Math 107 Section 7981 - Quiz #5 Answer Sheet Template Multiple Choice Questions Short Answer Section 1. ______ 13. ____________________________ 2. ______ 14. ____________________________ 3. ______ 15. ____________________________ 4. ______ 16. ____________________________ 5. ______ 17. ____________________________ 6. ______ 18. ____________________________ 7. ______ 19. ____________________________ 8. ______ 20. ____________________________ 9. ______ 10. ______ 11. ______ 12. ______ Math\t107-7981\t-\tQuiz\t#5\t-\tSchultz\t-\tDue\tMay\t1,\t2016\t-\tpage\t1\tof 5 Follow\tthese\tdirections\tcarefully. This\tquiz\tis\tdue\tby\t11:59\tEastern\ttime\ton\tMay\t1,\t2016. o This\tis\tan\timportant\tassignment,\tcounting\t12%\tof\tyour\tgrade. o Submit\tthis\tassignment\tin\tyour\tassignments\tfolder\tby\tthe\tdue\tdate. Answer\tall\tthe\tquestions. There\tare\t28\tproblems\ton\t5\tpages. o There\tare\t100\tpoints\tpossible. The\t5th\tpage\tis\textra\tcredit\tworth\t25\textra points. o No\twork\tis\trequired\tfor\tthe\tMULTIPLE\tCHOICE\tSECTION\tor\tfor\tthe\tSHORT ANSWER\tSECTION. There\tis\tno\tpartial\tcredit\tfor\tthese\tproblems,\tso\tplease consider\tyour\tanswers\tcarefully. o Show\tall\twork\tfor\tthe\tLONG\tANSWER\tSECTION. There\tis\tpartial\tcredit\tfor these\tproblems Use\tthe\ttemplate\tprovided\tto\tanswer\tthe\tMULTIPLE\tCHOICE\tand\tSHORT\tANSWER questions.\tPlease\tuse\ta\tseparate\tdocument\tto\tanswer\tthe\tlong\tanswer\tsection. You may\ttype\tup\tyour\tanswers\tin\ta\tdocument\t(like\tWord\tor\tExcel)\tor\thandwrite\tyour answers\tand\tscan\tthem\tin\t-\tup\tto\tyou. Submit\tyour\tassignment\tas\tan\tattachment. Under\tno\tcircumstances\tshould\tyou type\tyour\tanswers\tinto\ta\ttext\tbox. MULTIPLE\tCHOICE\tQUESTIONS. (36\tpts\ttotal, 3\tpts\teach) 1. Determine\tlog $ 27. A. 3/2 C. 3 B. 1/3 D. None\tof\tthese. 2. Solve 3 5 = 1 . A. Both\t2\tand\t3 C. 3\tonly B. 2\tonly D. No\tsolutions. 3. Determine\tthe\tdomain\tof = 0 5 6. A. , 2 [3, ) C. , 1 [6, ) B. [2,3] D. [1,6] 8 4. For\tthe\tfunction = 2 1,\tits\tinverse, 9: ,\tis: : 3 B. A. 2 1 21 8 C. D. :;< 8 0 :;= 8 0 Math\t107-7981\t-\tQuiz\t#5\t-\tSchultz\t-\tDue\tMay\t1,\t2016\t-\tpage\t2\tof 5 5. The\tfunction\tgraphed\tbelow\thas\tan\tinverse. T. F. True False 6. 7. 8. 9. 10. 11. The\tdomain\tof = A. [2, ) B. (2, ) The\tdomain\tof = 9@= is: A. (, ) B. [0, ) C. D. (, 0] , 0 The\trange\tof = 9@= is: A. (0, ) B. [0, ) C. D. (, 0] , 0 The\tinverse\tof = 9@= is: A. B. : B CDE C. ln\t(8) D D. None\tof\tthese. ln\t( ) The\tdomain\tof = ln 2 is: A. (2, ) C. B. (, 2] D. (, ) , 2 Express\tas\ta\tsingle\tlogarithm:\t3 log + 2 + log 1 log A. J KLM(=);: B. log 2\tis: C. (, 2] D. , 2 < = 8 ;@ < =;0 8 C. log D. log\t(3 + 7 ) < Math\t107-7981\t-\tQuiz\t#5\t-\tSchultz\t-\tDue\tMay\t1,\t2016\t-\tpage\t3\tof 5 12. Which\tof\tthe\tfollowing\tbest\tdescribes\tthis\tgraph? A. It\tis\tthe\tgraph\tof\ta\tquadratic\tfunction. B. It\tis\tthe\tgraph\tof\ta\tfunction\tand\tit\tis\tone-to-one. C. It\tis\tthe\tgraph\tof\ta\tfunction\tand\tit\tis\tnot\tone-to-one. 5.1 Function Composition 371 D. It\tis\tnot\tthe\tgraph\tof\ta\tfunction. 