View of /trunk/NationalProblemLibrary/Rochester/setLimitsRates1TangentVelocity/s1_1_2.pg

    1 ##DESCRIPTION
    2 ##  Find slope of secant lines of a curve, then guess the slope of
    3 ##  a tangent line
    4 ##ENDDESCRIPTION
    5 
    6 ##KEYWORDS('Calculus')
    7 ##Tagged by ynw2d
    8 
    9 ##DBsubject('Calculus')
   10 ##DBchapter('Limits and Derivatives')
   11 ## DBsection('Tangents, Velocities, and Other Rates of Change')
   12 ## TitleText1('Calculus: Early Transcendentals')
   13 ## EditionText1('1')
   14 ## AuthorText1('Rogawski')
   15 ## Section1('2.1')
   16 ## Problem1('19')
   17 ## TitleText2('Calculus: Early Transcendentals')
   18 ## EditionText2('1')
   19 ## AuthorText2('Rogawski')
   20 ## Section2('2.1')
   21 ## Problem2('19')
   22 
   23 DOCUMENT();        # This should be the first executable line in the problem.
   24 
   25 loadMacros(
   26 "PG.pl",
   27 "PGbasicmacros.pl",
   28 "PGchoicemacros.pl",
   29 "PGanswermacros.pl",
   30 "PGauxiliaryFunctions.pl",
   31 "PGcourse.pl"
   32 );
   33 
   34 TEXT(beginproblem());
   35 $showPartialCorrectAnswers = 1;
   36 
   37 $a1 = random(3,8,1);
   38 $x0 = random(2,5,1);
   39 $y0 = $x0*$x0 + $x0 + $a1;
   40 $x1 = $x0 + 0.1;
   41 $x01 = $x0 + 0.01;
   42 $x9 =  $x0 - 0.1;
   43 $x99 = $x0- 0.01;
   44 
   45 TEXT(EV2(<<EOT));
   46 The point \( P($x0 , $y0 ) \) lies on the curve
   47 \( y = x^2 + x + $a1 \).  If \( Q \) is the point
   48 \( (x, x^2 + x + $a1 ) \), find the slope of the secant line
   49 \( PQ \) for the following values of \( x \).
   50 $BR
   51 If \( x= $x1 \), the slope of \( PQ \) is:
   52 \{ans_rule(25) \}
   53 $BR
   54 EOT
   55 
   56 $ans = $x0 +$x1 + 1;
   57 ANS(num_cmp($ans));
   58 
   59 TEXT(EV2(<<EOT));
   60 and if \( x= $x01 \), the slope of \( PQ \) is:
   61 \{ans_rule(25) \}
   62 $BR
   63 EOT
   64 
   65 $ans = $x0 +$x01 + 1;
   66 ANS(num_cmp($ans));
   67 
   68 TEXT(EV2(<<EOT));
   69 and if \( x= $x9 \), the slope of \( PQ \) is:
   70 \{ans_rule(25) \}
   71 $BR
   72 EOT
   73 
   74 $ans = $x0 +$x9 + 1;
   75 ANS(num_cmp($ans));
   76 
   77 TEXT(EV2(<<EOT));
   78 and if \( x= $x99 \), the slope of \( PQ \) is:
   79 \{ans_rule(25) \}
   80 $BR
   81 EOT
   82 
   83 $ans = $x0 +$x99 + 1;
   84 ANS(num_cmp($ans));
   85 
   86 TEXT(EV2(<<EOT));
   87 Based on the above results, guess the slope of the tangent
   88 line to the curve at \( P($x0 , $y0 ) \).
   89 \{ans_rule(25) \}
   90 $BR
   91 EOT
   92 
   93 $ans = 2*$x0 + 1;
   94 ANS(num_cmp($ans, relTol=>1));
   95 
   96 ENDDOCUMENT();        # This should be the last executable line in the problem.
   97