[npl] / trunk / NationalProblemLibrary / WHFreeman / Rogawski_Calculus_Early_Transcendentals_Second_Edition / 10_Infinite_Series / 10.7_Taylor_Series / 10.7.29.pg Repository: Repository Listing bbplugincoursesdistsnplrochestersystemwww

# View of /trunk/NationalProblemLibrary/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/10_Infinite_Series/10.7_Taylor_Series/10.7.29.pg

Tue Nov 8 15:17:41 2011 UTC (2 years, 4 months ago) by aubreyja
File size: 2019 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('29')
10 # Author('Nick Hamblet')
11 # Institution('W.H.Freeman')
12
13 DOCUMENT();
17
19
20 $c = random(2,7,1); 21 22$f = Compute("1/x");
23
24 $series = Formula("(-1)^n/($c)^(n+1) * (x-$c)^n"); 25$series->{test_points} = [ [ 1, random(.5,9,.5) ],
26                            [ 2, random(.5,9,.5) ],
27                            [ 3, random(.5,9,.5) ],
28                            [ 4, random(.5,9,.5) ],
29                            [ 5, random(.5,9,.5) ],
30                            [ 6, random(.5,9,.5) ],
31                            [ 7, random(.5,9,.5) ],
32                            [ 8, random(.5,9,.5) ],
33                            [ 9, random(.5,9,.5) ] ];
34
35 $interval = Interval("(0,$c+$c)"); 36 37 Context()->texStrings; 38 BEGIN_TEXT 39 \{ beginproblem() \} 40 \{ textbook_ref_exact("Rogawski ET 2e", "10.7","29") \} 41$PAR
42 Find the Taylor series, centered at $$c=c$$, for the function
43 $f(x) = f.$
44 $PAR 45 $$\displaystyle f(x)=\sum_{n=0}^{\infty}$$ \{ans_rule()\}.$BR
46 The interval of convergence is: \{ans_rule()\}.
47 END_TEXT
48 Context()->normalStrings;
49
50 ANS($series->cmp,$interval->cmp);
51
52 Context()->texStrings;
53 SOLUTION(EV3(<<'END_SOLUTION'));
54 $PAR 55$SOL
56 \$PAR
57 Write
58 $f = \frac{1}{c+(x-c)}=\frac{1}{c}\cdot \frac{1}{1+\left(\frac{x-c}{c}\right)}$
59 and then substitute $$-\left(\frac{x-c}{c}\right)$$ for $$x$$ in the Maclaurin series for $$\frac{1}{1-x}$$ to obtain
60 $f = \frac{1}{c}\sum_{n=0}^{\infty} \left[-\left(\frac{x-c}{c}\right)\right]^n = \sum_{n=0}^{\infty} \frac{(-1)^n}{c^{n+1}}(x-c)^n.$
61 This series is valid for $$\left|-\left(\frac{x-c}{c}\right)\right|<1$$, or $$|x-c|<c$$.
62
63 END_SOLUTION
64
65 ENDDOCUMENT();