RiemannSums1
Jump to navigation
Jump to search
This problem has been replaced with a newer version of this problem
Dynamically Generated Graphs with Riemann Sums
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
| PG problem file | Explanation |
|---|---|
|
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: |
