Fix bug 4689

Author: gajennings

OpenProblemLibrary/Wiley/setAnton_Section_3.1/Anton3_1_Q18.pg

@@ -34,32 +34,34 @@ TEXT(beginproblem());
 
 ###################################
 # Setup
-     
-Context()->variables->add(y=>'Real',dx=>'Real',dy=>'Real'); 
-$a=random(-8,8,1);
-
-$f=Formula("x*cos(y)")->reduce;
-$g=Formula("$a*y")->reduce;
-$F=Formula("cos(y)-x*sin(y)*(dy/dx)")->reduce;
-$G=Formula("$a*(dy/dx)")->reduce;
-$Y=Formula("$a+x*sin(y)")->reduce;
-$X=Formula("cos(y)")->reduce;
-$dF=Formula("-sin(y)*(dy/dx)*($Y)-[sin(y)+x*cos(y)*(dy/dx)]*($X)")->reduce;
-$dG=Formula("($Y)^2")->reduce;
-$dd1=$dF/$dG;
-$df=Formula("(dy/dx)*[$a*sin(y)+x*sin^2(y)+x*cos^2(y)]+sin(y)*($X)")->reduce;
-$dd2=-$df/$dG;
-$DF=Formula("cos(y)*[sin(y)*($a)+x]+($X)*(sin(y)*($Y))")->reduce;
-$DG=Formula("($Y)^3");
-$dd3=$DF/$DG;
-$ans=Formula("-cos(y)*([2*$a*sin(y)+x*(1+sin^2(y))]/($Y)^3)")->reduce;
-
-$dydx=Formula("$X/$Y");;
+
+Context("Numeric");     
+Context()->variables->add(y=>'Real');
+
+$a=non_zero_random(-8,8,1);
+
+# If the answer is tested at random values of x and y 
+# a student might substitute cos(y)->x/a in the 
+# implicit solution and have it marked "incorrect" 
+# so we'll numerically generate a series of test points along the 
+# implicit curve and test it on those points.  
+
+$ans=Formula("-cos(y)*([2*$a*sin(y)+x*(1+sin(y)^2)]/($a+x*sin(y))^3)")->reduce;
+
+$ans->{test_points} = [ [0,0],
+  [$a*0.2, 0.19616428118783816 ],[-$a*0.2, -0.19616428118783816 ], 
+  [$a*0.4, 0.37255949583211007], [-$a*0.4, -0.37255949583211007],
+  [$a*0.6, 0.5205326392380185],  [-$a*0.6, -0.5205326392380185], 
+  [$a*0.8, 0.641134282813549],   [-$a*0.8, -0.641134282813549],
+  [$a,     0.7390851332151331 ], [-$a,     -0.7390851332151331 ]
+];
+
 ###################################
 # Main text
 Context()->texStrings;
 BEGIN_TEXT
-Find \(\frac{d^2y}{dx^2}\) by implicit differentiation when \[$f=$g\]
+Find \(\frac{d^2y}{dx^2}\) by implicit differentiation when 
+\[x\cos(y)=$a y\]
 
 $PAR
 \(\frac{d^2y}{dx^2}=\) \{ans_rule(50) \}
@@ -75,8 +77,15 @@ ANS($ans->cmp);
 Context()->texStrings;
 SOLUTION(EV3(<<'END_SOLUTION'));
 $PAR SOLUTION $PAR
-Differentiating gives \[$F=$G\] \[\left($Y\right)\frac{dy}{dx}=$X\] \[\frac{dy}{dx}=$dydx\]
-\[\frac{d^2y}{dx^2}=\frac{d}{dx}\left[$dydx\right]=$dd1\]\[=$dd2\]\[=$dd3=$ans\]
+Differentiating gives 
+\[\cos(y)-x\sin(y)\frac{dy}{dx}=$a\frac{dy}{dx}\] 
+\[\left($a + x\sin(y)\right)\frac{dy}{dx}=\cos(y)\] 
+\[\frac{dy}{dx}=\frac{\cos(y)}{$a+x\sin(y)}\]
+so
+\[\frac{d^2y}{dx^2}=\frac{d}{dx}\left(\frac{\cos(y)}{$a+x\sin(y)}\right)=\frac{-\frac{dy}{dx}\sin(y)\big($a+x\sin(y)\big) - \big(\sin(y)+x\cos(y)\frac{dy}{dx}\big)\cos(y)}{($a+x\sin(y))^2}\]
+\[=\frac{-\frac{dy}{dx}\big($a\sin(y)+x\sin^2(y)+x\cos^2(y)\big)-\sin(y)\cos(y)}{($a+x\sin(y))^2}\]
+\[=-\frac{\cos(y)\big($a\sin(y)+x\big)}{($a+x\sin(y))^3}-\frac{\sin(y)\cos(y)}{($a+x\sin(y))^2}\]
+\[=$ans\]
 END_SOLUTION
 Context()->normalStrings;