| 1 | #DESCRIPTION |
1 | # This file is just a pointer to the file |
| 2 | #KEYWORDS('derivatives', 'mean value theorem') |
2 | # |
| 3 | ## tsuda tagged and PAID on 2-20-2004 |
3 | # "Library/Rochester/setDerivatives12MVT/c3s2p1.pg" |
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4 | # |
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5 | # You may want to change your problem set to use that problem |
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6 | # directly, especially if you want to make a copy of the problem |
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7 | # for modification. |
| 4 | |
8 | |
| 5 | ## DBsubject('Calculus') |
9 | DOCUMENT(); |
| 6 | ## DBchapter('Applications of Differentiation') |
10 | includePGproblem("Library/Rochester/setDerivatives12MVT/c3s2p1.pg"); |
| 7 | ## DBsection('The Mean Value Theorem') |
11 | ENDDOCUMENT(); |
| 8 | ## Date('6/3/2002') |
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| 9 | ## Author('') |
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| 10 | ## Institution('') |
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| 11 | ## TitleText1('Calculus: Early Transcendentals') |
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| 12 | ## EditionText1('6') |
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| 13 | ## AuthorText1('Stewart') |
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| 14 | ## Section1('4.2') |
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| 15 | ## Problem1('12') |
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| 16 | |
12 | |
| 17 | # Validate mean value theorem in special cases |
13 | ## These tags keep this problem from being added to the NPL database |
| 18 | #ENDDESCRIPTION |
14 | ## |
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15 | ## DBsubject('ZZZ-Inserted Text') |
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16 | ## DBchapter('ZZZ-Inserted Text') |
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17 | ## DBsection('ZZZ-Inserted Text') |
| 19 | |
18 | |
| 20 | DOCUMENT(); # This should be the first executable line in the problem. |
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| 21 | |
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| 22 | loadMacros("PG.pl", |
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| 23 | "PGbasicmacros.pl", |
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| 24 | "PGchoicemacros.pl", |
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| 25 | "PGanswermacros.pl", |
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| 26 | "PGauxiliaryFunctions.pl", |
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| 27 | "PGcourse.pl"); |
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| 28 | |
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| 29 | TEXT(beginproblem()); |
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| 30 | $showPartialCorrectAnswers=1; |
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| 31 | |
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| 32 | sub f1 { |
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| 33 | my $x = shift; |
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| 34 | $A3*$x**3 + $A2*$x**2 + $A1*$x +$A0; |
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| 35 | } |
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| 36 | $discrim = 0; |
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| 37 | |
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| 38 | while ($discrim == 0 ) { |
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| 39 | $A3= random(1,3,1)*(random(0,1,1) ? 1: -1); |
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| 40 | $A2 = non_zero_random(-4,4,1); |
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| 41 | $A1 = non_zero_random(-4,4,1); |
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| 42 | $A0 = non_zero_random( -4 ,4,1); |
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| 43 | $left = random(-4,4,1); |
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| 44 | $right = $left + random (1,8,1); |
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| 45 | |
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| 46 | $avg = (f1($right) - f1($left) ) /($right - $left); |
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| 47 | $a= 3*$A3; |
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| 48 | $b=2*$A2; |
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| 49 | $c = $A1 - $avg; |
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| 50 | $discrim = $b**2 - 4*$a*$c; |
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| 51 | } |
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| 52 | |
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| 53 | $root1 = ( -$b - sqrt($discrim) ) /(2*$a); |
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| 54 | $root2 = ( -$b + sqrt($discrim) ) /(2*$a); |
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| 55 | TEXT(EV2(<<EOT)); |
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| 56 | Consider the function |
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| 57 | \[ f(x) = !{$A3}x^3 ? {$A2}x^2 ? {$A1}x ? {$A0} \] |
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| 58 | Find the average slope of this function on the interval \( ( $left , $right ) \). |
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| 59 | \{ans_rule(20) \} $BR |
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| 60 | EOT |
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| 61 | |
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| 62 | if ($left<= $root1 and $root1 <= $right and $left < $root2 and $root2 <= $right) { # both roots are in interval) |
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| 63 | TEXT(EV2(<<EOT)); |
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| 64 | By the Mean Value Theorem, we know there exists a \( c \) in the open interval |
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| 65 | \( ( $left, $right ) \) such that \( f'(c) \) is equal to this mean slope. |
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| 66 | Find the two values of \( c \) in the interval which work, enter the smaller root first: |
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| 67 | |
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| 68 | \{ ans_rule(10) \}\( $LTE \) \{ ans_rule(10) \} |
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| 69 | EOT |
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| 70 | |
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| 71 | ANS(num_cmp([$avg,num_sort($root1, $root2)], relTol=>.1)); |
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| 72 | } |
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| 73 | elsif ($left<$root1 and $root1<$right ) { #root 1 is in the interval |
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| 74 | TEXT(EV2(<<EOT)); |
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| 75 | By the Mean Value Theorem, we know there exists a \( c \) in the open interval |
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| 76 | \( ( $left, $right ) \) such that \( f'(c) \) is equal to this mean slope. |
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| 77 | Find the value of \( c \) in the interval which works |
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| 78 | \{ ans_rule(10) \} |
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| 79 | EOT |
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| 80 | ANS(num_cmp([$avg,$root1], relTol=>.1)); |
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| 81 | } |
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| 82 | elsif ($left<$root2 and $root2<$right ) { #root 1 is in the interval |
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| 83 | TEXT(EV2(<<EOT)); |
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| 84 | By the Mean Value Theorem, we know there exists a \( c \) in the open interval |
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| 85 | \( ( $left, $right ) \) such that \( f'(c) \) is equal to this mean slope. |
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| 86 | Find the value of \( c \) in the interval which works |
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| 87 | \{ ans_rule(10) \} |
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| 88 | EOT |
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| 89 | ANS(num_cmp([$avg,$root2], relTol=>.1)); |
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| 90 | } |
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| 91 | else { |
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| 92 | TEXT("Error in formulating problem -- inform instructor please."); |
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| 93 | } |
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| 94 | |
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| 95 | ENDDOCUMENT(); # This should be the last executable line in the problem. |
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