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Tue Nov 8 15:17:41 2011 UTC (2 years, 1 month ago) by aubreyja
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Rogawski problems contributed by publisher WHFreeman. These are a subset of the problems available to instructors who use the Rogawski textbook. The remainder can be obtained from the publisher.

    1 ## 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('12')
10 ## Author('Keith Thompson')
11 ## Institution('W.H.Freeman')
12
13 DOCUMENT();
19
20 #$showPartialCorrectAnswers=1; 21 22 Context()->variables->add(N=>'Real'); 23 24$start=random(4,9,1);
25 $smm=$start-2;#added for solution, LAD
26 $sm=$start-1;
27 $sp=$start+1;
28
29 $S_N = Formula("1/($sm)-1/(N+$sm)"); 30$ans2=1/($start-1); 31 32 Context()->texStrings; 33 BEGIN_TEXT 34 \{ beginproblem() \} 35 \{ textbook_ref_exact("Rogawski ET 2e", "10.2","12") \} 36$PAR
37
38 Write $$S=\sum\limits_{n=start}^\infty \frac{1}{n(n-1)}$$ as a telescoping series and find its sum.
39
40 $PAR $$S_N$$ = \{ans_rule()\} 41$PAR $$S$$ =  \{ans_rule()\}
42 END_TEXT
43
44 Context()->normalStrings;
45
46 ANS($S_N->cmp); 47 ANS(Real($ans2)->cmp);
48 Context()->texStrings;
49 SOLUTION(EV3(<<'END_SOLUTION'));
50 $PAR 51$SOL
52 By partial fraction decomposition
53 $\frac{1}{n(n-1)}=\frac{1}{n-1}-\frac{1}{n},$
54 so
55 $S=\sum_{n=start}^\infty \frac{1}{n(n-1)}=\sum_{n=start}^\infty \left(\frac{1}{n-1}-\frac{1}{n}\right).$
56
57 The general term in the sequence of partial sums for this series is
58 $S_N=\left(\frac{1}{sm}-\frac{1}{start}\right) + \left(\frac{1}{start}-\frac{1}{sp}\right) + \cdots + \left(\frac{1}{N+smm}-\frac{1}{N+sm}\right)=\frac{1}{sm}-\frac{1}{N+sm};$
59
60 thus,
61
62 $S=\lim_{N\rightarrow \infty} S_N = \lim_{N\rightarrow \infty}\left(\frac{1}{sm}-\frac{1}{N+sm}\right)=\frac{1}{sm}.$
63
64 END_SOLUTION
65
66 ENDDOCUMENT();