# SamsExampleProblem

From WeBWorK

# PG problems are essentially Perl source files, with one exception: Perl # backslashes (\) are replaced by double-tildes (~~) since TeX uses # backslashes. For more about the format of PG files, consult the Translator.pm # documentation at: # <http://webwork.maa.org/doc/cvs/pg_HEAD/lib/WeBWorK/PG/Translator.html> # The first section in this file is a list of tags for the database problem # library project. For more about the tagging format, look at: # <http://hobbes.la.asu.edu/webwork-stuff/Tagging.html> ##DESCRIPTION ## Plots a piecewise function made up of a horizontal line, a diagonal line, and ## a parabola and asks the student to determine the derivative at various ## interesting points. ##ENDDESCRIPTION ## DBsubject('Calculus') ## DBchapter('Limits and Derivatives') ## DBsection('Definition of the Derivative') ## KEYWORDS('calculus', 'derivatives', 'slope') ## TitleText1('Calculus') ## EditionText1('1') ## AuthorText1('Rogawski') ## Section1('3.1') ## Problem1('11') ## Author('Sam Hathaway') ## Institution('W.H.Freeman') # The DOCUMENT() call sets up initial values for PG internals. It should always # be the first executable line in the problem. DOCUMENT(); # loadMacros() calls load macro files (which are Perl source files) into the # problem environment. These first three files are required by all problems. # Documentation: # <http://devel.webwork.rochester.edu/doc/cvs/pg_HEAD/macros/PG.pl.html> # <http://devel.webwork.rochester.edu/doc/cvs/pg_HEAD/macros/PGbasicmacros.pl.html> # <http://devel.webwork.rochester.edu/doc/cvs/pg_HEAD/macros/PGanswermacros.pl.html> loadMacros("PG.pl","PGbasicmacros.pl","PGanswermacros.pl"); # Parser.pl is the main macro file for MathObjects. This file defines macros # such as Real() and Formula(). You'll load it in pretty much all problems. # Parser docs are available at: # <http://devel.webwork.rochester.edu/twiki/bin/view/Webwork/MathObjects> loadMacros("Parser.pl"); # This is our macro file that provides the textbook_ref_exact() and # textbook_ref_corr() macros. You'll load it in all problems. loadMacros("freemanMacros.pl"); # This macro file contains the ceil(), floor(), max(), and (min) macros, which # we use in this problem. If you're not using macros from this package, you do # not need to load it. loadMacros("PGauxiliaryFunctions.pl"); # This macro file contains the init_graph() and plot_functions() macros. We need # these to create the graph for this problem. Don't load this unless you need # it. Documentation at: # <http://devel.webwork.rochester.edu/doc/cvs/pg_HEAD/macros/PGgraphmacros.pl.html> loadMacros("PGgraphmacros.pl"); # These are the values in the book version of the problem. Note that they are # commented out. #$base = 1; #$rise = 2; #$vertex_y = -2.25; # Here are the randomized values. Each student will get a different (but # persistent) value from each of these calls. $base = random(1,3,1); $rise = random(1,2,1); $vertex_y = random(1,4,0.25)*list_random(-1,1); # Create a MathObject formula for the initial height of the first horizontal # line. $horiz_line = Formula($base); # Represent the slope of the diagonal line. Set the reduceConstants flag of the # MathObjects context to 0, so that "$rise/2" is not reduced. Context()->flags->set(reduceConstants=>0); $slope = Formula("$rise/2"); # This is the formula for the diagonal line. $diag_line = Formula("$slope*(x-3)+$base"); # This is the formula for the parabola. $par = Formula("-($vertex_y/4)(x-5)(x-9)+$base+$rise"); # Get the y-value of the vertex of the porabola. We know that it occurs at x=7, # so we evaluate the $par formula with at x=7. $real_vertex = $par->eval(x=>7); # Now we set some temporary variables that we'll pass into init_graph below: # Minimum and maximum x and y values to graph. $xmin = -1; $ymin = min(-1, ceil($real_vertex)-1); $xmax = 9; $ymax = max(5, $base+$rise+1, floor($real_vertex)+1); # We want grid lines on each integer value, but we have to specify the total # number of grid lines on each axis, so we just use the x and y range. $xrange = $xmax-$xmin; $yrange = $ymax-$ymin; # Size of the graph in pixels. $xsize = $xrange*25; $ysize = $yrange*25; # init_graph returns a graph object that we can then add functions to. $graph = init_graph( $xmin, $ymin, $xmax, $ymax, grid => [$xrange,$yrange], axes => [0,0], size => [$xsize,$ysize], ); # Add three functions to the graph. The language used in specifying plots is # described in the PGgraphmacros.pl docs. When a MathObject is used in double- # quotes, it is stringified into