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| 1 : | gage | 704 | #DESCRIPTION |
| 2 : | #KEYWORDS('Integrals', 'Area and Distance') | ||
| 3 : | # Interpreting Riemann sums in terms of area | ||
| 4 : | #ENDDESCRIPTION | ||
| 5 : | |||
| 6 : | ## DBsubject('Calculus') | ||
| 7 : | ## DBchapter('Integrals') | ||
| 8 : | ## DBsection('Area and Distance') | ||
| 9 : | ## Date('5/10/2008') | ||
| 10 : | ## Author('Paul Pearson') | ||
| 11 : | ## Institution('University of Rochester') | ||
| 12 : | ## TitleText1('Calculus: Early Transcendentals') | ||
| 13 : | ## EditionText1('6') | ||
| 14 : | ## AuthorText1('Stewart') | ||
| 15 : | ## Section1('5.1') | ||
| 16 : | ## Problem1('2,3,4') | ||
| 17 : | |||
| 18 : | DOCUMENT(); | ||
| 19 : | loadMacros("PGbasicmacros.pl", | ||
| 20 : | "PGchoicemacros.pl", | ||
| 21 : | "PGanswermacros.pl", | ||
| 22 : | "PGgraphmacros.pl", | ||
| 23 : | "PGnumericalmacros.pl", | ||
| 24 : | "extraAnswerEvaluators.pl" | ||
| 25 : | ); | ||
| 26 : | |||
| 27 : | TEXT(beginproblem()); | ||
| 28 : | |||
| 29 : | |||
| 30 : | # Construct a graph for the left endpoint Riemann sum, | ||
| 31 : | # define the function to be graphed, and add it to the graph | ||
| 32 : | $graphL = init_graph(-1,-1,9,9,ticks=>[10,10],axes=>[0,0],pixels=>[400,400]); | ||
| 33 : | $c = random(12,17,1); # a constant for scaling the function | ||
| 34 : | $f = FEQ("$c / x for x in <-1,9> using color:blue and weight:2"); | ||
| 35 : | $ftex = "\frac{$c}{x}"; | ||
| 36 : | # the parentheses around $fRefL are necessary | ||
| 37 : | ($fRefL) = plot_functions( $graphL, $f ); | ||
| 38 : | |||
| 39 : | |||
| 40 : | # Generate arrays of x and y values for the Riemann sum. | ||
| 41 : | # There are n+1 entries in each array so that we can use | ||
| 42 : | # only one pair of arrays for both the left and the right | ||
| 43 : | # endpoint Riemann sums. | ||
| 44 : | $a = random(2,4,1); # left endpoint of interval | ||
| 45 : | $b = $a+4; # right endpoint of interval | ||
| 46 : | $n = 8; # number of rectangles | ||
| 47 : | $deltax = ($b - $a)/$n; | ||
| 48 : | foreach $k (0..$n) { @x[$k] = $a + $k * $deltax; } | ||
| 49 : | foreach $k (0..$n) { @y[$k] = &{$fRefL->rule}(@x[$k]); } | ||
| 50 : | |||
| 51 : | |||
| 52 : | # Graph the left endpoint Riemann sum | ||
| 53 : | $lightblue = $graphL->im->colorAllocate(148,201,255); | ||
| 54 : | $darkblue = $graphL->im->colorAllocate(100,100,255); | ||
| 55 : | # Create arrays of pixel references for x and y values | ||
| 56 : | foreach $k (0..8) { | ||
| 57 : | @xpixL[$k] = $graphL->ii(@x[$k]); | ||
| 58 : | @ypixL[$k] = $graphL->jj(@y[$k]); | ||
| 59 : | } | ||
| 60 : | $xaxisL = $graphL->jj(0); | ||
| 61 : | # Plot the rectangles in the Riemann sum | ||
| 62 : | foreach $k (0..$n-1) { | ||
| 63 : | $graphL->im->filledRectangle(@xpixL[$k],@ypixL[$k],@xpixL[$k+1],$xaxisL,$lightblue); | ||
| 64 : | $graphL->im->rectangle(@xpixL[$k],@ypixL[$k],@xpixL[$k+1],$xaxisL,$darkblue); | ||
| 65 : | } | ||
| 66 : | $graphL->lb(new Label ( 8.5,0,'x','black','right','top')); | ||
| 67 : | $graphL->lb(new Label ( -0.25,8.5,'y','black','right','top')); | ||
| 68 : | |||
| 69 : | |||
| 70 : | # Construct a graph for the right endpoint Riemann sum | ||
| 71 : | $graphR = init_graph(-1,-1,9,9,ticks=>[10,10],axes=>[0,0],pixels=>[400,400]); | ||
| 72 : | # the parentheses around $fRefR are necessary | ||
| 73 : | ($fRefR) = plot_functions( $graphR, $f ); | ||
| 74 : | |||
| 75 : | |||
| 76 : | # Graph the right endpoint Riemann sum | ||
| 77 : | $lightblue = $graphR->im->colorAllocate(148,201,255); | ||