50. (g g)( 2) 51. (g f )( 2) 52. g(f (g(0))) SHORT\tANSWER\tQUESTIONS. (24\tpts\ttotal,\t3\tpts\teach) 53. f (f (f ( 1))) 54. f (f (f (f (f (1))))) 55. (g g g)(0) | {z } n times Questions\t13,\t14,\t15\tand\t16\tinvolve\tthe\tfunctions\tgraphed\tbelow. Determine\teach,\tor state\tthat\tit\tcannot\tbe\tfound\tusing\tthe\tinformation\tgiven. In Exercises 56 - 61, use the graphs of y = f (x) and y = g(x) below to nd the function value. y y 4 4 3 3 2 2 1 1 1 2 3 4 1 x y = f (x) 56. (g f )(1) 2 3 4 x y = g(x) 57. (f g)(3) 58. (g f )(2) 13. ( )(1) 59. (f g)(0) 60. (f f )(1) 61. (g g)(1) 14. ( )(1) 62. The volume V of a cube is a function of its side length x. Let's assume that x = t + 1 is also a function of time t, where x is measured in inches and t is measured in minutes. Find 15. ( )(2) a formula for V as a function of t. 16. ( )(4) 63. Suppose a local vendor charges $2 per hot dog and that the number of hot dogs sold per hour x is given by x(t) = 4t2 + @=9Q 92, where t is the number of hours since 10 AM, 0 t 4. 20t + 17. Find\tx: 3 = 9 (a) Find an expression for the revenue per hour R as a function of x. 18. Suppose\tthat\t$5,000\tis\tinvested\tin\tan\taccount\tat\tan\tannual\tinterest\trate\tof\t5.1% (b) Find and simplify (R x) (t). What does this represent? compounded\tcontinuously. How\tlong\t(to\tthe\tnearest\ttenth\tof\ta\tyear)\twill\tit\ttake\tfor (c) Whatthe\tinvestment\tto\tdouble\tin\tsize? is the revenue per hour at noon? 64. Discuss with your classmates how 'real-world' processes such as lling out federal income tax forms or computing your nal course grade could be viewed as a use of function composition. Find a process for which composition with itself (iteration) makes sense. Math\t107-7981\t-\tQuiz\t#5\t-\tSchultz\t-\tDue\tMay\t1,\t2016\t-\tpage\t4\tof 5 19. 20. Water\tinitially\tat\t200\tis\tleft\tin\ta\troom\tof\ttemperature\t60\tto\tcool. After\tt minutes,\tthe\ttemperature\tT\tof\tthe\twater\tis\tgiven\tby = 60 + 140 9V.V@WX . Find the\ttemperature\tof\tthe\twater\t20\tminutes\tafter\tit\tis\tleft\tto\tcool. Round\tto\tthe\tnearest degree. Find\tthe\tvalue\tof\tthe\tlogarithm: log Y : @: . LONG\tANSWER\tPROBLEMS. 21. 22. (40\tpts\ttotal) Solve\tthe\tequation 10 4 = 2 1. The\tquadratic\tyou\tneed\tto\tend\tup\tsolving\tcan\tbe\tfactored. Reminders. 1. When\tyou\tsquare\tboth\tsides,\tbe\tcareful. is\tnot + or . You\tshould\tnot\tbe\tmaking\tthat\tmistake\tanymore. FOIL. There is\ta\tmiddle\tterm. 2. Make\tsure\tyou\tcheck\tyour\tanswers. (10\tpts) Let = 2 0 and = 3. All\tthe\tparts\trefer\tto\tthese\ttwo\tfunctions\tf\tand g.\tDetermine\teach\tcomposition\tof\tfunctions\tbelow. (Note\tI\tam\tnot\tasking\tyou\tto multiply\tthese\tfunctions. Please\tuse\tcomposition\tof\tfunctions. Multiplying\tthem\twill get\tno\tcredit,\tso\tplease\tdon't\tdo\tthat.) a. ( )() b. ( )() 23. For\tthe\tfunction = =;Y Q90= (4\tpts) (10\tpts) : a. Determine\tthe\tdomain\tof\tf\tin\tinterval\tnotation. (3\tpts) b. Determine\tthe\tinverse\tfunction, 9: . (10\tpts) c. Determine\tthe\trange\tof\tf\t(Hint.