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# 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
# documentation at:
# <>

# 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:
# <>

## 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.
## 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.

# loadMacros() calls load macro files (which are Perl source files) into the
# problem environment. These first three files are required by all problems.
# Documentation:
# <>
# <>
# <>

# 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:
# <>

# This is our macro file that provides the textbook_ref_exact() and
# textbook_ref_corr() macros. You'll load it in all problems.

# 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.

# 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:
# <>

# 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.
$slope = Formula("$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 docs. When a MathObject is used in double-
# quotes, it is stringified into the quasi-TI notation that WeBWorK uses.
    "$horiz_line for x in [0,3] using color:red and weight:2",
    "$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.

# 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.
\{ beginproblem() \}

# We use our textbook_ref_exact() macro to print "From Rogawski ET, section 3.1
# problem 11".
\{ textbook_ref_exact("Rogawski ET", "3.1","11") \}
#  $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).
Let \( f(x) \) be the function whose graph is shown below.

# 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, which is always loaded. Documentation at:
# <>
\{ image(insertGraph($graph)) \}

# Now we ask the question.
Determine \( f'(a) \) for \( a = 1,2,4,7 \).

# And generate the answer blanks. $BR is a single-line break.
\( f'(1) = \) \{ans_rule()\}
\( f'(2) = \) \{ans_rule()\}
\( f'(4) = \) \{ans_rule()\}
\( f'(7) = \) \{ans_rule()\}

# Switch back to normal strings now that we're done with the text block.

# 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.

# Switch back to TeX stringification.

# 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.
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 \( f'(7) = 0 \).

# This finishes everything up. It should always be the last executable line in
# the file.
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