 16 A mathematician wishes to mail popcorn in a cylindrical package of length It and a circular base of radius r Because of the post office regulations7 the length of the cylinder plus the circumference of the base cannot be more than 108 inches a Express the volume V of the package in terms of the radius r b Find the dimensions of the cylindrical package with maximum volume Justify your answer 17 What is the maximum possible area of a rectangle with a base that lies on the z axis and with two upper vertices lying on the graph of y 9 7 m2 Justify your answer 18 Suppose that a rectangular box with open top and square base is to be made using two different materials The material for the base cost 2 per square foot and the material for the four sides costs 1 per square foot Find the dimensions of the box of greatest volume subject to the condition that 96 is spent for the material What is the maximum volume Justify your answer 19 Find the points on the hyperbola 2 7 y2 16 closest to the point 0 2 20 Amy is setting up a lemonade stand The cost for making z glasses of lemonade is 5 1 002m dollars Previous experience indicates that she can sell 80 glasses of lemonade at a price of 050 per glass and that for each 010 increase in price she will sell 4 fewer glasses At what price should the lemonade be sold to maximize the pro t 21 The velocity of a wave of length L in deep water is UK l L C C L where K and C are known positive constants What is the length of the wave that gives the minimum velocity 22 Given the cost function in dollars 2 C36 7 2 78000 nd a the average cost function b the marginal cost function c d the minimum average cost the production level that minimizes the average cost 23 Given the cost function and the demand function C35 16 000 500m 716252 0004353 pm 1700 7 7m nd a the production level that maximize the revenue and the maximum revenue b the production level that maximize the pro t and the maximum pro t c the consumer surplus when the sale level is z 100 24 The manager of a 100 unit apartment complex knows from experience that all units will be occupied if the rent is 800 per month A market survey suggests that on the average one additional unit will remain vacant for each 10 increase in rent What rent should the manager charge to maximize revenue 25 a Give the iterative formula for Newton s method for approximating a root of the equation x O b Use Newton s method with initial approximation ml 2 to estimate the solution of the equation 3 7 2x 7 8 0 in the interval 27 3 accurate to eight decimal places Give your sequence of approximations 26 A major league pitcher can throw a baseball with an initial velocity of 144 ftsec If he throws the ball straight up7 how high will it go Neglect air resistance and use 9 732 ft sec2 27 A particle moves along the y axis so that its velocity at any time t 2 0 is given by vt tcos 25 At time t 07 the position of the particle is y 3 a For what intervals of 257 0 g t g 57 is the particle moving upward Write an expression for the acceleration at of the particle in terms of t c Write an expression for the position yt of the particle in terms oft d For t gt 07 nd the position of the particle the rst time when the velocity of the particle is zero 28 A particle with velocity at any time t given by vt 5 moves along a straight line How far does the particle move from time t 0 to t 2 29 a State the trapezoidal approximation T6 for an arbitrary function f on the interval la bl b In a three hour trip7 the velocity of a car at each half hours was recorded as follows l TimeHours l l VelocityMPH l 5l2l25l3l 55 50 l35l 30 0 Estimate the distance traveled by using the trapezoidal approximation T6 0 0 l5l 1 l1 l40l c Estimate the average velocity of the car during the trip 30 Calculate a the trapezoidal approximation T67 b the midpoint approximation M67 c 3 the Simpson s approximation to 2 him 0 31 Let R be the region enclosed by the curve y ln m the z axis7 and the lines z 1 and z 5 Use the trapezoidal rule with n 4 to approximate R 32 A population of honeybees increased at a rate of Mt bees per week7 where the graph of r is as shown Use the a trapezoidal rule7 b Simpson s rule7 with six subintervals to estimate the increase in the bee population during the rst 24 weeks 33 If 2 f t 3 for all tin 1 5 then if is concave upward 0n 1 5 t is decreasing 0n 1 5 f 8 8 f5 f112 f f 3 3 0 341f mm 5 and mm 3 then 2 7 3fxdm 0 2 2 a 2 b 710 c 10 d 7 2 e 6 35 The graph of x consists of two straight lines and a semicircle Use it to evaluate each integral a jngltzgtdz b jg4fzdz c C7xdz d 8dx 36 Find fm a fx5x472m57 w b f m12sinzicosz f03 c f z3ew5smx f01 f 02 d f 37 f07 f12 37 Find the derivative of the function x 1 a 35 tzsintdt b Mu 1 3mm 0 2 7 c gm1m cost2dt d W Lie m dm 38 Evaluate the integral a esmmcosmdx b n c 0sin7rmdm d mm e msinzdm f 0 6mz222dz g A32e 3md h fmda i 07r2cosmsi dm Mam klXMQZW lt1gtltzewelwgtdm w m gtzd nzlnlzldz oOe 3sin3zdz r p 12 lt dz q 01zcm2 dz 1lnzdz dz 2 2 3 7 4cosu sW 032 sec 2 WW u AZ 1sinu2 u v18Wdz w 521 coszdz X dz 1 39 Let xtantdt What is the value of F 005 0 a 005 b 02237 c 22351 d 2241 e 00075 d 32 40 7 costht dz 0 