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Tue Nov 8 15:17:41 2011 UTC (18 months, 1 week ago) by aubreyja
File size: 1665 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('18')
10 ## Author('Christopher Sira')
11 ## Institution('W.H.Freeman')
12
13 DOCUMENT();
19 $context = Context(); 20 21$t = random(1, 15);
22 $den = ($t**2 + 1)**2;
23 $con =$t**2;
24
25 $exp =$t * (4/$den + 9*$con)**.5;
26
27 $ans = Formula("$exp");
28
29
30 Context()->texStrings;
31 BEGIN_TEXT
32 \{ beginproblem() \}
33 \{ textbook_ref_exact("Rogawski ET 2e", "11.2","18") \}
34 $PAR 35 Determine the speed $$s(t)$$ of a particle with a given trajectory at a time $$t_0$$ (in units of meters and seconds). 36 $c(t) = (\ln(t^2 + 1), t^3), \, t = t$ 37$PAR
38 \{ ans_rule() \}
39 $PAR 40 END_TEXT 41 Context()->normalStrings; 42 43 ANS($ans->cmp);
44
45 Context()->texStrings;
46 SOLUTION(EV3(<<'END_SOLUTION'));
47 $PAR 48$SOL
49 We have $$x = \ln(t^2 + 1), \, y = t^3$$, so $$x' = \frac{2t}{t^2 + 1}$$ and $$y' = 3t^2$$.  The speed of the particle at time t is thus
50 $\frac{ds}{dt} = \sqrt{x'(t)^2 + y'(t)^2} = \sqrt{ \frac{4t^2}{ \left( t^2 + 1 \right) ^2 } + 9t^4 } = t \sqrt{\frac{4}{ \left( t^2 + 1 \right) ^2 } + 9t^2}$
51 The speed at time $$t = t$$ is
52 $\frac{ds}{dt} \mid _{t = t} = t \sqrt{ \frac{4}{den} + 9 \cdot con } = ans$
53 END_SOLUTION
54
55 ENDDOCUMENT();