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Tue Nov 8 15:17:41 2011 UTC (2 years, 5 months ago) by aubreyja
File size: 2467 byte(s)
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('')
3 # DBsection('')
4 # KEYWORDS('')
5 # TitleText1('Calculus: Early Transcendentals')
6 # EditionText1('2')
7 # AuthorText1('Rogawski')
8 # Section1('10.7')
9 # Problem1('14')
10 # Author('Emily Price')
11 # Institution('W.H.Freeman')
12 DOCUMENT();
13
14
15
17
21
22
24
25 #Book Values
26 #$power1 = 2 27 #$power2 = 1
28
29 $power1 = random(2, 9); 30$power2 = random(2, 4);
31
32 Context()->variables->set(x=>{limits=>[.1,10]});
33 Context()->variables->set(n=>{limits=>[1,5]});
34
35 $finalpower = Formula("2*$power1*n - $power2")->reduce; 36$interimpower = Formula("2*$power1")->reduce; 37$mcterms = Formula("(-1)^(n+1)*x^($finalpower)/(2*n)!")->reduce; 38$denominator = Formula("x^($power2)")->reduce; 39$interval = Interval("(-inf,inf)"); #added LAD
40
41 $point1=.5; 42$point2=1.5;
43 $point3=2.5; 44$mcterms->{test_points} = [[1,$point1],[2,$point2],[3,$point3]]; 45 46 Context()->texStrings; 47 48 BEGIN_TEXT 49 \{ beginproblem() \} 50 \{ textbook_ref_exact("Rogawski ET 2e", "10.7", "14") \} 51$PAR
52 Find the Maclaurin series and corresponding interval of convergence of the following function.
53 $PAR 54 $f(x) = \frac{1 - \cos (x^{power1})}{denominator}$ 55$PAR
56 $$f(x) = \sum\limits_{n=1}^{\infty}$$ \{ ans_rule() \}
57 $PAR 58 The interval of convergence for this power series is: \{ans_rule() \} 59 END_TEXT 60 Context()->normalStrings; 61 62 #Answer Check Time! 63 ANS($mcterms->cmp);
64 ANS($interval->cmp); 65 66 Context()->texStrings; 67 SOLUTION(EV3(<<'END_SOLUTION')); 68$PAR
69 \$SOL
70
71 Substituting $$x^{power1}$$ for $$x$$ in the Maclaurin series for $$\cos x$$ gives
72 $\cos (x^{power1}) = \sum_{n=0}^{\infty} (-1)^n \frac{(x^{power1})^{2n}}{(2n)!} = 1 + \sum_{n=1}^{\infty} (-1)^n \frac{x^{interimpower n}}{(2n)!}$
73 Thus,
74 $1 - \cos (x^{power1}) = 1 - \left( 1 + \sum_{n=1}^{\infty} (-1)^n \frac{x^{interimpower n}}{(2n)!} \right) = \sum_{n=1}^{\infty} (-1)^{n+1} \frac{x^{interimpower n}}{(2n)!}$
75 and
76 $\frac{1 - \cos (x^{power1})}{denominator} = \frac{1}{denominator} \sum_{n=1}^{\infty} (-1)^{n+1} \frac{x^{interimpower n}}{(2n)!} = \sum_{n=1}^{\infty} (-1)^{n+1} \frac{x^{finalpower}}{(2n)!} .$
77 This series is valid for all $$x\ne0$$.  Thus, the interval of convergence is $$(-\infty,0)\cup (0,\infty)$$.
78
79
80
81 END_SOLUTION
82
83 ENDDOCUMENT()