a cos2 z 10 cos2 z2 c sin2 z2 d 2z cos2 z2 e z2 cos2 z2 b 41 For all real 97 l2zl dz is 0 a iblbl b b2 c 7122 d blbl e none of these 0 42 If the function g has a continuous derivative on 07 c then gz dz 0 a 90 790 0 995 0 0 Wm 790l 1903 e 9 940 k 43 If 2kz 7 mm 18 then k 0 a 79 b 73 c 3 d 9 e 18 1 44 Let Fz xtZ 2tdt 1 a Find 10 Find the domain of F c Find the length of the curve y for 1 g z 2 ah 45 If f is a continuously differentiable function for all real z7 then fz dz is 4 a a 0 0 ND 0 Ha d f 0 8 Ha 46 Let a lt c lt b and let 9 be differentiable on 0 1 Which of the following is NOT necessarily true a Ab9d ACQWMH Abgz dz b There exists a number d in 11 such that g d M b 7 a c imam 2 0 d hm 91 9a zac b b e If k is a real numloer7 then kgz dz k gz dz 1 I 2 47 If f is an even and continuous function7 ie7 f7z fz for all z7 then fzdz 1 a212fzdx b 1fxdx c0 d e Noneofabove 71 12 48 The gure shows the graphs of f f and ft dt ldentify each graph7 and explain 0 your choices 49 Evaluate the improper integral or show it is divergent a 1 d mOomd cO de 7 13 7 13 5 13 01 m 24 1 2 7600 1 5 dm y d 7 d e f 7 d 712m1 1xlnm 2vy72y 50 A publisher estimates that a book will be sold at the rate of Mt 1670005415 books per year7 where t is the number of years from now Find the total number of books that will ever be sold up to t oo 51 Let R be the shaded region in the rst quadrant enclosed by the y axis and the curves of y sinz and y cosm7 for 0 g x 7r4 a Set up the de nite integral for the area of R and evaluate it exactly b Find the centroid of 733 of R c Set up the integral for the volume of the solid generated when R is revolved about the z axis and evaluate it exactly d Set up de nite integrals to compute the perimeter of R Do not compute the integrals 52 Two cars7 A and B7 start side by side and accelerate from rest The gure shows the graphs of their velocity functions a Which car is ahead after one minute Explain b What is the meaning of the area of the shaded region c d Estimate the time at which the cars are again side by side Which car is ahead after two minutes Explain 53 Find the area of the region bounded by the given curves aym276x7 y12m72m2 bx72y707 y276y7m0 54 Find the centroid of the region shown 55 Find the volume of the spherical dome obtained by rotating the region between the graph of y VRZ 7 2 and the z axis7 R 7 h g x R7 about the z axis 56 Let R be the region enclosed by the curves of y z and y Find the volume of the solid obtained by rotating R about a the z axis7 b the y axis7 c d the line y 71 the line z 71 57 The amount of pollution in a lake z years m 2 1 after the closing of a chemical plant is Pm 100z tons Find the average amount of pollution between 1 and 10 years after the closing 58 Consider the function x 1 2 on the interval 07 2 Find a number 0 in 07 2 so that the area of the rectangle with base on 07 2 and height fc is equal to the area under the curve of f in the given interval 59 Compute the length of the curve given by z 5t sint and y at cost for 0 g t g 7139 60 A particle is moved along the z axis by a force that measures 4x2 pounds at a point z feet from the origin Find the work done in moving the particle a distance of 10 feet from the origin 61 A crane is lifting a 1500 lb transformer from the ground level to the third oor which is 30 feet above the ground level A 60 foot cable connects the transformer to the top of the crane The cable weighs 5 lb per foot How much work is done in lifting the transformer 30 feet above the ground 62 The graph of a differentiable function f on the closed interval 17 7 is shown in the gure 12 Let W mm for 1 g m g 7 1 Justify your answer 1 63 The graph of f is shown below In the right frame7 sketch the graph of ft dt on the interval 0 1 Be sure to label the local extrema and in ection points a 64 The gure shows the two shaded regions R enclosed by the curves of x 2 and gz 2m in the rst quadrant 7 x a Use a calculator to estimate the x coordinates of the two points of intersections of the curves of f and g 10 Express the total area of R with de nite integrals You don t have to evaluate it 7r2 65 When using the substitution method of integrating7 the integral sin3 xcosxdx is equal to 0 1 the integral 743 du where u 0 a cosx b 7cosx c sinx d 7 sinx e None of these 10 Z2 1 66 Let S denote a Riemann right hand sum of x2 dx Which of the following 1 0 statements is trlue 1 1 1 aS dex bSlt dex cSgt dex 0 0 0 1 d S 7 x2 dx e None of these 0 67 If fx2cosxdx x 7 2xsinxdx7 then x a 2sinx 2xcos x b x2 sinx c 2xcos x 7 x2sin x d 4x cosx 7 2xsin x e 2 7 x2cos x 7 4sin x 68 Let Fx Awesmtdt Then F 7r a 1 b 0 c 27139 d 71392 e None of these on 711 is d 061562 e 7061562 69 The average value of x tan 1x a 04388 b 704388 c 0 1 70 tanilxdx 0 7r 7r74ln2 7r4ln2 7r72ln2 7r2ln2 1 Wf f df 8f 7r2 71 Ls edo 0 x1s1n0 a 72W 7 1 b 7M e M d W571 6 WA 1 72 The function g has a continuous second derivative on the interval 717 4 The graph of g is displayed in the gure The graph of g on the interval 17 2 is contained in the shaded rectangle Circle those that are true Emwx u 3st 6st is E E u v A 913E s 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