| 78 : | $darkblue = $graphR->im->colorAllocate(100,100,255); | ||
| 79 : | # Create arrays of pixel references for x and y values | ||
| 80 : | foreach $k (0..8) { | ||
| 81 : | @xpixR[$k] = $graphR->ii(@x[$k]); | ||
| 82 : | @ypixR[$k] = $graphR->jj(@y[$k]); | ||
| 83 : | } | ||
| 84 : | $xaxisR = $graphR->jj(0); | ||
| 85 : | # Plot the rectangles in the Riemann sum | ||
| 86 : | foreach $k (1..$n) { | ||
| 87 : | $graphR->im->filledRectangle(@xpixR[$k-1],@ypixR[$k],@xpixR[$k],$xaxisR,$lightblue); | ||
| 88 : | $graphR->im->rectangle(@xpixR[$k-1],@ypixR[$k],@xpixR[$k],$xaxisR,$darkblue); | ||
| 89 : | } | ||
| 90 : | $graphR->lb(new Label ( 8.5,0,'x','black','right','top')); | ||
| 91 : | $graphR->lb(new Label ( -0.25,8.5,'y','black','right','top')); | ||
| 92 : | |||
| 93 : | |||
| 94 : | |||
| 95 : | BEGIN_TEXT | ||
| 96 : | |||
| 97 : | The rectangles in the graph below illustrate a left endpoint Riemann sum for \( \displaystyle f(x) = $ftex \) on the interval \( \lbrack $a, $b \rbrack \). | ||
| 98 : | $BR | ||
| 99 : | The value of this left endpoint Riemann sum is \{ans_rule(30)\} | ||
| 100 : | $BR | ||
| 101 : | $BR | ||
| 102 : | |||
| 103 : | $BCENTER | ||
| 104 : | \{ begintable(1) \} | ||
| 105 : | \{ row( image( insertGraph($graphL), height=>400, width=>400, tex_size=>800 ) ) \} | ||
| 106 : | \{ row("Left endpoint Riemann sum for \( y = $ftex \) on \( \lbrack $a, $b \rbrack \)") \} | ||
| 107 : | \{ endtable() \} | ||
| 108 : | $ECENTER | ||
| 109 : | |||
| 110 : | $BR | ||
| 111 : | $HR | ||
| 112 : | $BR | ||
| 113 : | |||
| 114 : | The rectangles in the graph below illustrate a right endpoint Riemann sum for \( \displaystyle f(x) = $ftex \) on the interval \( \lbrack $a, $b \rbrack \). | ||
| 115 : | $BR | ||
| 116 : | The value of this right endpoint Riemann sum is \{ans_rule(30)\} | ||
| 117 : | $BR | ||
| 118 : | $BR | ||
| 119 : | |||
| 120 : | $BCENTER | ||
| 121 : | \{ begintable(1) \} | ||
| 122 : | \{ row( image( insertGraph($graphR), height=>400, width=>400, tex_size=>800 ) ) \} | ||
| 123 : | \{ row("Right endpoint Riemann sum for \( y = $ftex \) on \( \lbrack $a, $b \rbrack \)") \} | ||
| 124 : | \{ endtable() \} | ||
| 125 : | $ECENTER | ||
| 126 : | |||
| 127 : | END_TEXT | ||
| 128 : | |||
| 129 : | $LeftRiemannSum = 0; | ||
| 130 : | foreach $k (0..$n-1) { $LeftRiemannSum = $LeftRiemannSum + @y[$k]; } | ||
| 131 : | $LeftRiemannSum = $deltax * $LeftRiemannSum; | ||
| 132 : | ANS(num_cmp($LeftRiemannSum)); | ||
| 133 : | |||
| 134 : | $RightRiemannSum = 0; | ||
| 135 : | foreach $k (1..$n) { $RightRiemannSum = $RightRiemannSum + @y[$k]; } | ||
| 136 : | $RightRiemannSum = $deltax * $RightRiemannSum; | ||
| 137 : | ANS(num_cmp($RightRiemannSum)); | ||
| 138 : | |||
| 139 : | |||
| 140 : | SOLUTION(EV3(<<'END_SOLUTION')); | ||
| 141 : | $BR | ||
| 142 : | $HR | ||
| 143 : | ${BBOLD}Solution:${EBOLD} | ||
| 144 : | $BR | ||
| 145 : | $BR | ||
| 146 : | |||
| 147 : | (A) The left endpoint Riemann sum is | ||
| 148 : | \( f(@x[0]) \cdot 0.5 + f(@x[1]) \cdot 0.5 + \cdots + f(@x[$n-1]) \cdot 0.5 | ||
| 149 : | = ( @y[0] + @y[1] + \cdots + @y[7] ) \cdot 0.5 = $LeftRiemannSum.\) | ||
| 150 : | $BR | ||
| 151 : | $BR | ||
| 152 : | |||
| 153 : | (B) The right endpoint Riemann sum is | ||
| 154 : | \( f(@x[1]) \cdot 0.5 + f(@x[2]) \cdot 0.5 + \cdots + f(@x[$n]) \cdot 0.5 | ||
| 155 : | = ( @y[1] + @y[2] + \cdots + @y[$n] ) \cdot 0.5 = $RightRiemannSum.\) | ||
| 156 : | |||
| 157 : | END_SOLUTION | ||
| 158 : | |||
| 159 : | |||
| 160 : | ENDDOCUMENT(); |
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