ANALYT GEOM & CALC 1
ANALYT GEOM & CALC 1 MAC 2311
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Date Created: 09/19/15
MAC 2311 Test Four Review Fall 2008 Exam covers Lectures 27 i 34 Sections 47 55 1 Use Newton7s method to approximate x correct to 3 decimal places Use the initial approximation 1 2 lllustrate how the method works to nd x2 by graphing your function and the tangent line at z 2 2 Find the third approximation 3 of Newton7s method to approximate the point of intersection of the graphs y 1 and y 2 3 Evaluate each integral 12 33 54 lnsz a dx b dx c dx x x2 2x 5 1 z esz d 3csc2z cot2x dx e de f dx cos x 0 g2d h sz4dx i cotxd 1 s1n z 4 r 7 61 fa 4 True or false a If f i e 7 then T dx i 3fx C 1 d 1 b gm dt g glttgt dt c fltzgtgltzgt dz U M dz U gltzgt dz 1 1 5 The slope of the tangent line to the curve y f at any point is given by 2 sing cos2g lf f0 17 nd the function r z2 z lt 0 6 F1nd the area under the graph of fx w on 723ln 4 e z 2 0 7 E1 tBf d f 1 vauae 1 x x1 sidlnxim 8 Find 71 if fm f 72 3 and f73 2 9 Find the maximum and minimum values of f azz 2x on 722 and use them 2 to nd upper and lower bounds for the de nite integral 3 x2 2x dx 72 m If g lnt2 21d1t7 nd g e Determine the intervals on which g is 0 increasing and decreasing H d W1 Evaluate 7 xfsin t2 dt d m4 12 H 9 H r H CH 4 Consider the area given by Suppose that f is an even function and g is odd 10 5 5 10 lf g dx 10 g dx 5 f dx 12 and f dx 207 nd 5 0 75 0 a ifltzgtgltzgtidz b 273gltzgtidz 5 c the area of the region bounded by the s axis and g on 7510 V L 3 I 9 2 Write the following limit of Riemann sums as a de nite integral lim E 7 1 2 TLANX 111 77 n 3 Evaluate the de nite integral 2 7 2x dx using a Riemann sum with n subintervals 1 of equal width and letting right endpoint of the subinterval ml1 Then check your answer by evaluating the de nite integral using the fundamental theorem s 0 mdx a Approximate the area using a Riemann sum with n 4 and using the left endpoint of the interval xi47m as b Use the midpoint rule with n 2 to approximate the integral c Find the exact value of the integral A particle moves along a straight line with velocity 11t t2 7 2t 7 8 Find the displacement of the particle on the time interval 07 5 Find the distance traveled by the particle on the same interval A certain population is increasing at a rate of 500 100t per year Find the total change in population from the 4th to the 10th year A farmer wishes to fence an area next to his barn as sketched below He needs a wire fence that costs 1 per linear foot in front of the barn and and wooden fencing that costs 2 per foot on the other sides Find z and y so that he can enclose the maximum area if his budget for materials is 4400 For what values of z and y will the right triangle inscribed in a semicircle of radius 2 have maximum area What is the maximum area A cardboard box with a square base and open top is to be constructed to hold 32 cu in Find the dimensions of the box that will minimize the amount of material required to construct the box H to 03 F U a 1 00 to H O H H 2 Consider the area given by 0 The price of a particular model of a Toyota is increasing at a rate of MAC 2311 Spring 2009 Some Review Problems Lecture 34 Final covers lectures 1 34 Evaluate each integral E12 3x 3 8 lnx2 a dx b dx c d xx22x5 1 x 52x71 e de 0 h xxx 4dx 74 52x f dx d 3csc2xcot2xdx i cotxdx g The slope of the tangent line to the curve y x at any point is given by 2 sin lt2 cos2 lf O 17 nd the function cos x 1 sin2 x 2 Find the area under the graph of x m m lt 0 on 7231n 4 5 x 2 0 5 i 2 2 Evaluate 1 xdx if a x m and b x i i H 2 Find f711ff x W f 72 3 and 1073 2 d V5 Evaluate 7 t2 sin t2 dt dx m3 Suppose that f is an even function and g is odd 10 5 5 10 lf gx dx 107 gx dx 57 dx 12 and dx 2O7 nd 5 0 75 0 10 a W gltzgtidz 75 b the area of region bounded by the x axis and gx on 757 10 V L 3 9 2 Write the following limit of Riemann sums as a de nite integral lim 2 7 1 2 Hoe 111 n n 1 7 d x2 2 a Approximate the area using a Riemann sum with n 4 and using the midpoint of the interval xi17 as b Find the exact value of the integral A particle moves along a straight line with velocity vt t2 7 2t 7 8 Find the displacement of the particle on the time interval 07 57 and the total distance traveled by the particle A certain population is increasing at a rate of 500 1002f per year Find the total change in population from the 4th to the 10th year 3t 2 thousand dollars 25 years after its introduction If the base price of the car was 18000 when it was rst introduced7 what will the base price be of the same model two