aimmath-commit Mailing List for AiM Assessment in Mathematics (Page 22)

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Update of /cvsroot/aimmath/AIM/WEB-INF/maple/aim
In directory sc8-pr-cvs1:/tmp/cvs-serv28505

Modified Files:
      Tag: develop_2_1
	Int.mpl Util1.mpl 
Log Message:
Util1.mpl: added 
  - NiceFrac ... for fine-tuning of fractions
  - simplifyabs ... simplifies the result of diff(ln(abs(u)), u) to 1/u
Int.mpl: some improvements (using the above) + added some new features
  for checking for integration constants with indefinite integrals.    - GG


Index: Int.mpl
===================================================================
RCS file: /cvsroot/aimmath/AIM/WEB-INF/maple/aim/Int.mpl,v
retrieving revision 1.2.2.1
retrieving revision 1.2.2.2
diff -C2 -d -r1.2.2.1 -r1.2.2.2
*** Int.mpl	10 Jul 2003 20:10:03 -0000	1.2.2.1
--- Int.mpl	19 Jul 2003 17:37:34 -0000	1.2.2.2
***************
*** 119,131 ****
    @a['AnswerNote']@.  The function returns a list of the form
    @[mark,feedback,note]@.
   ",
   proc(a,
        Q::`aim/Int/Problem/Indefinite`,
        #optional
!       val_::numeric
        )
  
!   local q,argtype,ans,diffans,rightans,integrand,var,
!         feedback,note,mark,t,val,u,base,expt,newvar,ns;
  
    if type(a,`aim/Question/Attempt`) then
--- 119,140 ----
    @a['AnswerNote']@.  The function returns a list of the form
    @[mark,feedback,note]@.
+   By default, the student's answer need not have an integration constant,
+   but to insist that one be included pass as 4th argument 
+   the optional argument @intconst_@ with value @\"insist\"@, and in this case
+   the student is penalised (by multiplying by a factor of 0.9).
+   To suggest there should be an integration constant but not penalise
+   pass the optional argument @intconst_@ with value @\"alert\"@.
+   In order for there to be a 4th argument one is obliged, in this case,
+   to pass the 3rd argument @val_@.
   ",
   proc(a,
        Q::`aim/Int/Problem/Indefinite`,
        #optional
!       val_::numeric,
!       intconst_::string
        )
  
!   local q,argtype,ans,diffans,rightans,integrand,var,subsvars,varstr,
!         feedback,note,mark,t,val,intconst,u,base,expt,newvar,ns;
  
    if type(a,`aim/Question/Attempt`) then
***************
*** 140,144 ****
    fi;
  
!   val := `if`(nargs > 2, val_,1);
  
    q := `aim/Int/Problem/Split`(Q);
--- 149,159 ----
    fi;
  
!   if nargs >= 4 then
!     val, intconst := val_, intconst_;
!   elif nargs = 3 then
!     val, intconst := val_, "ignore";
!   else
!     val, intconst := 1, "ignore";
!   fi;
  
    q := `aim/Int/Problem/Split`(Q);
***************
*** 150,155 ****
    note := "";
  
!   diffans :=
!    traperror(simplify(diff(ans,var),assume = positive));
  
    if diffans = lasterror then
--- 165,170 ----
    note := "";
  
!   diffans := traperror(simplify(simplifyabs(diff(ans,var))));
!   t := `aim/CheckVars`(ans, integrand);
  
    if diffans = lasterror then
***************
*** 159,163 ****
     mark := 0;
  
!   elif traperror(simplify(diffans - integrand)) = 0 then
     feedback := "";
     note := "";
--- 174,188 ----
     mark := 0;
  
