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Wed Jul 18 00:35:17 2007 UTC (5 years, 10 months ago) by jjholt
File size: 2328 byte(s)
Changed Stew4e to Stew 6e.


    1 ##DESCRIPTION
2 ## Anti-derivatives --
3 ##ENDDESCRIPTION
4 ##KEYWORDS('derivatives', 'antiderivatives', 'distance,velocity,acceleration')
5
6 ## Shotwell cleaned
7 ## lcao , PAID on 11-24-2003
8
9 ## DBsubject('Calculus')
10 ## DBchapter('Applications of Differentiation')
11 ## DBsection('Antiderivatives')
12 ## Date('6/3/2002')
13 ## Author('')
14 ## Institution('')
15 ## TitleText1('Calculus: Early Transcendentals')
16 ## EditionText1('6')
17 ## AuthorText1('Stewart')
18 ## Section1('4.9')
19 ## Problem1('2')
20 #TYPE('word problem')
21
22 DOCUMENT();        # This should be the first executable line in the problem.
23
25 "PGbasicmacros.pl",
27 "PGauxiliaryFunctions.pl"
28 );
29
30 TEXT(beginproblem());
31 $showPartialCorrectAnswers = 1; 32 33$a1 = random(2,10,1);
34 $b1 = random(2,10,1); 35$c1 = random(2,10,1);
36 $d1 = random(1,10,1); 37 38 BEGIN_TEXT 39 Consider the function $$f(x) = a1 x^3 - b1 x^2 + c1 x - d1$$. An antiderivative of $$f(x)$$ is $$F(x) = A x ^4 +B x^3 +C x^2 +D x$$.$BR$BR 40 Solve for $$A, B,$$ and $$C$$.$BR
41 $$A =$$ \{ans_rule(20) \} $BR 42 $$B =$$ \{ans_rule(20)\}$BR
43 $$C =$$ \{ans_rule(20)\} $BR 44 $$D=$$ \{ans_rule(20)\}$BR
45
46 END_TEXT
47
48 $ans1 = "$a1/4";
49 $ans2 = "-($b1/3)";
50 $ans3 = "$c1/2";
51 $ans4 = -$d1;
52
53 ANS(num_cmp($ans1)); 54 ANS(num_cmp($ans2));
55 ANS(num_cmp($ans3)); 56 ANS(num_cmp($ans4));
57
58 SOLUTION(EV3(<<'EOF'));
59 $SOL$BR
60 This is just an exercise in the rules of finding derivatives of
61 polynomials.  It is, however, somewhat more difficult since we are
62 working in terms of antiderivatives.  First, recall that the reason we
63 call $$F(x)$$ the antiderivative of $$f(x)$$ is that the derivative
64 of $$F(x)$$ is $$f(x)$$.  So in this case, let's find the derivative of
65 $$F(x)$$ with the constants A,B,C, and D in place.
66 $BR$BR
67 $68 F'(x) = 4Ax^3 + 3Bx^2 + 2Cx + D 69$
70 $BR$BR
71 Now, matching this up with $$f(x)$$, we see that we must have
72 the following equalities if $$f(x)$$ is to be the derivative of $$F(x)$$.
73 $BR$BR
74 $75 \begin{array} 76 & a1 = 4A, \quad -b1 = 3B \\ 77 & c1 = 2C, \quad -d1 = D \\ 78 \end{array} 79$
80 $BR$BR
81 So solving, we get:
82 $BR$BR
83 $84 \begin{array} 85 & A = \frac{a1}{4} = ans1} & 86 & B = -\frac{b1}{3} = ans2} & 87 & C = \frac{c1}{2} = ans3} & 88 & D = ans4 \\ & 89 \end{array} 90$
91
92 EOF
93
94 ENDDOCUMENT();        # This should be the last executable line in the problem.