| … | |
… | |
| 23 | loadMacros( |
23 | loadMacros( |
| 24 | "PG.pl", |
24 | "PG.pl", |
| 25 | "PGbasicmacros.pl", |
25 | "PGbasicmacros.pl", |
| 26 | "PGchoicemacros.pl", |
26 | "PGchoicemacros.pl", |
| 27 | "PGanswermacros.pl", |
27 | "PGanswermacros.pl", |
| 28 | "PGauxiliaryFunctions.pl" |
28 | "PGauxiliaryFunctions.pl", |
|
|
29 | "PGgraphmacros.pl" |
| 29 | ); |
30 | ); |
| 30 | |
31 | |
| 31 | $showPartialCorrectAnswers = 1; |
32 | $showPartialCorrectAnswers = 1; |
| 32 | |
33 | |
| 33 | $pi = 4*arctan(1); |
34 | $pi = 4*arctan(1); |
| … | |
… | |
| 45 | \(x=0\) to \(x=$a\pi\). |
46 | \(x=0\) to \(x=$a\pi\). |
| 46 | $BR |
47 | $BR |
| 47 | Hint: Notice that this region consists of two parts. |
48 | Hint: Notice that this region consists of two parts. |
| 48 | $BR |
49 | $BR |
| 49 | \{ans_rule(45)\} |
50 | \{ans_rule(45)\} |
| 50 | $PAR |
|
|
| 51 | END_TEXT |
51 | END_TEXT |
| 52 | |
52 | |
| 53 | ANS(num_cmp($ans)); |
53 | ANS(num_cmp($ans)); |
| 54 | |
54 | |
| 55 | ##set $PG_environment{'textbook'} in webworkCourse.ph |
55 | ##set $PG_environment{'textbook'} in webworkCourse.ph |
| … | |
… | |
| 59 | This is similar to Example 1 in Section 5.8 of the text. |
59 | This is similar to Example 1 in Section 5.8 of the text. |
| 60 | END_TEXT |
60 | END_TEXT |
| 61 | } |
61 | } |
| 62 | } |
62 | } |
| 63 | |
63 | |
|
|
64 | $soln_leftx = -0.5; |
|
|
65 | $soln_rightx = $b+0.5; |
|
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66 | $soln_boty = -0.5; |
|
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67 | $soln_topy = ($g > $f*sin($b)) ? $g+0.5 : $f*sin($b)+0.5; |
|
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68 | |
|
|
69 | $graph = init_graph($soln_leftx,$soln_boty,$soln_rightx,$soln_topy, |
|
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70 | 'axes'=>[0,0], |
|
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71 | 'grid'=>[$soln_rightx-$soln_leftx,$soln_topy-$soln_boty], |
|
|
72 | 'size'=>[400,400]); |
|
|
73 | plot_functions($graph,FEQ( |
|
|
74 | "$g*cos(x) for x in [0,$b] using color=blue and weight=2" |
|
|
75 | )); |
|
|
76 | plot_functions($graph,FEQ( |
|
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77 | "$f*sin(x) for x in [0,$b] using color=red and weight=2" |
|
|
78 | )); |
|
|
79 | |
|
|
80 | $graphtext = image(insertGraph($graph)); |
|
|
81 | |
|
|
82 | $soln_fog = $f / $g; |
|
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83 | |
|
|
84 | &SOLUTION(EV3(<<'EOT')); |
|
|
85 | |
|
|
86 | $SOL $BR $BR |
|
|
87 | You can examine what this situation looks like by viewing the following graph: |
|
|
88 | $BR $graphtext $BR $BR |
|
|
89 | |
|
|
90 | We note that the functions cross, and we need to calculate that point at which |
|
|
91 | they cross, so we can integrate on each side of it separately. We need to do |
|
|
92 | those two integrals separately because in the once case one function is |
|
|
93 | greater, and in the other case the other function is greater. This matters |
|
|
94 | because the integral will need to contain an expression of the difference |
|
|
95 | between the functions, so we'll need to know which is greater to decide in what |
|
|
96 | order to take that difference. $BR $BR |
|
|
97 | |
|
|
98 | So we wish to discover the point at which \(y = $g \cos(x)\) crosses |
|
|
99 | \(y = $f \sin(x)\). So we set them equal to one another and solve for \(x\): |
|
|
100 | \[ |
|
|
101 | \begin{align*} |
|
|
102 | $g \cos(x) = $f \sin(x) |
|
|
103 | &\iff \frac{\cos(x)}{\sin(x)} = \frac{$f}{$g} \\\\ |
|
|
104 | &\iff \tan(x) = $soln_fog \\\\ |
|
|
105 | &\iff x = \tan^{-1}($soln_fog) |
|
|
106 | \end{align*} |
|
|
107 | \] |
|
|
108 | $BR $BR |
|
|
109 | |
|
|
110 | So then we simply integrate from 0 to \(\tan^{-1}($soln_fog)\) in one section, |
|
|
111 | and then from there to \($a\pi\) in another section. For ease of notation, |
|
|
112 | let's say \(C = \tan^{-1}($soln_fog)\). |
|
|
113 | \[ |
|
|
114 | \begin{align*} |
|
|
115 | & \int_0^{C} ( $g \cos(x) - $f \sin(x) ) \; dx |
|
|
116 | + \int_{C}^{$a\pi} ( $f \sin(x) - $g \cos(x) ) \; dx \\\\ |
|
|
117 | =& \left ( $g \sin(x) + $f \cos(x) ) \right|_0^{C} |
|
|
118 | + \left ( $g \sin(x) + $f \cos(x) ) \right|_{C}^{$a\pi} \\\\ |
|
|
119 | =& $g \sin(C) + $f \cos(C) - $g \sin(0) - $f \cos(0) - $f \cos($a\pi) |
|
|
120 | - $g \sin($a\pi) + $f \cos(C) + $g \sin(C) \\\\ |
|
|
121 | =& $answer |
|
|
122 | \end{align*} |
|
|
123 | \] |
|
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124 | |
|
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125 | EOT |
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126 | |
|
|
127 | |
|
|
128 | |
| 64 | ENDDOCUMENT(); # This should be the last executable line in the problem. |
129 | ENDDOCUMENT(); # This should be the last executable line in the problem. |
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