!   # Sometimes it's best to compare the integrals and sometimes it's best to
!   # differentiate and see that it gives back the integrand
!   # ... if a student transforms the integrand in some way and then
!   # integrates, as is typical of some trig. integrals, differentiation and
!   # comparing with the integrand can sometimes fail no matter how much help
!   # you give Maple. simplifyabs here takes care of examples like int(1/u, u)
!   # where the student gives the answer: ln(abs(u)) + C
!   elif traperror(simplify(simplifyabs(diff(simplify(ans - rightans))))) = 0 or 
!        traperror(expand(simplify(diffans - integrand))) = 0 or 
!        traperror(expand(simplify(diffans - integrand), assume = positive)) = 0
!        then
     feedback := "";
     note := "";
***************
*** 171,176 ****
       __("In fact, the derivative of your answer is:"),
       "\n",
!      sprintf("<latex>\\[\\frac{d}{d%a}\\left[%s\\right] = %s\\]</latex>",
!       var,LaTeX(ans),LaTeX(diff(ans,var))),
       "\n",
       __("so you must have done something wrong."),
--- 196,201 ----
       __("In fact, the derivative of your answer is:"),
       "\n",
!      sprintf("<latex>\\[\\frac{d}{d%a}\\bigl(%s\\bigr) = %s\\]</latex>",
!       var,LaTeX(ans),LaTeX(simplifyabs(diff(ans,var)))),
       "\n",
       __("so you must have done something wrong."),
***************
*** 180,198 ****
     mark := 0;
  
!    t := `aim/CheckVars`(ans,Q);
! 
!    if t['Extra'] <> [] then
      feedback :=
       feedback,
!       __("Your answer involves the following variable(s) that do not occur in the question or the right answer:"),
!       " <font color='red'><tt>",
!       `Util/CommaJoin`(op(t['Extra'])),
!       "</tt></font>\n<br/>\n",
!       __("Note that variable names are case-sensitive; you should not enter <font color='green'><tt>A</tt></font> for <font color='green'><tt>a</tt></font> or vice-versa."),
!       "<br/><br/>\n";
!      note := __("extra vars");
!     if ({op(t['Extra'])} intersect {"u","v","w","U","V","W"}) <> {} then
       feedback :=
!       feedback,__("#Explain integration backsubstitution"),"\n<br/>\n";
      fi;
     elif type([integrand],[`^`]) then
--- 205,240 ----
     mark := 0;
  
!    if t['Missing'] = [] and nops(t['Extra']) > 1 then
      feedback :=
       feedback,
!       __("Your answer seems to include at least one superfluous integration constant, one is enough!");
!    fi;
! 
!    if t['Missing'] <> [] then
!     if t['Extra'] <> [] then
       feedback :=
!       feedback,
!        __("Your answer involves the following variable(s) that do not occur in the integrand:"),
!        " <font color='red'><tt>",
!        `Util/CommaJoin`(op(t['Extra'])),
!        "</tt></font>\n<br/>\n",
!        __("Note that variable names are case-sensitive; you should not enter <font color='green'><tt>A</tt></font> for <font color='green'><tt>a</tt></font> or vice-versa. One of the extra variables could be intended as the integration constant but where is the original variable of integration?"),
!        "<br/><br/>\n";
!      note := __("extra vars");
!      if ({op(t['Extra'])} intersect {"u","v","w","U","V","W"}) <> {} then
!       feedback :=
!        feedback,__("#Explain integration backsubstitution"),"\n<br/>\n";
!      fi;
!      #The following is a possible alternative for the preceding if ... fi
!      #subsvars := {op(t['Extra'])} intersect {"u","v","w","U","V","W"};
!      #if subsvars <> {} then
!      # varstr := convert(var, string);
!      #feedback :=
!      # feedback, "It looks as if you made a substitution ", subsvars[1],
!      # " = f(", varstr, ") for some function f. ", 
!      # "Your final answer should be expressed in terms of the original ",
!      # "integration variable by substituting f(", varstr, ") for ",
!      # subsvars[1], "\n<br/>\n";
!      #fi;
      fi;
     elif type([integrand],[`^`]) then
***************
*** 219,222 ****
--- 261,283 ----
    fi;
  
