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7 - B/MapleTA MATH 135 1119 Algebra for Honours Mathematics...

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Unformatted text preview: B/MapleTA' MATH 135 1119 Algebra for Honours Mathematics : Grades Sfitem Homegage » Class Homepage » Gradebook » Student Grades » Assignment Details aw Maulesoft' Welcome Bill Zimou Peng MyProfile Gradebook Help Logout VWew Panel Refresh View Rows: 20 F Grade Data l" User Data , Startdate l" Starttime f1? Lastname f1? Given Name '— Class Name Enddate I" Endtime _‘ Ml _‘ Login Exportto CSV Style: Numerlc #Attempts l" Duration _‘ Email _‘ StudentlD Showing Best grades, Completed assignments, All users Student Details vAssignment 7 Score: Duration: 17 min Started: 9/22/11 8:48:02 PM 9/ 1 0.0 Flnished: 9/22/11 9:05:44 PM V Bill Zimou Peng Login: bzpeng Email: [email protected] Student ID: 20430954 Page:1of1Rows:1-50f5 - 0 1 Question Grade: Weighted Grade: F 2 Question Grade: Weighted Grade: Com plete d: Assignments 10 Active: 0 To Be Reviewed: 0 Passed: 8 1.0 (1/1.0) 1.0 mm) Question Your res ponse Look up the definition of continuous at the point X on page 85 of the text Howto Read and Do Proofs, 5th ed., by Daniel Solow and choose the bestcompletion forthe statement: The function f ofone real variable is continous at the point X ifand only if Ve>0 36>0 3 Vy where |x—y|{§, |f[x}—f[y}|<e (100%) Comment: The definition of continuous at a point X is given in the text as: The function fofone real variable is continuous at a point X ifand onlyif, for every real number E > 0 ,there is a real number 5‘) U such that, for all real numbers y with I X — y I < 5, IflX}-fly}|<é- The correct option is given in symbols that corresponds to the text description above: Ve>0 38>0 3 Vywherelx—y|<§,|f[X}—f[y)|<e Your response Which ofthe following symbolic statements correctly represents the following English Statement? Fora set S of real numbers, no matter which element is chosen in the set S, you can find another element that is strictlylarger. Vxe S, EIyE S, y>x.(1oo%) Comment: The correctanswer is: VXES,EIyES,y>x And this statment reads as: "Forallelements X in S,there exists an element y in 3,5uch that )2' is strictlygreaterthan X." ‘ 3 Question Grade: Weighted Gade: ‘ 4 Question Grade: Weighted Gade: 0.15 (314.0) 1 .0 (313.0) When we say "For all elements X in S" it is equivalentto saying "no matter which elementis chosen in the set S", and we have labelled this arbitraryelementas X. When we say "there exists an element y in 3“ it is equivalentto saying “you can find another element", exceptwe explicitlystate that the elementwill be found in the set S as well (and we have labelled this element as y ). When we say "such that y is strictlygreater than X" this matches the end ofthe given statementwhich, in the words "that is strictlylarger“, implies that the element we can find in S is strictlylargerthan the arbitrarilychosen element of Your response The following questions deal with functions that are onto. To saythat a real-talued function f of one real variable is onto means thatfor every real number y,there is a real number X such that such that flX} = y. Is the object X modified byan existential quantifier? (25%) What is the "something that happens" associated with the number )2? there is a real number x such that fix}: y (25%) Suppose you are required to show that g is an onto function. Which ofthe following would be mostsuitable fora structure ofyour proof? Let y E IR . . . Construct X as follows . .(25%) Is the function tan[X} onto? No (0%) S . Correct response The following questions deal with functions that are onto. To saythata real-Valued function f of one real variable is onto means that for every real number y , there is a real number X such that such that flX} = y. Is the object X modified by an existential quantifier? What is the "something that happens" associated with the number )2? there is a real number X such that flX} = y Suppose you are required to show that g is an onto function. Which ofthe following would be mostsuitable for a structure ofyour proof? Let y E [R . . . Construct X as follows Is the function tan[X} onto? Yes Total grade: 1 .OX1I4 +1.0X1I4 +1.0X1I4 + 0.0X1I4 = 25% + 25% + 25% + 0% Comment: i 2} Incorrect The object X is modifed bythe existential quantifier‘there exists“ or El .This is because we saythat"there is a real number X" which is another wayofsaying "there exists a real number X". Once we have chosen a real number y ,the "something that happens" with it is that we can find a real number X such that fix} = y. Ifwe are asked to constructa proofto show thata function g is onto, we should firstchoose our arbitrary y E R, then oonstructa suitable X based on that choice of y . Thus, a suitable structure forthe proofis LetyelR.. .Construct X as follows . .. When trying to determine ifa function is onto, we can ask ourselves what the range ofthatfunction is. lfthe range ofthe function is all real numbers, then the function is onto. This is because ifevery real number is in the range of f, then for anyreal number y,there mustbe some X such that flX} = y, otherwise )J is notin the range of f which contradicts thatlhe range of fis all real numbers. The range of tan[X} is all real numbers, and thus tan[X} is onto. Your res ponse The following questions deal with a convergentsequence. The sequence of real numbers {xi} ]“ convergesto Xif Ve E lR,e> 0, EIJ'E 2,3213 v k E 2, ic>j, le— X|< 5. Correct (Note that sequences are covered in AppendixD ofthe text Howto Read and Do Proofs, 5th ed., by Daniel Solow.) Whatis the "universe ofdiscourse“forthe object j? Z (33%) ...
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