the quasi-TI notation that WeBWorK uses. plot_functions($graph, "$horiz_line for x in [0,3] using color:red and weight:2", "$diag_line for x in [3,5] using color:red and weight:2", "$par for x in [5,9] using color:red and weight:2", ); # This changes how MathObjects are stringified. Instead of quasi-TI syntax, # we switch to TeX stringification. Context()->texStrings; # A BEGIN_TEXT...END_TEXT block is replaced by the preprocessor with: # TEXT(EV3(<<'END_TEXT')); # ... # END_TEXT # The <<'END_TEXT' part is the beginning of a Perl here document. Anyway, this # is just a convenience function, so you don't have to type that whole thing. # Basically whenever you see BEGIN_TEXT, you can read it as # TEXT(EV3(<<'END_TEXT')); # # TEXT() is the basic macro that outputs (well, accumulates actually) problem # "text", which can be HTML of TeX depending on the display mode. # # EV3() is a macro that interpretes it's contents according to these rules: # * Perl variables are evaluated in double-quoted string context. That is, # they get stringified. # * \{ EXPR \} blocks are replaced by the result of evaluating the perl code # inside. # * \( TEX \) blocks are replaced by equations described by the TEX code # within. Depending on the display mode, this can be plain text, an image, a # jsMath block, or raw TeX. # * \[ TEX \] blocks are treated similarly to \( TEX \) blocks, except that # the display math is used instead of inline math. # # In the original version of this problem, there was one BEGIN_TEXT/END_TEXT # block, but because I have to comment, I'm going to split it up into multiple # blocks. # beginproblem() prints the point value of the problem, the beginning of the # HTML form (in HTML-based display modes), and other header-type stuff. BEGIN_TEXT \{ beginproblem() \} END_TEXT # We use our textbook_ref_exact() macro to print "From Rogawski ET, section 3.1 # problem 11". BEGIN_TEXT \{ textbook_ref_exact("Rogawski ET", "3.1","11") \} END_TEXT # $PAR is a double-line break. It contains <P> in HTML-based modes and \par in # TeX mode. Note the use of \( ... \) to typeset f(x). BEGIN_TEXT $PAR Let \( f(x) \) be the function whose graph is shown below. END_TEXT # We insert the graph that we generated before. insertGraph() actually generates # the image, and it returns the pathname of the image. image() takes that image # and actually generates code needed to place the image in the output. These # macros are in dangerousMacros.pl, which is always loaded. Documentation at: # <http://devel.webwork.rochester.edu/doc/cvs/pg_HEAD/macros/dangerousMacros.pl.html> BEGIN_TEXT $PAR \{ image(insertGraph($graph)) \} END_TEXT # Now we ask the question. BEGIN_TEXT $PAR Determine \( f'(a) \) for \( a = 1,2,4,7 \). END_TEXT # And generate the answer blanks. $BR is a single-line break. BEGIN_TEXT $BR \( f'(1) = \) \{ans_rule()\} $BR \( f'(2) = \) \{ans_rule()\} $BR \( f'(4) = \) \{ans_rule()\} $BR \( f'(7) = \) \{ans_rule()\} END_TEXT # Switch back to normal strings now that we're done with the text block. Context()->normalStrings; # The ANS() macro takes an "answer evaluator" as its argument. An answer # evaluator is a perl function that gets passed the student's answer and # determines if it is correct or not. We don't have to write them ourselves, # because MathObjects know how to generate them, using the ->cmp method. These # ANS() calls must be in the same order as the ans_rule() calls above. ANS(Real(0)->cmp); ANS(Real(0)->cmp); ANS($slope->cmp); ANS(Real(0)->cmp); # Switch back to TeX stringification. Context()->texStrings; # SOLUTION() works like TEXT() except that it's only shown if the "show # solutions" flag is given. $SOL evaluates to "Solution: " in bold. Note the # MathObjects embedded in math expressions in the solution. Remember that they # are stringifying to their TeX representations. SOLUTION(EV3(<<'END_SOLUTION')); $PAR $SOL Remember that the value of the derivative of \( f \) at \( x=a \) can be interpreted as the slope of the line tangent to the graph of \( y = f(x) \) at \( x=a \). From the figure, we see that the graph of \( y = f(x) \) is a horizontal line (that is, a line with zero slope) on the interval \( 0 \le x \le 3 \). Accordingly, \( f'(1) = f'(2) = 0 \). On the interval \( 3 \le x \le 5 \), the graph of \( y = f(x) \) is a line of slope \( $slope \); thus, \( f'(4) = $slope \). Finally, the line tangent to the graph of \( y = f(x) \) at \( x=7 \) is horizontal, so \( f'(7) = 0 \). END_SOLUTION # This finishes everything up. It should always be the last executable line in # the file.