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Fri Jun 29 18:55:12 2012 UTC (10 months, 3 weeks ago) by jj
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```Duplicate problems.
```

```    1 ## DESCRIPTION
2 ##   The Mean Value Theorem
3 ## ENDDESCRIPTION
4
5 ## KEYWORDS('Mean Value', 'Slope')
6 ## Tagged by nhamblet
7
8 ## DBsubject('Calculus')
9 ## DBchapter('Applications of Differentiation')
10 ## DBsection('The Mean Value Theorem')
11 ## Date('')
12 ## Author('')
13 ## Institution('Rochester')
14 ## TitleText1('Calculus: Early Transcendentals')
15 ## EditionText1('6')
16 ## AuthorText1('Stewart')
17 ## Section1('4.2')
18 ## Problem1('12')
19
20
21 DOCUMENT();        # This should be the first executable line in the problem.
22
24            "PGbasicmacros.pl",
25            "PGchoicemacros.pl",
27 "PGauxiliaryFunctions.pl");
28
29 TEXT(beginproblem());
31
32 sub f1 {
33     my \$x = shift;
34     \$A3*\$x**3 + \$A2*\$x**2 + \$A1*\$x +\$A0;
35     }
36 \$discrim = 0;
37
38 while (\$discrim == 0 ) {
39   \$A3=  random(1,3,1)*(random(0,1,1) ? 1: -1);
40   \$A2 = non_zero_random(-4,4,1);
41   \$A1 = non_zero_random(-4,4,1);
42   \$A0 = non_zero_random( -4 ,4,1);
43   \$left = random(-4,4,1);
44   \$right = \$left + random (1,8,1);
45
46   \$avg = (f1(\$right) - f1(\$left) ) /(\$right - \$left);
47   \$a= 3*\$A3;
48   \$b=2*\$A2;
49   \$c = \$A1 - \$avg;
50   \$discrim = \$b**2 - 4*\$a*\$c;
51   }
52
53 \$poly = nicestring([\$A3, \$A2, \$A1, \$A0]);
54
55 \$root1 = ( -\$b - sqrt(\$discrim) ) /(2*\$a);
56 \$root2 = ( -\$b + sqrt(\$discrim) ) /(2*\$a);
57 TEXT(EV2(<<EOT));
58 Consider the function
59 \[ f(x) = \$poly\]
60 Find the average slope of this function on the interval \( ( \$left , \$right ) \).
61 \{ans_rule(20) \} \$BR
62 EOT
63
64 if (\$left<= \$root1 and \$root1 <= \$right and \$left < \$root2 and \$root2 <= \$right) { # both roots are in interval)
65   TEXT(EV2(<<EOT));
66 By the Mean Value Theorem, we know there exists a \( c \) in the open interval
67 \( ( \$left, \$right ) \) such that \( f'(c) \) is equal to this mean slope.
68 Find the two values of \( c \) in the interval which work, enter the smaller root first:
69
70 \{ ans_rule(10) \}\(  \$LTE \)  \{ ans_rule(10) \}
71 EOT
72
73     ANS(num_cmp([\$avg,num_sort(\$root1, \$root2)], relTol=>.1));
74     }
75 elsif (\$left<\$root1 and \$root1<\$right ) { #root 1 is in the interval
76   TEXT(EV2(<<EOT));
77 By the Mean Value Theorem, we know there exists a \( c \) in the open interval
78 \( ( \$left, \$right ) \) such that \( f'(c) \) is equal to this mean slope.
79 Find the  value of \( c \) in the interval which works
80 \{ ans_rule(10) \}
81 EOT
82   ANS(num_cmp([\$avg,\$root1], relTol=>.1));
83   }
84 elsif (\$left<\$root2 and \$root2<\$right ) { #root 1 is in the interval
85   TEXT(EV2(<<EOT));
86 By the Mean Value Theorem, we know there exists a \( c \) in the open interval
87 \( ( \$left, \$right ) \) such that \( f'(c) \) is equal to this mean slope.
88 Find the  value of \( c \) in the interval which works
89 \{ ans_rule(10) \}
90 EOT
91   ANS(num_cmp([\$avg,\$root2], relTol=>.1));
92   }
93 else  {
94   TEXT("Error in formulating problem -- inform instructor please.");
95   }
96
97 ENDDOCUMENT();        # This should be the last executable line in the problem.
```