[npl] / trunk / NationalProblemLibrary / WHFreeman / Rogawski_Calculus_Early_Transcendentals_Second_Edition / 14_Differentiation_in_Several_Variables / 14.5_The_Gradient_and_Directional_Derivatives / 14.5.7.pg Repository: Repository Listing bbplugincoursesdistsnplrochestersystemwww

Tue Nov 8 15:17:41 2011 UTC (18 months, 1 week ago) by aubreyja
File size: 1859 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('The Gradient and Directional Derivatives')
4 ## KEYWORDS('calculus')
5 ## TitleText1('Calculus: Early Transcendentals')
6 ## EditionText1('2')
7 ## AuthorText1('Rogawski')
8 ## Section1('14.5')
9 ## Problem1('7')
11 ## Institution('W.H.Freeman')
12
13 DOCUMENT();
14
21
22 TEXT(beginproblem());
23
24 Context()->texStrings;
25 $a=non_zero_random(-3,3,1); 26$b=non_zero_random(-3,3,1);
27 $c=non_zero_random(-3,3,1); 28 29$context = Context();
30 $context->variables->add(y=>'Real'); 31$context->variables->add(z=>'Real');
32
33 $f=Formula("x^($a)*y^($b)*z^($c)")->reduce();
34 $fx=Formula("$a*x^($a-1)*y^($b)*z^($c)")->reduce(); 35$fy=Formula("$b*x^($a)*y^($b-1)*z^($c)")->reduce();
36 $fz=Formula("$c*x^($a)*y^($b)*z^($c-1)")->reduce(); 37 38 BEGIN_TEXT 39 \{ textbook_ref_exact("Rogawski ET 2e", "14.5","7") \} 40$PAR
41 Calculate the gradient of $$h(x,y,z)=f$$
42 $PAR 43 $$\nabla h=$$ \{ans_rule()\} 44$BR
45 END_TEXT
46
47 Context()->normalStrings;
48 Context("Vector");
49 ANS(Vector(Formula($fx),Formula($fy),Formula($fz))->cmp); 50 Context("Numeric"); 51 Context()->texStrings; 52 53 SOLUTION(EV3(<<'END_SOLUTION')); 54$PAR
55 $SOL 56$BR
57 We compute the partial derivatives of $$h(x,y,z)=f$$, obtaining
58 $\frac{\partial{h}}{\partial{x}}=fx \quad \frac{\partial{h}}{\partial{y}}=fy \quad \frac{\partial{h}}{\partial{z}}=fz$
59 The gradient vector is thus
60 $\nabla h=\left<\frac{\partial{h}}{\partial{x}},\frac{\partial{h}}{\partial{y}},\frac{\partial{h}}{\partial{z}}\right>= 61 \left<fx, fy, fz \right>$
62
63 \$BR
64 END_SOLUTION
65
66 ENDDOCUMENT();