RiemannSums1

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Dynamically Generated Graphs with Riemann Sums

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This PG code shows how to make dynamically generated graphs with shaded (filled) Riemann sums.

• File location in OPL: FortLewis/Authoring/Templates/IntegralCalc/RiemannSums1.pg
• PGML location in OPL: FortLewis/Authoring/Templates/IntegralCalc/RiemannSums1_PGML.pg

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PG problem file	Explanation
Problem tagging data	Problem tagging:
DOCUMENT(); loadMacros( "PGstandard.pl", "PGchoicemacros.pl", "PGgraphmacros.pl", "MathObjects.pl", "weightedGrader.pl", "unionTables.pl", ); TEXT(beginproblem()); install_weighted_grader(); $refreshCachedImages = 1; $showPartialCorrectAnswers = 1;	Initialization:
# Construct a graph for the left endpoint Riemann sum, # define the function to be graphed, and add it to the graph $graphL = init_graph(-1,-1,9,9,ticks=>[10,10],axes=>[0,0],pixels=>[250,250]); $graphL->lb('reset'); foreach my $i (1..8) { $graphL->lb( new Label($i,-0.5,$i, 'black','center','middle')); $graphL->lb( new Label(-0.5,$i,$i, 'black','center','middle')); } $graphL->lb(new Label ( 8.5,0.25,'x','black','center','middle')); $graphL->lb(new Label ( 0.25,8.5,'y','black','center','middle')); $c = random(8,12,1); # a constant for scaling the function $f = FEQ("x**2/$c for x in <-1,9> using color:blue and weight:2"); $ftex = "\frac{x^2}{$c}"; # the parentheses around $fRefL are necessary ($fRefL) = plot_functions( $graphL, $f ); # Generate arrays of x and y values for the Riemann sum. # There are n+1 entries in each array so that we can use # only one pair of arrays for both the left and the right # endpoint Riemann sums. $a = random(2,4,1); # left endpoint of interval $b = $a+2; # right endpoint of interval $n = 4; # number of rectangles $deltax = ($b - $a)/$n; foreach $k (0..$n) { $x[$k] = $a + $k * $deltax; } foreach $k (0..$n) { $y[$k] = &{$fRefL->rule}($x[$k]); } # Graph the left endpoint Riemann sum $lightblue = $graphL->im->colorAllocate(148,201,255); $darkblue = $graphL->im->colorAllocate(100,100,255); # Create arrays of pixel references for x and y values foreach $k (0..$n) { $xpixL[$k] = $graphL->ii($x[$k]); $ypixL[$k] = $graphL->jj($y[$k]); } $xaxisL = $graphL->jj(0); # Plot the rectangles in the Riemann sum foreach $k (0..$n-1) { $graphL->im->filledRectangle($xpixL[$k],$ypixL[$k],$xpixL[$k+1],$xaxisL,$lightblue); $graphL->im->rectangle($xpixL[$k],$ypixL[$k],$xpixL[$k+1],$xaxisL,$darkblue); } # Construct a graph for the right endpoint Riemann sum $graphR = init_graph(-1,-1,9,9,ticks=>[10,10],axes=>[0,0],pixels=>[250,250]); $graphR->lb('reset'); foreach my $i (1..8) { $graphR->lb( new Label($i,-0.5,$i, 'black','center','middle')); $graphR->lb( new Label(-0.5,$i,$i, 'black','center','middle')); } $graphR->lb(new Label ( 8.5,0.25,'x','black','center','middle')); $graphR->lb(new Label ( 0.25,8.5,'y','black','center','middle')); # the parentheses around $fRefR are necessary ($fRefR) = plot_functions( $graphR, $f ); # Graph the right endpoint Riemann sum $lightblue = $graphR->im->colorAllocate(148,201,255); $darkblue = $graphR->im->colorAllocate(100,100,255); # Create arrays of pixel references for x and y values foreach $k (0..