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Tue Nov 8 15:17:41 2011 UTC (2 years, 5 months ago) by aubreyja
File size: 2632 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('Infinite Sequences and Series')
3 ## DBsection('Infinite Sequences and Series')
4 ## KEYWORDS('calculus', 'series')
5 ## TitleText1('Calculus: Early Transcendentals')
6 ## EditionText1('2')
7 ## AuthorText1('Rogawski')
8 ## Section1('10.1')
9 ## Problem1('31')
10 ## Author('Danny Glin')
11 ## Institution('W.H.Freeman')
12
13 DOCUMENT();
17
18 Context()->texStrings;
19 Context()->flags->set(reduceConstants=>0);
20 Context()->variables->are(n=>'Real',t=>'Real',M=>'Real');
21
22 #$a=0 23 #$b=1
24
25 $a = random(0,3,1); 26$b = $a + random(1,3,1); 27 28$an=Formula("(n+$a)/(n+$b)")->reduce;
29
30 $ans1=Real(999*$b-1000*$a); 31$ans2=Real(99999*$b-100000*$a);
32
33 $an1=Formula("(n+$a-(n+$b))/(n+$b)");
34 $ab=$a-$b; 35$an2=Formula("$ab/(n+$b)");
36 $ab2=-$ab;
37 $an3=Formula("$ab2/(n+$b)"); 38$an4=$an3->substitute(n=>M); 39 40$ans3=Formula("$ab2/t-$b");
41
42 BEGIN_TEXT
43 \{ beginproblem() \}
44 \{ textbook_ref_exact("Rogawski ET 2e", "10.1","31") \}
45 $PAR 46 Let $$a_n=an$$. Find the smallest number $$M$$ such that: 47$PAR
48 $BBOLD (a)$EBOLD $$|a_n-1|\le 0.001$$ for $$n\ge M$$$PAR 49 $$M=$$\{ans_rule(10)\} 50 END_TEXT 51 SOLUTION(EV3(<<'END_SOLUTION')); 52$PAR
53 $SOL 54 We have 55 $\left|a_n-1\right|=\left|an-1\right|=\left|an1\right|=\left|an2\right|=an3$ 56 Therefore $$|a_n-1|\le 0.001$$ provided $$an3\le 0.001$$, that is, $$n\ge ans1$$. It follows that we can take $$M=ans1$$. 57 END_SOLUTION 58 59 60 BEGIN_TEXT 61$PAR
62 $BBOLD (b)$EBOLD $$|a_n-1|\le 0.00001$$ for $$n\ge M$$$PAR 63 $$M=$$\{ans_rule(10)\} 64 END_TEXT 65 66 SOLUTION(EV3(<<'END_SOLUTION')); 67$PAR
68 $SOL 69 By part (a), $$|a_n-1|\le 0.00001$$ provided $$an3\le 0.00001$$, that is, $$n\ge ans2$$. It follows that we can take $$M=ans2$$. 70 END_SOLUTION 71 72 BEGIN_TEXT 73$PAR
74 $BBOLD (c)$EBOLD Now use the limit definition to prove that $$\displaystyle\lim_{n\to\infty}a_n=1$$.  That is, find the smallest value of $$M$$ (in terms of $$t$$) such that $$\left|a_n-1\right|<t$$ for all $$n>M$$.$BR 75 (Note that we are using $$t$$ instead of $$\epsilon$$ in the definition in order to allow you to enter your answer more easily).$BR
76 $$M=$$\{ans_rule(20)\} (Enter your answer as a function of $$t$$)
77 END_TEXT
78
79 SOLUTION(EV3(<<'END_SOLUTION'));
80 $PAR 81$SOL
82 Using part (a), we know that $|a_n-1|=an3<t,$ provided $$n>ans3$$.  Thus to complete the proof, let $$t>0$$ and take $$M=ans3$$.  Then, for $$n>M$$, we have
83 $|a_n-1|=an3<\{an4\}=t.$
84 END_SOLUTION
85
86
87 ANS($ans1->cmp); 88 ANS($ans2->cmp);
89 ANS(\$ans3->cmp);
90
91
92 Context()->texStrings;
93
94 ENDDOCUMENT();


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