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| 1 : | jjholt | 255 | ##KEYWORDS('Calculus') |
| 2 : | ##Tagged by ynw2d | ||
| 3 : | jj | 143 | |
| 4 : | jjholt | 255 | ##DBsubject('Calculus') |
| 5 : | jjholt | 446 | ## DBchapter('Limits and Derivatives') |
| 6 : | jjholt | 255 | ##DBsection('Continuity') |
| 7 : | jj | 143 | |
| 8 : | DOCUMENT(); # This should be the first executable line in the problem. | ||
| 9 : | |||
| 10 : | loadMacros( | ||
| 11 : | "PG.pl", | ||
| 12 : | "PGbasicmacros.pl", | ||
| 13 : | "PGchoicemacros.pl", | ||
| 14 : | "PGanswermacros.pl", | ||
| 15 : | "PGgraphmacros.pl", | ||
| 16 : | "PGauxiliaryFunctions.pl" | ||
| 17 : | ); | ||
| 18 : | |||
| 19 : | TEXT(beginproblem()); | ||
| 20 : | $showPartialCorrectAnswers = 1; | ||
| 21 : | |||
| 22 : | $a=random(-3,3,1); | ||
| 23 : | $b=non_zero_random(-2,3,1); | ||
| 24 : | $c=random(-3,2,1); | ||
| 25 : | $m1=random(-1,1,0.5); | ||
| 26 : | $m2=($b - $a)/2; | ||
| 27 : | $m3=($c - $b - 1)/2; | ||
| 28 : | $m4=random(-1,1,0.5); | ||
| 29 : | @slice = NchooseK(3,3); | ||
| 30 : | |||
| 31 : | @colors = ("blue", "red", "green"); | ||
| 32 : | @sc = @colors[@slice]; #scrambled colors | ||
| 33 : | @sa = ('A','B','C')[@slice]; | ||
| 34 : | |||
| 35 : | $f1 = FEQ("${m1}*(x+1)+$a for x in [-2,-1) using color:$sc[0] and weight:2"); | ||
| 36 : | $f2 = FEQ("${m2}*(x-1)+$b for x in (-1,1) using color=$sc[0] and weight:2"); | ||
| 37 : | $f3 = FEQ("${m3}*(x-3)+$c for x in [1,3) using color=$sc[0] and weight=2"); | ||
| 38 : | $f4 = FEQ("1+$a for x in [-1,-1] using color=$sc[0] and weight=2"); | ||
| 39 : | $f5 = FEQ("${m4}*(x-3)+$c for x in (3,4] using color=$sc[0] and weight=2"); | ||
| 40 : | |||
| 41 : | $graph = init_graph(-3,-6,5,6,'axes'=>[0,0],'grid'=>[8,12]); | ||
| 42 : | |||
| 43 : | ($f1Ref,$f2Ref,$f3Ref,$f4Ref,$f5Ref) = plot_functions($graph,$f1,$f2,$f3,$f4,$f5); | ||
| 44 : | |||
| 45 : | TEXT(EV2(<<EOT)); | ||
| 46 : | Let F be the function below.$PAR | ||
| 47 : | EOT | ||
| 48 : | |||
| 49 : | TEXT(image( insertGraph($graph) , height=>200, width=>200)); | ||
| 50 : | |||
| 51 : | TEXT(EV2(<<EOT)); | ||
| 52 : | $BR | ||
| 53 : | $BR | ||
| 54 : | Evaluate each of the following expressions. $PAR | ||
| 55 : | Note: Enter 'DNE' if the limit does not exist or is not defined. $PAR | ||
| 56 : | |||
| 57 : | a) \( \displaystyle \lim_{x \to -1^-} F(x) \) = \<ans_rule(4)\> | ||
| 58 : | $PAR | ||
| 59 : | |||
| 60 : | b) \( \displaystyle \lim_{x \to -1^+} F(x) \) = \<ans_rule(4)\> | ||
| 61 : | $PAR | ||
| 62 : | |||
| 63 : | c) \( \displaystyle \lim_{x \to -1} F(x) \) = \<ans_rule(4)\> | ||
| 64 : | $PAR | ||
| 65 : | |||
| 66 : | d) \( F(-1) \) = \<ans_rule(4)\> | ||
| 67 : | $PAR | ||
| 68 : | |||
| 69 : | e) \( \displaystyle \lim_{x \to 1^-} F(x) \) = \<ans_rule(4)\> | ||
| 70 : | $PAR | ||
| 71 : | |||
| 72 : | f) \( \displaystyle \lim_{x \to 1^+} F(x) \) = \<ans_rule(4)\> | ||
| 73 : | $PAR | ||
| 74 : | |||
| 75 : | g) \( \displaystyle \lim_{x \to 1} F(x) \) = \<ans_rule(4)\> | ||
| 76 : | $PAR | ||
| 77 : | |||
| 78 : | h) \( \displaystyle \lim_{x \to 3} F(x) \) = \<ans_rule(4)\> | ||
| 79 : | $PAR | ||
| 80 : | |||
| 81 : | i) \( F(3) \) = \<ans_rule(4)\> | ||
| 82 : | $PAR | ||
| 83 : | |||
| 84 : | EOT | ||
| 85 : | |||
| 86 : | $ap1 = 1 + $a; | ||
| 87 : | $bp1 = 1 + $b; | ||
| 88 : | |||
| 89 : | ANS(num_cmp($a, strings=>['DNE'])) ; | ||
| 90 : | ANS(num_cmp($a, strings=>['DNE'])) ; | ||
| 91 : | ANS(num_cmp($a, strings=>['DNE'])) ; | ||
| 92 : | ANS(num_cmp($ap1, strings=>['DNE'])); | ||
| 93 : | ANS(num_cmp($b, strings=>['DNE'])) ; | ||
| 94 : | ANS(num_cmp($bp1, strings=>['DNE'])); | ||
| 95 : | ANS(num_cmp('DNE', strings=>['DNE'])) ; | ||
| 96 : | ANS(num_cmp($c, strings=>['DNE'])) ; | ||
| 97 : | ANS(num_cmp('DNE', strings=>['DNE'])) ; | ||
| 98 : | |||
| 99 : | ENDDOCUMENT(); # This should be the last executable line in the problem. | ||
| 100 : |
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