$n) { $xpixR[$k] = $graphR->ii($x[$k]); $ypixR[$k] = $graphR->jj($y[$k]); } $xaxisR = $graphR->jj(0); # Plot the rectangles in the Riemann sum foreach $k (1..$n) { $graphR->im->filledRectangle($xpixR[$k-1],$ypixR[$k],$xpixR[$k],$xaxisR,$lightblue); $graphR->im->rectangle($xpixR[$k-1],$ypixR[$k],$xpixR[$k],$xaxisR,$darkblue); }	Setup:
Context()->texStrings; BEGIN_TEXT \{ ColumnTable( "Suppose \( \displaystyle f(x) = $ftex \).". $BR. $BR. "(a) The rectangles in the graph on the left illustrate a left endpoint Riemann sum for \( f(x) \) on the interval \( $a \leq x \leq $b \). The value of this left endpoint Riemann sum is ". NAMED_ANS_RULE('optional1',30). ", and it is an ". NAMED_POP_UP_LIST('optional2',['?','overestimate of', 'equal to','underestimate of','there is ambiguity']). " the area of the region enclosed by \(\displaystyle y = f(x) \), the x-axis, and the vertical lines \(x = $a\) and \(x = $b\).". $BR. $BR. "(b) The rectangles in the graph on the right illustrate a right endpoint Riemann sum for \( f(x) \) on the interval \( $a \leq x \leq $b \). The value of this right endpoint Riemann sum is ". NAMED_ANS_RULE('optional3',30). ", and it is an ". NAMED_POP_UP_LIST('optional4',['?','overestimate of', 'equal to','underestimate of','there is ambiguity']). " the area of the region enclosed by \(\displaystyle y = f(x) \), the x-axis, and the vertical lines \(x = $a\) and \(x = $b\)." , $BCENTER. BeginTable(1). AlignedRow( [image( insertGraph($graphL), height=>250, width=>250, tex_size=>450 ), image( insertGraph($graphR), height=>250, width=>250, tex_size=>450 )] ). TableSpace(5,0). AlignedRow( ["Left endpoint Riemann sum", "Right endpoint Riemann sum"] ). EndTable(). $ECENTER , indent => 0, separation => 30, valign => "TOP" ); \} END_TEXT Context()->normalStrings;	Main Text:
$LeftRiemannSum = 0; foreach $k (0..$n-1) { $LeftRiemannSum = $LeftRiemannSum + $y[$k]; } $LeftRiemannSum = Real("$deltax * $LeftRiemannSum"); NAMED_WEIGHTED_ANS('optional1',$LeftRiemannSum->cmp(),45); NAMED_WEIGHTED_ANS('optional2',str_cmp("underestimate of"),5); $RightRiemannSum = 0; foreach $k (1..$n) { $RightRiemannSum = $RightRiemannSum + $y[$k]; } $RightRiemannSum = Real("$deltax * $RightRiemannSum"); NAMED_WEIGHTED_ANS('optional3',$RightRiemannSum->cmp(),45); NAMED_WEIGHTED_ANS('optional4',str_cmp("overestimate of"),5);	Answer Evaluation:
Context()->texStrings; BEGIN_SOLUTION (A) The left endpoint Riemann sum is \( f($x[0]) \cdot 0.5 + f($x[1]) \cdot 0.5 + \cdots + f($x[$n-1]) \cdot 0.5 = ( $y[0] + $y[1] + \cdots + $y[7] ) \cdot 0.5 = $LeftRiemannSum.\) $BR $BR (B) The right endpoint Riemann sum is \( f($x[1]) \cdot 0.5 + f($x[2]) \cdot 0.5 + \cdots + f($x[$n]) \cdot 0.5 = ( $y[1] + $y[2] + \cdots + $y[$n] ) \cdot 0.5 = $RightRiemannSum.\) END_SOLUTION Context()->normalStrings; COMMENT('MathObject version'); ENDDOCUMENT();	Solution:

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