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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
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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:150f5  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 realtalued 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 realValued 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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 Spring '08
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 Calculus

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