View of /trunk/NationalProblemLibrary/ma122DB/set10/s4_10_2.pg

    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 
   24 loadMacros(
   25 "PGbasicmacros.pl",
   26 "PGanswermacros.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.