\tthis\tis\tthe\tdomain\tof\tthe\tinverse\tfunction). (3\tpts) Math\t107-7981\t-\tQuiz\t#5\t-\tSchultz\t-\tDue\tMay\t1,\t2016\t-\tpage\t5\tof 5 Extra\tcredit\tproblem. Each\tof\tthese\tis\tworth\t5\tpoints. For\tthis\tproblem,\tyou\tshould\tthink\tabout\twhat\thappens\tto\ta\thot\tbeverage\t(like\tcoffee\tor tea)\tif\tyou\tlet\tit\tsit\tat\troom\ttemperature\tfor\ta\tlong\ttime. How\thot\tis\tthe\tcoffee\tat\tfirst? Then,\tyou\tlet\tit\tsit.\tHow\thot\tis\tit\tafter\tsome\ttime\thas\tpassed? Think\tabout\tthis\tin\tterms\tof real\tlife. Everyone\tknows\twhat\thappens\twhen\tyou\tthink\tyour\tcoffee,\ttea,\tor\tcocoa\tis\thot, take\ta\tbig\tsip,\tand\tthen\trealize\tit\thas\tbeen\tsitting\tfor\ta\twhile. You've\tbeen\ttaking\tMath\t107,\tand\tyou've\tgot\tyour\tnerd\ton. You\tare\ttrying\tto\tfit\tan exponential\tmodel\tof\tthe\tform = `= to\tthe\tfollowing\tdata\tthat\tyou\tworked\thard\tto collect. You\tstuck\ta\tthermometer\tin\ta\tcoffee\tcup\tevery\t10\tminutes,\tand\tyou\ttook\ta temperature\treading\tof\tthe\tcoffee\t(in\tdegrees\tCelsius). The\tcoffee\tcup\tis\tsitting\tin\ta\troom\tat room\ttemperature,\twhich\tfor\tus\twe'll\ttake\tto\tbe\tabout\t20\tdegrees\tCelsius. After\tan\thour\tof measurements,\tyou\tgot\tbored,\tand\tyou\twrote\tdown\tthe\tfollowing\ttable. x y (in\tminutes) (in\tdegrees\tCelsius) 0 97 10 71.5 20 56.2 30 41.3 40 35.5 50 29.4 60 24.9 24. Someone\ttries\tto\tconvince\tyou\tthat\twhen\tx\tis\t70,\ty\tis\t40.\tJust\tby\texamining\tthe\ttable -\tno\tmath\t-\tdo\tyou\tbelieve\tthe\tperson? Why\tor\twhy\tnot? Write\tone\tor\ttwo\tshort sentences\tto\tanswer\tthis\tquestion. 25. Fit\tan\texponential\tregression\tmodel\tof\tthe\tform = `= to\tthis\tdata,\tand determine\tA\tand\tk.\tYou'll\tneed\tto\tuse\tsome\tkind\tof\ttechnology\tto\tdo\tthis,\twhether\tit be\ta\tgraphing\tcalculator,\tMicrosoft\tExcel,\tan\tonline\tcalculator\tlike\tDesmos,\tor\tsome other\tregression\ttool\t-\tthere\tare\tmany\tfreely\tavailable\tonline.\tThe\tway\tthis\tis\tset\tup, k\tis\tnegative. 26. Use\tyour\tregression\tequation\tto\tdetermine\tthe\ttemperature\tof\tthe\tcoffee\twhen\tx\tis 70\tminutes. 27. Use\tyour\tregression\tequation\tto\tdetermine\tthe\ttime,\tin\tminutes,\twhen\tthe temperature\tof\tthe\tcoffee\tis\ty\t=\t70\tdegrees. 28. (This\tis\tmore\tconceptually\tchallenging\tthan\tthe\tothers.) You\tshould\tsee\tthat\tthe\texponential\tmodel\tyou\tfound\tfits\tthe\tdata\tvery\twell. When\tI\tgraphed\tmy\tmodel\twith\tthis\tdata,\tI\tgot\t0 = 0.98478. But\tis\tthe\tmodel = `= a\tgood\tmodel\tfor\tthis\tphysical\tsituation? Why\tor why\tnot?\t(Hint: Think\tabout\twhat\tthe\ttemperature\tof\tthe\tcoffee\twill\tbe\tafter\ta\treally long\ttime. What\tdoes\tyour\tmodel\tgive\tfor\tthe\ttemperature\tafter,\tsay,\t1\tweek,\tor 10,000\tminutes? Is\tthis\trealistic?)