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| 1 : | aubreyja | 2584 | ## DBsubject('Calculus') |
| 2 : | ## DBchapter('Limits and Derivatives') | ||
| 3 : | ## DBsection('Definition of the Derivative') | ||
| 4 : | ## KEYWORDS('calculus', 'derivatives', 'slope') | ||
| 5 : | ## TitleText1('Calculus: Early Transcendentals') | ||
| 6 : | ## EditionText1('2') | ||
| 7 : | ## AuthorText1('Rogawski') | ||
| 8 : | ## Section1('10.2') | ||
| 9 : | ## Problem1('42') | ||
| 10 : | ## Author('Keith Thompson') | ||
| 11 : | ## Institution('W.H.Freeman') | ||
| 12 : | |||
| 13 : | DOCUMENT(); | ||
| 14 : | loadMacros("PG.pl","PGbasicmacros.pl","PGanswermacros.pl"); | ||
| 15 : | loadMacros("Parser.pl"); | ||
| 16 : | loadMacros("freemanMacros.pl"); | ||
| 17 : | loadMacros("PGauxiliaryFunctions.pl"); | ||
| 18 : | loadMacros("PGgraphmacros.pl"); | ||
| 19 : | |||
| 20 : | #$showPartialCorrectAnswers=1; | ||
| 21 : | Context()->variables->add(n=>'Real'); | ||
| 22 : | $base=random(2,9,1); | ||
| 23 : | $num=random(2,9,1); | ||
| 24 : | |||
| 25 : | $ans1=$base-$num/100; | ||
| 26 : | $ans2=$num*(1/9-1/256); | ||
| 27 : | $ans3=$num/4-$num/9; | ||
| 28 : | $ans4=Formula("$num*(2*n-1)/((n*(n-1))**2)"); | ||
| 29 : | $ans5=$base; | ||
| 30 : | Context()->texStrings; | ||
| 31 : | BEGIN_TEXT | ||
| 32 : | \{ beginproblem() \} | ||
| 33 : | \{ textbook_ref_exact("Rogawski ET 2e", "10.2","42") \} | ||
| 34 : | $PAR | ||
| 35 : | Let \(S=\sum\limits_{n=1}^\infty a_n\) be an infinite series such that \(S_N=$base-\frac{$num}{N^2}\). | ||
| 36 : | $PAR | ||
| 37 : | $BBOLD (a) $EBOLD What are the values of \(\sum\limits_{n=1}^{10} a_n\) and \(\sum\limits_{n=4}^{16} a_n\)? | ||
| 38 : | $BR \(\sum\limits_{n=1}^{10} a_n=\) \{ans_rule()\} | ||
| 39 : | $BR \(\sum\limits_{n=4}^{16} a_n=\) \{ans_rule()\} | ||
| 40 : | $PAR | ||
| 41 : | $BBOLD (b) $EBOLD What is the value of \(a_3\)? | ||
| 42 : | $BR \(a_3=\) \{ans_rule()\} | ||
| 43 : | $PAR | ||
| 44 : | $BBOLD (c) $EBOLD Find a general formula for \(a_n\). | ||
| 45 : | $BR \(a_n=\) \{ans_rule()\} | ||
| 46 : | $PAR | ||
| 47 : | $BBOLD (d) $EBOLD Find the sum \(\sum\limits_{n=1}^\infty a_n\). | ||
| 48 : | $BR \(\sum\limits_{n=1}^\infty a_n=\) \{ans_rule()\} | ||
| 49 : | END_TEXT | ||
| 50 : | |||
| 51 : | Context()->normalStrings; | ||
| 52 : | |||
| 53 : | ANS(Real($ans1)->cmp); | ||
| 54 : | ANS(Real($ans2)->cmp); | ||
| 55 : | ANS(Real($ans3)->cmp); | ||
| 56 : | ANS($ans4->cmp); | ||
| 57 : | ANS(Real($ans5)->cmp); | ||
| 58 : | Context()->texStrings; | ||
| 59 : | SOLUTION(EV3(<<'END_SOLUTION')); | ||
| 60 : | $PAR | ||
| 61 : | $SOL | ||
| 62 : | $BBOLD (a) $EBOLD | ||
| 63 : | \[\sum_{n=1}^{10} a_n=S_{10}=$base-\frac{$num}{10^2}=$ans1;\] | ||
| 64 : | \[\sum_{n=4}^{16} a_n=(a_1+\ldots+a_{16})-(a_1+a_2+a_3)=S_{16}-S_3\] | ||
| 65 : | \[=\left($base-\frac{$num}{16^2}\right)-\left($base-\frac{$num}{3^2}\right)=\frac{$num}{9}-\frac{$num}{256}\approx $ans2.\] | ||
| 66 : | |||
| 67 : | $BBOLD (b) $EBOLD | ||
| 68 : | \[a_3=(a_1+a_2+a_3)-(a_1+a_2)=S_3-S_2=\] | ||
| 69 : | \[\left($base-\frac{$num}{3^2}\right)-\left($base-\frac{$num}{2^2}\right)=\frac{$num}{4}-\frac{$num}{9}\approx $ans3.\] | ||
| 70 : | |||
| 71 : | $BBOLD (c) $EBOLD Since \(a_n=S_n-S_{n-1}\), we have: | ||
| 72 : | \[a_n=S_n-S_{n-1}=\left($base-\frac{$num}{n^2}\right)-\left($base-\frac{$num}{(n-1)^2}\right)=\frac{$num}{(n-1)^2}-\frac{$num}{n^2}\] | ||
| 73 : | \[=\frac{$num(n^2-(n-1)^2)}{(n(n-1))^2}=\frac{$num(n^2-n^2+2n-1)}{(n(n-1))^2}=\frac{$num(2n-1)}{(n(n-1))^2}.\] | ||
| 74 : | |||
| 75 : | $BBOLD (d) $EBOLD The sum \(\sum\limits_{n=1}^\infty a_n\) is the limit of the partial sums \(\{S_N\}\). Hence; | ||
| 76 : | \[\sum_{n=1}^\infty a_n = \lim_{N\rightarrow \infty} S_N = \lim_{N\rightarrow \infty} \left($base - \frac{$num}{N^2}\right) = $base.\] | ||
| 77 : | |||
| 78 : | END_SOLUTION | ||
| 79 : | ENDDOCUMENT(); |
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