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| 1 : | jjholt | 197 | ## DESCRIPTION |
| 2 : | ## Calculus | ||
| 3 : | ## ENDDESCRIPTION | ||
| 4 : | jj | 144 | |
| 5 : | jjholt | 197 | ## KEYWORDS('integral' 'summation' 'limit') |
| 6 : | ## Tagged by tda2d | ||
| 7 : | |||
| 8 : | ## DBsubject('Calculus') | ||
| 9 : | ## DBchapter('Techniques of Integration') | ||
| 10 : | jjholt | 483 | ## DBsection('Approximating Integrals') |
| 11 : | jjholt | 197 | ## Date('') |
| 12 : | ## Author('') | ||
| 13 : | ## Institution('Rochester') | ||
| 14 : | ## TitleText1('') | ||
| 15 : | ## EditionText1('') | ||
| 16 : | ## AuthorText1('') | ||
| 17 : | ## Section1('') | ||
| 18 : | ## Problem1('') | ||
| 19 : | |||
| 20 : | jj | 144 | DOCUMENT(); # This should be the first executable line in the problem. |
| 21 : | |||
| 22 : | loadMacros( | ||
| 23 : | "PG.pl", | ||
| 24 : | "PGbasicmacros.pl", | ||
| 25 : | "PGchoicemacros.pl", | ||
| 26 : | "PGanswermacros.pl", | ||
| 27 : | "PGauxiliaryFunctions.pl" | ||
| 28 : | ); | ||
| 29 : | |||
| 30 : | TEXT(beginproblem()); | ||
| 31 : | $showPartialCorrectAnswers = 1; | ||
| 32 : | |||
| 33 : | $a = random(1,4,2); | ||
| 34 : | $a2 = $a*2; | ||
| 35 : | $b = random(2,9,1)*random(-1,1,2); | ||
| 36 : | $c = random(3,5,1); | ||
| 37 : | |||
| 38 : | BEGIN_TEXT | ||
| 39 : | |||
| 40 : | The interval \([0,$c]\) is partitioned into \(n\) equal subintervals, and a number \(x_i\) is arbitrarily chosen in the \(i^{th}\) subinterval for each \(i\). Then | ||
| 41 : | $BR | ||
| 42 : | \( \displaystyle \lim_{n\to \infty} \sum_{i=1}^{n}\frac{$a2 x_i + $b}{n} = \) \{ans_rule(30)\} | ||
| 43 : | END_TEXT | ||
| 44 : | |||
| 45 : | $answer = $a*$c + $b; | ||
| 46 : | |||
| 47 : | ANS(num_cmp($answer)); | ||
| 48 : | |||
| 49 : | SOLUTION(EV3(<<'EOF')); | ||
| 50 : | $SOL $BR | ||
| 51 : | Let's interpret the sum as a Riemann sum. | ||
| 52 : | $BR Recall that the Riemann sum | ||
| 53 : | for a function \(f(x)\) on the interval \([0,$c]\) has the form | ||
| 54 : | \( \displaystyle \sum_{i=1}^n f(x_i) \frac{$c}{n} \) since the length of each subinterval is \( \displaystyle \Delta x = \frac{$c}{n} \). | ||
| 55 : | $BR $BR | ||
| 56 : | \( \displaystyle \sum_{i=1}^{n}\frac{$a2 x_i + $b}{n} = | ||
| 57 : | \sum_{i=1}^{n}\frac{$a2 x_i + $b}{$c} \cdot \frac{$c}{n} \), therefore the given sum is the Riemann sum for \(\displaystyle f(x) = \frac{$a2 x + $b}{$c}\). | ||
| 58 : | $BR $BR The limit of the Riemann sum as \(n\) approaches infinity is the integral of the function \(f(x)\) from \(0\) to \($c\), thus | ||
| 59 : | $BR $BR | ||
| 60 : | \( \displaystyle \lim_{n\to \infty} \sum_{i=1}^{n}\frac{$a2 x + $b}{$c} \cdot \frac{$c}{n} = \int_0^{$c} \frac{$a2 x + $b}{$c} dx = \frac{1}{$c} \int_0^{$c} ($a2 x + $b) dx = | ||
| 61 : | \left. \frac{1}{$c} \left( $a x^2 + $b x \right) \right|_0^{$c} = | ||
| 62 : | \frac{1}{$c} \left( $a \cdot $c^2 + $b\cdot $c\right) = $answer \) | ||
| 63 : | |||
| 64 : | EOF | ||
| 65 : | |||
| 66 : | ENDDOCUMENT(); # This should be the last executable line in the problem. | ||
| 67 : |
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