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| 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(<<EOT)); | ||
| 43 : | The function | ||
| 44 : | \[ f(x) = $A3 x^3 + $A2 x^2 + $A1 x + $A0 \] | ||
| 45 : | is decreasing on the interval ( \{ans_rule(5)\} , \{ans_rule(5) \} ). | ||
| 46 : | $BR | ||
| 47 : | It is increasing on the interval ( \(-\infty\), \{ans_rule(5) \} ) and the | ||
| 48 : | interval ( \{ans_rule(5) \}, \(\infty \) ). | ||
| 49 : | $BR | ||
| 50 : | The function has a local maximum at \{ans_rule(10) \}. | ||
| 51 : | |||
| 52 : | EOT | ||
| 53 : | |||
| 54 : | ANS(num_cmp([$a,$b,$a,$b,$a], relTol=>.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. |
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