+   if intconst <> "ignore" and nops(t['Extra']) < 1 then
+     if member("C", t['Correct']) then
+       # strange choice of integration variable!!
+       feedback := feedback,
+         __("You neglected to include an integration constant.");
+     else
+       feedback := feedback,
+         __("You neglected to include an integration constant, i.e. you should have included, e.g.: + C.");
+     fi;
+     if intconst = "insist" then
+       mark := mark * 0.9;
+     else
+       feedback := feedback,
+         __("There was no penalty for omitting it.");
+     fi;
+     feedback := feedback, "\n<br>\n";
+     note := note, __("no integration constant");
+   fi;
+ 
    feedback := cat(feedback);
    note := cat(note);
***************
*** 268,271 ****
--- 329,336 ----
     feedback := "";
     note := "";
+   elif `aim/TestSymbolic`(ans,-q['RightAnswer']) then
+    mark := 0;
+    feedback := __("Possibly you evaluated the limits in the wrong order.");
+    note := __("Reversed limits");
    else
     mark := 0;
***************
*** 310,326 ****
  
  `Package/Assign`(
   `aim/DefInt/ShowSubs`,
!  "It is assumed that @Q@ is an indefinite integration problem, and
    that @F@ is a solution of the corresponding indefinite problem.
    The output is a LaTeX string enclosed in a <latex> tag,
!   expressing something like this:
    <pre><font color='green'>
  
          4
!        /            |   3  | 4      |  3 |   |  3 |
!       |    2        |  x   |        | 4  |   | 2  |   64   8   56
!       |   x  dx  =  | ---- |     =  | -- | - | -- | = -- - - = --
!       |             |  3   |        | 3  |   | 3  |   3    3   3
!      /              |      | x=2    |    |   |    |
         2
  
--- 375,410 ----
  
  `Package/Assign`(
+  `aim/DefInt/NiceInt`,
+  "Given @Q@, a definite integration problem, returns
+   a LaTeX string for the integration problem that has its
+   limits set more nicely and with a little whitespace before
+   the d@x@, where @x@ is the integration variable
+  ",
+  proc(Q::`aim/Int/Problem/Definite`,F)
+   local q, a, b;
+   q := `aim/Int/Problem/Split`(Q);
+   a, b := op( q['Limits'] );
+   cat( "\\int_{", NiceFrac(a, 1, "textstyle"), "}",
+             "^{", NiceFrac(b, 1, "textstyle"), "} ",
+             LaTeX( op(1, q['Integral']) ), " d", LaTeX( q['Variable'] ) );
+  end
+ ):
+ 
+ ######################################################################
+ 
+ `Package/Assign`(
   `aim/DefInt/ShowSubs`,
!  "It is assumed that @Q@ is a definite integration problem, and
    that @F@ is a solution of the corresponding indefinite problem.
    The output is a LaTeX string enclosed in a <latex> tag,
!   expressing something like this, except on multiple lines:
    <pre><font color='green'>
  
          4
!        /            |   3  | 4      /  3 \\     /  3 \\
!       |    2        |  x   |       |  4   |   |  2   |   64   8   56
!       |   x  dx  =  | ---- |     = |  --  | - |  --  | = -- - - = --
!       |             |  3   |       |  3   |   |  3   |   3    3   3
!      /              |      | x=2    \\    /     \\    /
         2
  
***************
*** 328,353 ****
   ",
   proc(Q::`aim/Int/Problem/Definite`,F)
!   local q,limits,a,b,x,Fa,Fb,L0,L1,L2,L3,s;
  
    q := `aim/Int/Problem/Split`(Q);
!   limits := q['Limits'];
!   a := op(1,limits);
!   b := op(2,limits);
    x := q['Variable'];
    Fa  := limit(F,x=a);
    Fb  := limit(F,x=b);
  
!   L0 := LaTeX(q['Integral']);
  
!   L1 :=
!    cat("\\left[ ", LaTeX(F), "\\right]_{",LaTeX(a),"}^{",LaTeX(b),"}");
  
!   L2 :=
!    cat("\\left( ", LaTeX(Fb), "\\right) - \\left(", LaTeX(Fa), "\\right)");
  
