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Tue Nov 8 15:17:41 2011 UTC (18 months, 2 weeks ago) by aubreyja
File size: 2893 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('Differentiation in Several Variables')
3 ## DBsection('Optimization in Several Variables')
4 ## KEYWORDS('calculus')
5 ## TitleText1('Calculus: Early Transcendentals')
6 ## EditionText1('2')
7 ## AuthorText1('Rogawski')
8 ## Section1('14.7')
9 ## Problem1('37')
11 ## Institution('W.H.Freeman')
12 ## UsesAuxiliaryFiles('14-7-35-image.png')
13
14 DOCUMENT();
15
22
23 TEXT(beginproblem());
24
25 Context()->texStrings;
26 $context = Context(); 27$context->variables->add(y=>'Real');
28 $a=random(1,5,1); 29$b=random(2,8,1);
30 $aminb=$a-$b; 31 32$f=Formula("$a*x^3-$b*y")->reduce();
33 $fx=Formula("3*$a*x^2")->reduce();
34 $fy=Real(-$b);
35
36 $f1=Formula("$a*x^3")->reduce();
37 $f2=Formula("$a-$b*y")->reduce(); 38$f3=Formula("$a*x^3-$b")->reduce();
39 $f4=Formula("-$b*y")->reduce();
40
41 $fmin=-$b;
42 $fmax=$a;
43
44 BEGIN_TEXT
45 \{ textbook_ref_exact("Rogawski ET 2e", "14.7","37") \}
46 $PAR 47 Determine the global extreme values of the function 48 $$f(x,y)=f, \quad 0 \le x,y \le 1$$ 49$PAR
50 $$f_{min}=$$\{ans_rule()\} $BR 51 $$f_{max}=$$\{ans_rule()\} 52$BR
53 END_TEXT
54
55 Context()->normalStrings;
56 ANS(Real($fmin)->cmp); 57 ANS(Real($fmax)->cmp);
58 Context()->texStrings;
59
60 SOLUTION(EV3(<<'END_SOLUTION'));
61 $PAR 62$SOL
63 $BR 64 First we find the critical points 65 $f_x(x,y)=fx=0, \quad f_y(x,y)=fy$ 66 The two equations have no solutions, hence there are no critical points. 67$BR
68 The extreme values occur either at the critical points or at a point on the boundary of the domain. $BR 69 To check the boundary we consider each edge of the square $$0\le x,y \le 1$$ separately.$BR
70 \{image("14-7-35-image.png", width=>150, height=>150)\}
71 $BR 72 The segment $$\mathrm{OA}$$: On this segment $$y=0, 0\le x\le 1$$, and $$f$$ takes the values $$f(x,0)=f1$$.$BR
73 The minimum value is $$f(0,0)=0$$ and the maximum value is $$f(1,0)=a$$. $PAR 74 The segment $$\mathrm{AB}$$: On this segment $$x=1, 0\le y\le 1$$, and $$f$$ takes the values $$f(1,y)=f2$$.$BR
75 The minimum value is $$f(1,1)=aminb$$ and the maximum value is $$f(1,0)=a$$. $PAR 76 The segment $$\mathrm{BC}$$: On this segment $$y=1, 0\le x\le 1$$, and $$f$$ takes the values $$f(x,1)=f3$$.$BR
77 The minimum value is $$f(0,1)=-b$$ and the maximum value is $$f(1,1)=aminb$$. $PAR 78 The segment $$\mathrm{OC}$$: On this segment $$x=0, 0\le y\le 1$$, and $$f$$ takes the values $$f(0,y)=f4$$.$BR
79 The minimum value is $$f(0,1)=-b$$ and the maximum value is $$f(0,0)=0$$. $PAR 80 81 The smallest value is $$f_{min}=fmin$$ and it is the global minimum of $$f$$ on the square.$BR
82 The global maximum is the largest value $$f_{max}=fmax$$.
83 \$BR
84 END_SOLUTION
85
86 ENDDOCUMENT();