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Tue Nov 8 15:17:41 2011 UTC (2 years, 1 month ago) by aubreyja
File size: 1584 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('Parametric Equations, Polar Coordinates, and Conic Sections')
3 ## DBsection('Arc Length and Speed')
4 ## KEYWORDS('calculus', 'parametric', 'polar', 'conic')
5 ## TitleText1('Calculus: Early Transcendentals')
6 ## EditionText1('2')
7 ## AuthorText1('Rogawski')
8 ## Section1('11.2')
9 ## Problem1('3')
10 ## Author('Christopher Sira')
11 ## Institution('W.H.Freeman')
12
13 DOCUMENT();
19 $context = Context(); 20 21 22$a = random(2, 9);
23 $b = random(2, 9); 24$a2 = 2 * $a; 25$b2 = 2 * $b; 26$a2sq = $a2 ** 2; 27$b2sq = $b2 ** 2; 28$coeff = Formula("$a2sq +$b2sq");
29
30 #$ans = Formula("$coeff**.5 * 8");
31 $ans = Compute("8*sqrt($a2 ** 2 + $b2 ** 2)"); 32 33 Context()->texStrings; 34 BEGIN_TEXT 35 \{ beginproblem() \} 36 \{ textbook_ref_exact("Rogawski ET 2e", "11.2","3") \} 37 Use equation 4 to calculate the length of the path over the given interval. 38 $(a t^2, b t^2 - 1), \, 0 \le t \le 4$ 39$PAR
40 \{ ans_rule() \}
41 $PAR 42 END_TEXT 43 Context()->normalStrings; 44 45 ANS($ans->cmp);
46
47 Context()->texStrings;
48 SOLUTION(EV3(<<'END_SOLUTION'));
49 $PAR 50$SOL
51 Since $$x = a t^2$$ and $$y = b t^2 - 1$$ we have $$x' = a2 t$$ and $$y' = b2 t$$.  By the formula for the arc length we get
52 $S = \int ^4 _0 \sqrt{x'(t)^2 + y'(t)^2} \, dt = \int ^4 _0 \sqrt{a2sq t^2 + b2sq t^2} \, dt = \sqrt{coeff} \int ^4 _0 t \, dt = 8\sqrt{coeff} = ans$
53 END_SOLUTION
54
55 ENDDOCUMENT();


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