    L3 := LaTeX(Fb - Fa);
  
    
!   s := cat(L0," = ",L1," = ",L2," = ",L3);
  
    # The next line tests whether Fb - Fa is visibly equal to 
--- 412,441 ----
   ",
   proc(Q::`aim/Int/Problem/Definite`,F)
!   local q,limits,a,b,x,Fa,Fb,L0,L1,L2,L3,LaTeXF,s;
  
    q := `aim/Int/Problem/Split`(Q);
!   a, b := op( q['Limits'] );
    x := q['Variable'];
    Fa  := limit(F,x=a);
    Fb  := limit(F,x=b);
  
!   limits := cat("_{", NiceFrac(a, 1, "textstyle"), "}",
!                 "^{", NiceFrac(b, 1, "textstyle"), "}");
  
!   L0 := cat( "\\int", limits, " ",
!              LaTeX( op(1, q['Integral']) ), " d", LaTeX(x) );
  
!   LaTeXF := LaTeX(F);
!   L1 := `if`( evalb( SearchText("frac", LaTeXF) > 0 ),
!               cat("\\left[ ", LaTeXF, " \\right]", limits),
!               cat("\\bigl[ ", LaTeXF, " \\bigr]", limits) );
! 
!   L2 := cat( "\\left(", LaTeX(Fb), "\\right) - ",
!              "\\left(", LaTeX(Fa), "\\right)");
  
    L3 := LaTeX(Fb - Fa);
  
    
!   s := cat(L0, " &= ", L1, "\\\\ &= ", L2, "\\\\ &= ", L3);
  
    # The next line tests whether Fb - Fa is visibly equal to 
***************
*** 355,362 ****
    # simplification, provided that the question is correct.
    if (Fb - Fa <> q['RightAnswer']) then
!    s := cat(s," = ",LaTeX(q['RightAnswer']));
    fi;
  
!   RETURN(cat("<latex>\\[ ",s," \\]</latex>\n"));
   end
  ):
--- 443,450 ----
    # simplification, provided that the question is correct.
    if (Fb - Fa <> q['RightAnswer']) then
!    s := cat(s,"\\\\ &= ", LaTeX(q['RightAnswer']));
    fi;
  
!   RETURN(cat("<latex>\\begin{eqnarray*} ",s," \\end{eqnarray*}</latex>\n"));
   end
  ):

