## DESCRIPTION ## This is Problem 2.4.24 from the APEX Calculus text. It covers differentiation of basic trig functions and the Quotient Rule. ## ENDDESCRIPTION ## DBsubject(Calculus - single variable) ## DBchapter(Differentiation) ## DBsection(Quotient rule (with trigonometric functions)) ## Level(2) ## Institution('Valdosta State University') ## Author('S. V. Ault') ## RevisedBy('F. J. Francis') ## TitleText1('APEX Calculus') ## AuthorText1('Hartman') ## EditionText1('3.0') ## Section1('2.4') ## Problem1('24') ## MO(1) ## Keywords('derivative', 'quotient rule', 'trig functions', 'ULETH-MATH1560', 'ULETH-MATH1565') ########################### # Initialization DOCUMENT(); # Load whatever macros you need for the problem loadMacros( # REQUIRED: Used for basic question and answer operations. "PGstandard.pl", # REQUIRED: Used for expression parsing. "MathObjects.pl", # Usually required for proper text formatting. "PGML.pl", # Used to provide contextual help for how to type inequalities/intervals. "AnswerFormatHelp.pl" ); # Sets up basic problem information. TEXT(beginproblem()); ############################# # Setup #-ULETH-# Context("Numeric")->noreduce('(-x)-y','(-x)+y');; $a = random(2,10,1)*random(-1,1,2);$b = random(1,10,1)*random(-1,1,2); $c = random(1,10,1)*random(-1,1,2);$f = Formula("$a*sin(x)")->reduce();$df = $f->D('x');$g = Formula("$b*cos(x)+$c")->reduce(); $dgu =$g->D('x'); $dg = Compute("$dgu")->reduce(); $func = Formula($f/$g)->reduce();$ans = $func->D('x'); #-ENDULETH-# ############################# # Problem Text #-ULETH-# BEGIN_PGML Compute the derivative of the given function. >>[f(x) = \frac{[$f]}{[$g]}.]<< [f'(x) = ] [___________________________________________________] [@ AnswerFormatHelp("formulas") @]* END_PGML #-ENDULETH-# ############################# # Solution #-ULETH-# BEGIN_PGML_SOLUTION You may compute this derivative using the Quotient Rule. [ \begin{array}{rcl} f'(x) &=& \displaystyle \frac{ ([$g])\frac{d}{dx}[ [$f] ] - ([$f]) \cdot \frac{d}{dx}[ [$g] ] }{([$g])^2}\\ &&\\ &=& \displaystyle \frac{ ([$g])([$df]) - ([$f])([$dg]) }{ ([$g])^2 }. \end{array} ] END_PGML_SOLUTION #-ENDULETH-# ############################# # Answer evaluation #-ULETH-# # Setting this to 1 means that students will receive feedback on whether their # answers are correct.$showPartialCorrectAnswers = 0; #-ENDULETH# ANS( \$ans->cmp() ); ############################# # Comments COMMENT(' Randomization provides 7200 different possible versions of this question.
Includes a solution set.
Made from a ULETH template.
'); ENDDOCUMENT();