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| 1 : | jj | 143 | #DESCRIPTION |
| 2 : | #KEYWORDS('continuity', 'theory') | ||
| 3 : | # properties of continuous and differentiable functions -- theory | ||
| 4 : | #ENDDESCRIPTION | ||
| 5 : | |||
| 6 : | DOCUMENT(); # This should be the first executable line in the problem. | ||
| 7 : | |||
| 8 : | loadMacros("PG.pl", | ||
| 9 : | "PGbasicmacros.pl", | ||
| 10 : | "PGchoicemacros.pl", | ||
| 11 : | "PGanswermacros.pl", | ||
| 12 : | "PGauxiliaryFunctions.pl"); | ||
| 13 : | |||
| 14 : | TEXT(beginproblem()); | ||
| 15 : | $showPartialCorrectAnswers = 0; | ||
| 16 : | |||
| 17 : | # continuity (for real numbers) | ||
| 18 : | |||
| 19 : | @questions = (); | ||
| 20 : | @answers = (); | ||
| 21 : | qa(~~@questions, ~~@answers, | ||
| 22 : | EV2( "If \( f(x) \) is a continuous function and the sequence \(a_{1}, a_{2}, a_{3}, ...\) converges to | ||
| 23 : | a finite limit, then the sequence \(f(a_{1}), f(a_{2}), f(a_{3}), ...\) also converges to a limit."), | ||
| 24 : | "T", | ||
| 25 : | |||
| 26 : | EV2( "If \( f(x) \) is a continuous function and the sequence \(f(a_{1}), f(a_{2}), f(a_{3}), ...\) converges to | ||
| 27 : | a finite limit, then the sequence \(a_{1}, a_{2}, a_{3}, ...\) also converges to a limit."), | ||
| 28 : | "F", | ||
| 29 : | |||
| 30 : | # continuous functions and max and min | ||
| 31 : | |||
| 32 : | EV2( "Every continuous function has a maximum value."), | ||
| 33 : | "F", | ||
| 34 : | |||
| 35 : | EV2( "Every continuous function whose domain is a bounded, closed | ||
| 36 : | interval has a maximum value."), | ||
| 37 : | "T", | ||
| 38 : | |||
| 39 : | EV2( "If a continuous function has a maximum value then it also has a minimum value."), | ||
| 40 : | "F", | ||
| 41 : | |||
| 42 : | EV2( "Every continuous function | ||
| 43 : | whose domain is a bounded, closed interval and which has a | ||
| 44 : | maximum value also has a minimum value."), | ||
| 45 : | "T", | ||
| 46 : | |||
| 47 : | EV2( "If a continuous function \(f(x)\) has a maximum value on an interval then the | ||
| 48 : | function \( -f(x) \) has a minimum on that same interval."), | ||
| 49 : | "T", | ||
| 50 : | |||
| 51 : | EV2( "If a continuous function has a maximum value then its domain must be a | ||
| 52 : | bounded, closed interval."), | ||
| 53 : | "F", | ||
| 54 : | |||
| 55 : | # differentiable functions and max-min | ||
| 56 : | EV2( "Every differentiable function is continuous."), | ||
| 57 : | "T", | ||
| 58 : | |||
| 59 : | EV2( "Every continuous function is differentiable."), | ||
| 60 : | "F", | ||
| 61 : | |||
| 62 : | EV2( "Every differentiable function has a maximum value."), | ||
| 63 : | "F", | ||
| 64 : | |||
| 65 : | EV2( "Every differentiable function whose domain is a bounded, closed | ||
| 66 : | interval has a maximum value."), | ||
| 67 : | "T", | ||
| 68 : | |||
| 69 : | EV2( "If a differentiable function has a maximum value then it also has a minimum value."), | ||
| 70 : | "F", | ||
| 71 : | |||
| 72 : | EV2( "Every differentiable function | ||
| 73 : | whose domain is a bounded, closed interval and which has a | ||
| 74 : | maximum value also has a minimum value."), | ||
| 75 : | "T", | ||
| 76 : | |||
| 77 : | EV2( "If a differentiable function \(f(x)\) has a maximum value on an interval then the | ||
| 78 : | function \( -f(x) \) has a minimum on that same interval."), | ||
| 79 : | "T", | ||
| 80 : | |||
| 81 : | EV2( "If a differentiable function has a maximum value then its domain must be a | ||
| 82 : | bounded, closed interval."), | ||
| 83 : | "F", | ||
| 84 : | |||
| 85 : | EV2( "If the linear approximation of a differentiable function is increasing at a point \( a \) | ||
| 86 : | then the function is also increasing near the point \( a \)."), | ||
| 87 : | "T", | ||
| 88 : | |||
| 89 : | EV2( "If a function is increasing near a point \(a \) then its linear approximation | ||
| 90 : | at \( a \) cannot be decreasing."), | ||
| 91 : | "T", | ||
| 92 : | |||
| 93 : | EV2( "If the linear approximation of a differentiable function is decreasing at a point \( a \) | ||
| 94 : | then the function could be constant near the point \( a \)."), | ||
| 95 : | "F", | ||
| 96 : | |||
| 97 : | EV2( "If the linear approximation of a differentiable function is constant at a point \( a \) then | ||
| 98 : | the function could be increasing near the point \( a \)."), | ||
| 99 : | "T", | ||
| 100 : | |||
| 101 : | EV2( "If the linear approximation of a differentiable function is constant at a point \( a \) then | ||
| 102 : | the function could be decreasing near the point \( a \)."), | ||
| 103 : | "T", | ||
| 104 : | ); | ||
| 105 : | $thisCourse = $inputs_ref->{course}; | ||
| 106 : | TEXT(EV2(<<EOT)); | ||
| 107 : | Enter a T or an F in each answer space below | ||
| 108 : | to indicate whether the corresponding statement is true or | ||
| 109 : | false. $PAR A good technique is to think of several examples, especially examples | ||
| 110 : | which might show that the statement is false! $PAR | ||
| 111 : | For reference you can find | ||
| 112 : | some definitions \{ &htmlLink("${htmlURL}continuitydefinitions.html","here")\}. | ||
| 113 : | $PAR | ||
| 114 : | You must get all of the answers correct to receive credit. | ||
| 115 : | EOT | ||
| 116 : | |||
| 117 : | @slice = NchooseK(scalar(@questions),4); | ||
| 118 : | |||
| 119 : | TEXT( | ||
| 120 : | &match_questions_list(@questions[@slice]) | ||
| 121 : | ); | ||
| 122 : | ANS(str_cmp([@answers[@slice]])); | ||
| 123 : | |||
| 124 : | ENDDOCUMENT(); # This should be the last executable line in the problem. |
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