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# Annotation of /trunk/NationalProblemLibrary/Rochester/setDerivatives10MaxMin/c3s3p1.pg

 1 : jjholt 184 ## DESCRIPTION 2 : ## Locating Increasing and Decreasing Intervals and Local Extrema 3 : ## ENDDESCRIPTION 4 : jj 145 5 : jjholt 184 ## KEYWORDS('Increasing', 'Decreasing', 'Local', 'Extrema') 6 : ## Tagged by nhamblet 7 : 8 : ## DBsubject('Calculus') 9 : ## DBchapter('Applications of Differentiation') 10 : ## DBsection('How Derivatives Affect the Shape of a Graph') 11 : ## Date('') 12 : ## Author('') 13 : ## Institution('Rochester') 14 : ## TitleText1('') 15 : ## EditionText1('') 16 : ## AuthorText1('') 17 : ## Section1('') 18 : ## Problem1('') 19 : 20 : jj 145 DOCUMENT(); # This should be the first executable line in the problem. 21 : 22 : loadMacros("PG.pl", 23 : "PGbasicmacros.pl", 24 : "PGchoicemacros.pl", 25 : "PGanswermacros.pl", 26 : "PGauxiliaryFunctions.pl"); 27 : 28 : TEXT(beginproblem()); 29 : $showPartialCorrectAnswers = 1; 30 : 31 :$a = random(-8,-1,1); 32 : $b = random(1,7,1); 33 : if ($a + $b==0) {$b = 8;} 34 : $c = 6*random(1,3,1); # keeps everything an integer 35 :$d = non_zero_random(-8,8,1); 36 : 37 : $A3=$c/3; 38 : $A2=-$c*($a+$b)/2; 39 : $A1=$c*$a*$b; 40 : $A0 =$d; 41 : 42 : TEXT(EV2(<.1)); 55 : 56 : $A22 =$A2*2; 57 : 58 : $apb =$a + $b; 59 :$ab = $a*$b; 60 : 61 : SOLUTION(EV3(<<'EOF')); 62 : $SOL$BR 63 : To find the intervals of increase and decrease, we have to find the intervals where the derivative is positive and where it is negative. 64 : $BR $$f'(x) = c x^2 + A22 x + A1$$. 65 :$BR The derivative is $$0$$ when $$c x^2 + A22 x + A1 = 0$$. 66 : $BR $$c ( x^2 - apb x + ab) = 0$$ 67 :$BR $$c (x- a)(x- b) = 0$$ 68 : $BR We have two roots: $$x=a$$ and $$x=b$$. 69 :$BR It is easy to check that $$f'(x)$$ is negative (and therefore $$f(x)$$ is decreasing) on the interval $$(a,b)$$. 70 : $BR $$f'(x)$$ is positive (and therefore $$f(x)$$ is increasing) on the interval 71 : $$(-\infty, a)$$ and on the interval $$(b, \infty)$$. 72 :$BR Since $$f'(x)$$ changes from positive to negative at $$a$$, $$f(x)$$ has a local maximum at $$a$$. 73 : 74 : EOF 75 : 76 : ENDDOCUMENT(); # This should be the last executable line in the problem.