View of /trunk/NationalProblemLibrary/Rochester/setLimitsRates1TangentVelocity/s1_1_4.pg

    1 ##DESCRIPTION
    2 ##  Find slopes of secant lines of a curve at a point, then guess the slope of
    3 ##  a tangent line at that point
    4 ##ENDDESCRIPTION
    5 
    6 ##KEYWORDS('Calculus')
    7 ##Tagged by ynw2d
    8 
    9 ##DBsubject('Calculus')
   10 ##DBchapter('Limits and Derivatives')
   11 ##DBsection('The Tangent and Velocity Problems')
   12 
   13 DOCUMENT();        # This should be the first executable line in the problem.
   14 
   15 loadMacros(
   16 "PG.pl",
   17 "PGbasicmacros.pl",
   18 "PGchoicemacros.pl",
   19 "PGanswermacros.pl",
   20 "PGauxiliaryFunctions.pl"
   21 );
   22 
   23 TEXT(beginproblem());
   24 $showPartialCorrectAnswers = 1;
   25 
   26 $a1 = random(2,5,1);
   27 $b1= random(2,5,1);
   28 $x0 = 1/$b1;
   29 $y0 = $a1*$b1;
   30 $x1 = $x0 + 0.1;
   31 $x01 = $x0 + 0.01;
   32 $x9 =  $x0 - 0.1;
   33 $x99 = $x0- 0.01;
   34 
   35 TEXT(EV2(<<EOT));
   36 The point \( P( $x0 , $y0 ) \) lies on the curve
   37 \( y = $a1 / x \).  If \( Q \) is the point
   38 \( (x,  $a1 / x ) \), find the slope of the secant line
   39 \( PQ \) for the following values of \( x \).
   40 $BR
   41 If \( x= $x1 \), the slope of \( PQ \) is:
   42 \{ans_rule(25) \}
   43 $BR
   44 EOT
   45 
   46 $ans =-$a1/($x0*$x1);
   47 ANS(num_cmp($ans));
   48 
   49 TEXT(EV2(<<EOT));
   50 and if \( x= $x01 \), the slope of \( PQ \) is:
   51 \{ans_rule(25)  \}
   52 $BR
   53 EOT
   54 
   55 $ans =-$a1/($x0*$x01);
   56 ANS(num_cmp($ans));
   57 
   58 TEXT(EV2(<<EOT));
   59 and if \( x= $x9 \), the slope of \( PQ \) is:
   60 \{ans_rule(25)  \}
   61 $BR
   62 EOT
   63 
   64 $ans =-$a1/($x0*$x9);
   65 ANS(num_cmp($ans));
   66 
   67 TEXT(EV2(<<EOT));
   68 and if \( x= $x99 \), the slope of \( PQ \) is:
   69 \{ans_rule(25) \}
   70 $BR
   71 EOT
   72 
   73 $ans =-$a1/($x0*$x99);
   74 ANS(num_cmp($ans));
   75 
   76 TEXT(EV2(<<EOT));
   77 Based on the above results, guess the slope of the tangent
   78 line to the curve at \( P($x0 , $y0 ) \).
   79 \{ans_rule(25) \}
   80 $BR
   81 EOT
   82 
   83 $ans =-$a1/($x0*$x0);
   84 
   85 ANS(num_cmp($ans, relTol=>4));
   86 
   87 ENDDOCUMENT();        # This should be the last executable line in the problem.
   88