Index: Util1.mpl
===================================================================
RCS file: /cvsroot/aimmath/AIM/WEB-INF/maple/aim/Attic/Util1.mpl,v
retrieving revision 1.1.2.1
retrieving revision 1.1.2.2
diff -C2 -d -r1.1.2.1 -r1.1.2.2
*** Util1.mpl	9 Jul 2003 09:59:48 -0000	1.1.2.1
--- Util1.mpl	19 Jul 2003 17:37:34 -0000	1.1.2.2
***************
*** 8,12 ****
  Package("aim/Util1","
   This package provides some extra utility functions initially created
!  as they were needed by or were useful with the #`aim/Trig`# package.
  "
  ):
--- 8,13 ----
  Package("aim/Util1","
   This package provides some extra utility functions initially created
!  as they were needed by or were useful with packages such as the 
!  #`aim/Trig`# package.
  "
  ):
***************
*** 148,151 ****
--- 149,248 ----
      s_ := cat(s_[1 .. pos - 1], s_[posn .. posb - 1], "/", s_[posb .. -1]);
    od;
+  end
+ ):
+ 
+ `Package/Assign`(
+  NiceFrac::string,
+  "Return a nicely formatted rendition of the fraction @n@/@d@ as a
+   LaTeX string. It accepts the following optional arguments
+   <dl>
+    <dt>@\"term\"@</dt>
+    <dd>if present, @n@/@d@ is treated as a term in a larger expression,
+        i.e. if @n@/@d@ = 0 then an empty string is returned or otherwise
+        the operational sign (+ or -) is prepended;</dd>
+    <dt>@\"coef\"@ (or @\"coeff\"@)</dt>
+    <dd>if present, @n@/@d@ is treated as the coefficient of a term,
+        i.e. if @n@/@d@ = 1 or -1 then only an operational sign is returned
+        (+, in the former case, or - otherwise); in any case, as with 
+        @\"term\"@ an operational sign is prepended;</dd>
+    <dt>@\"lcoef\"@ (or @\"lcoeff\"@)</dt>
+    <dd>if present, @n@/@d@ is treated as the coefficient of a leading term,
+        i.e. if @n@/@d@ = -1 then only the operational sign - is returned,
+        and if @n@/@d@ = 1 a null string is returned; in any case, an
+        operational sign is prepended only if @n@/@d@ is negative;</dd>
+    <dt>@\"nocancel\"@</dt>
+    <dd>if present, @n@/@d@ is not cancelled to lowest terms;</dd>
+    <dt>@\"textstyle\"@</dt>
+    <dd>if present, @n@/@d@ is formatted in textstyle, i.e. using / between
+        numerator and denominator
+        (by default, @n@/@d@ is formatted in displaystyle, i.e. in
+         @\\frac{..}{..}@ form).
+   </dl>",
+  proc(n, d)
+   local x, n_, d_, sign_, term, coef, lcoef, nocancel, textstyle;
+   if nargs = 1 or type(args[2], string) then
+     d_ := 1;
+   else
+     d_ := d;
+   fi;
+   term := member("term", [args]);
+   coef := member("coef", [args]) or member("coeff", [args]);
+   lcoef := member("lcoef", [args]) or member("lcoeff", [args]);
+   nocancel := member("nocancel", [args]);
+   textstyle := member("textstyle", [args]);
+   
+   x := n/d_;
+   sign_ := `if`(evalb( convert(signum(x), string)[1] = "-" ), 
+                "-", `if`(evalb(term or coef), "+", ""));
+   if sign_ = "-" then
+     x := -x;
+   fi;
+   if term or coef then
+     sign_ := cat(sign_, " ");
+   fi;
+ 
+   if not nocancel and term and x = 0 then
+     return "";
+   elif not nocancel and (coef or lcoef) and x = 1 then
+     return sign_;
+   elif not nocancel and denom(x) = 1 then
+     return cat(sign_, LaTeX(x));
+   else
+     if nocancel then
+       n_ := `if`(evalb( signum(n) = -1 ), -n, n);
+       d_ := `if`(evalb( signum(d_) = -1 ), -d_, d_);
+     else
+       n_, d_ := numer(x), denom(x);
+     fi;
+     if textstyle then
+       return cat(sign_, LaTeX(n_), "/", 
+                        `if`(evalb(nops(d_) = 1), "", "("),
+                        LaTeX(d_),
+                        `if`(evalb(nops(d_) = 1), "", ")"));
+     else
+       return cat(sign_, "\\frac{", LaTeX(n_), "}{", LaTeX(d_), "}");
+     fi;
+   fi;
+  end
+ ):
+ 
+ `Package/Assign`(
+  `simplifyabs`::algebraic,
+  "Returns @e@ with any subexpression pairs of form @abs(1, u)@,  @abs(u)@ 
+   replaced by @u@. These typically turn up when differentiating expressions
+   that contain absolute values, e.g. @diff(ln(abs(u)),u)@ gives
+   @abs(1,u)/abs(u)@ which #`simplifyabs`# simplifies to @1/u@. It is used
+   in #`aim/Int/Test`#.",
+  proc(e::algebraic)
+    local e_, absfns, x, u;
+    e_ := e;
+    absfns := selectfun(e_, abs);
+    for x in select(x -> evalb(nops(x) = 2), absfns) do
+      u := op(2, x);
+      if member(abs(x), absfns) then
+        e_ := subs( abs(1, u) = 1, abs(u) = u, e_ );
+      fi;
+    od;
+    return e_;
   end
  ):

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aimmath-commit Mailing List for AiM Assessment in Mathematics (Page 22)

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