years later 1 7 a39reaching Center MAC2311 Review Exam 3 v2 Page 1 of 3 The following document is a Review provided bythe AT Teaching Center s Math Study Center located in SE Broward Basement The AT Teaching Center provides WalkinTutoring as well as Individual Weekly Appointment Tutoring Services for avariety ofsubectsThe service is 100 free It is made with input from your instructor The solutions to the document are provided atthe Review For more information on Reviews or our other services please go to wwwteachingce nterufledu 1 The position in meters of a particle at time t is given by 4 t t3 79t2 15t712 where t is measured in seconds What is the displacement from 0 to 8 seconds a b Find the particles average rate of change from 2 to 4 seconds c Find a functionforthe velocity attime t d What is the particle39s instantaneous rate of change at 2 seconds e At which times is the particle at rest When is the particle moving in the positive direction f g Find a function forthe acceleration at time t ex How many local extrema does the function 1 X Wz haveand are they local minima maxima or neither Find the absolute minimum and maximum values for each of the following functions al 90 lnx2 x 1 overthe interval 11 bl 1 N4 2 overthe interval 12 Considerthe graph off x which is given below forthe continuousfunction x Where does 90 have critical numbers On what intervals is x increasingdecreasing On what intervals is x concave updown Slltetch the graph offx showingthe relative extrema and points ofinflection usingthe answers from parts althrough cl ngcrm 1 39reaching Center MAC2311 Review Exam 3 v2 Page 2 of 3 5 Ayoung gentieman is standing 10feet from a buiiding where his significant other is throwing his stuff out oftheir apartment window 20feet above the main apartment entrance which is horizontaiiy iined up with the gentieman s iine of site if he iooilts strait ahead However the gentieman can t take his eyes off ofthe faiiing objects especiaiiythe TV and his iine of sight makes an angie 9 reiative to him iooilting at the apartment entrance How fast is quot 39 i i his TV is h at a rate of30ftsecond at that position y d 39 its faiiing 6 Find the criticai numbers ofthe foiiowing functions Use the second derivative test to determine which criticai numbers represent iocai maximums and which represent iocai minimums ai x x224quot bi 21nx cl cos9 sin2 9 7 Approximate 1976 using differentiais 8 Considerthefoiiowingfunctions x2 2 1 x W gx xz 2xz Sketch a graph ofthe foiiowing function and iist the iocations ai The verticai and horizontai asymptotes bi Intervais where the function is increasing decreasing concave up and concave down cl Locai Minimum iocai Maximum and Points ofnfiection 9 Find the iimit You may use L Hospitai s Ruie if it appiies If it does not appiy state why equot ex l x a llmx m x z d llmx m 962 b ma a w e limx n l tan x sec x xcosx l f limx 0 sin 30 c limx m xx 10 How many infiection points doesthe function have overthe entire reai number iine gx 3x5 10x 1 10x 11 According to Hooke s Law the force exerted on a spring of mass m is given bythe foiiowing equation F kx where k is the spring constant and x is the distance thatthe spring is extended from its resting position Determine the veiocity ofthe spring whenthe rate ofchange ofthe force with respect to time is 10 Newtons and the spring constant is 3 N Gravei is being dumped from a conveyor beit at a rate of 30 ft3min As it faiis from the conveyor beit it forms a piie in the shape ofa cone whose base diameter and height are aiways equai How fast is the height ofthe piie increasing when the piie is 10ft high 13 What is the iinearization Lx ofthe function 3 x at a 8 1 a 39l eaching Center MAC2311 Review Exam 3 v2 Page 3 of 3 14 The radius ofa circuiar sphere is given as 24 cm with a maximum error in measurement of02 cm ai Use differentiaisto estimate the maximum error in the caicuiated voiume ofthe disk bi What is the reiative error What is the percentage error 15 Verifythe function satisfies the hypothesis ofthe Mean Vaiue Theorem on the intervai 02 Then find aii the numbers L that satisfythe conciusion ofthe Mean Vaiue Theorem 90 x3 x 1 16 Verifythe function satisfies the hypothesis of Roiie sTheorem onthe intervai 0211 Then finai aii the numbers L that satisfy the conciusion of Roiie s Theorem sinx cosx 17 A watertrough is 10 m iong and a crosssection hasthe shape ofan isosceiestrapezoid that is 03 m wide at the bottom 08 m wide atthe top and has a height 05 m fthe trough is being fiiied with water atthe rate of 02 mein how fast is the water ievei rising when the water is 03 m deep