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<!DOCTYPE html>
<html data-require="math math-format polynomials graphie interactive word-problems">
<head>
    <meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
    <title>Recognizing concavity</title>
    <script data-main="../local-only/main.js" src="../local-only/require.js"></script>
    <style type="text/css">
        .graph-caption {
            display: block;
            margin-top: 10px;
            margin-bottom: 30px;
        }
    </style>
</head>
<body>
<div class="exercise">
    <div class="vars" data-ensure="SOLUTION_INTERVAL != null &amp;&amp; SOLUTION_INTERVAL[1] - SOLUTION_INTERVAL[0] &gt; 1.2">
        <!-- Some hard-coded polynomial functions -->
        <var id="COEF, XRANGE, YRANGE">randFromArray([
            [[ 1, 0, -16, 0, 49], [-5, 5], [-100, 100]],
            [[-1, 0, 16, 0, -49], [-5, 5], [-250, 100]],
            [[ 1, 3, -14, -35, 21], [-5, 5], [-100, 200]],
            [[-1, -3, 14, 35, 0], [-5, 5], [-200, 100]],
            [[ 1, 4, -1, -4], [-5, 5], [ -40, 120]],
            [[ -1, -4, 1, 4], [-5, 5], [-150, 40]],
            [[ 1, 0, -16, 0], [-5, 5], [ -50, 50]],
            [[ -1, 0, 16, 0], [-5, 5], [ -70, 60]],
            [[ 1, 0, -22, 27], [-5, 5], [ -30, 70]],
            [[ -1, 0, 22, -27], [-5, 5], [ -70, 30]],
            [[ 1, 1, -13, -14], [-5, 5], [ -40, 40]],
            [[ -1, -1, 13, 14], [-5, 5], [ -40, 40]],
            [[ 1, 0, -12], [-5, 5], [ -20, 20]],
            [[ -1, 0, 12], [-5, 5], [ -20, 20]],
            [[ 1, -1, -6], [-5, 5], [ -20, 20]],
            [[ -1, 1, 6], [-5, 5], [ -30, 20]],
            [[ 1, 0, -3], [-5, 5], [ -20, 30]],
            [[ -1, 0, 3], [-5, 5], [ -30, 20]],
            [[ 1, 2, 0], [-5, 5], [ -10, 35]],
            [[ -1, -2, 0], [-5, 5], [ -40, 20]]
        ])</var>
        <var id="POLYNOMIAL">new Polynomial(0, COEF.length - 1, COEF.reverse())</var>

        <var id="FNX">function(x) {return POLYNOMIAL.evalOf(x);}</var>
        <var id="DDX">function(x) {return POLYNOMIAL.derivative().evalOf(x);}</var>
        <var id="D2DX">function(x) {return POLYNOMIAL.derivative().derivative().evalOf(x);}</var>

        <!-- Break XRANGE into intervals based on the roots of DDX -->
        <var id="DDX_INTERVALS">_.reduce(findRootsNumerically(DDX, XRANGE), function(intervals, root) {
            var last = _.last(intervals)
            return _.initial(intervals).concat([[last[0], root], [root, last[1]]]);
        }, [XRANGE])</var>
        <!-- Break XRANGE into intervals based on the roots of D2DX -->
        <var id="D2DX_INTERVALS">_.reduce(findRootsNumerically(D2DX, XRANGE), function(intervals, root) {
            var last = _.last(intervals)
            return _.initial(intervals).concat([[last[0], root], [root, last[1]]]);
        }, [XRANGE])</var>
        <!-- Find just the increasing intervals -->
        <var id="DDX_INTERVALS_POS">_.filter(DDX_INTERVALS, function(intv) {
            return DDX(intv[0] + (intv[1] - intv[0]) / 2) &gt; 0;
        })</var>
        <!-- Find just the decreasing intervals -->
        <var id="DDX_INTERVALS_NEG">_.filter(DDX_INTERVALS, function(intv) {
            return DDX(intv[0] + (intv[1] - intv[0]) / 2) &lt; 0;
        })</var>
        <!-- Find just the concave up intervals -->
        <var id="D2DX_INTERVALS_POS">_.filter(D2DX_INTERVALS, function(intv) {
            return D2DX(intv[0] + (intv[1] - intv[0]) / 2) &gt; 0;
        })</var>
        <!-- Find just the concave down intervals -->
        <var id="D2DX_INTERVALS_NEG">_.filter(D2DX_INTERVALS, function(intv) {
            return D2DX(intv[0] + (intv[1] - intv[0]) / 2) &lt; 0;
        })</var>
        <!-- Break XRANGE into intervals based on the roots of DDX and D2DX -->
        <!-- I.e. each interval has the same increasing-ness and concativity -->
        <var id="COMBINED_INTERVALS">_.reduce(
            sortNumbers(findRootsNumerically(DDX, XRANGE).concat(findRootsNumerically(D2DX, XRANGE))),
            function(intervals, root) {
                var last = _.last(intervals)
                return _.initial(intervals).concat([[last[0], root], [root, last[1]]]);
            },
            [XRANGE]
        )</var>
        <!-- Find intervals that are solutions -->
        <var id="SOLUTION_INTERVALS">_.filter(COMBINED_INTERVALS, function(intv) {
            return PREDICATE(intv[0] + (intv[1] - intv[0]) / 2);
        })</var>
        <!-- Sort to find the widest solution interval -->
        <var id="SOLUTION_INTERVAL">
            _.sortBy(SOLUTION_INTERVALS, function(intv) {
                return intv[0] - intv[1];
            })[0]
        </var>
    </div>

    <div class="solution" data-type="custom">
        <div class="instruction">
            Move the blue window to select part of the function.
        </div>
        <div class="guess">
            graph.slidingWindow.getX()
        </div>
        <div class="validator-function">
            var correct = _.reduce(_.range(guess, guess + 1, 0.02), function(correct, x) {
                return correct &amp;&amp; PREDICATE(x);
            }, true);
            if (!graph.moved &amp;&amp; !correct) {
                return ""
            }
            return correct;
        </div>
        <div class="show-guess">
            graph.slidingWindow.moveTo(guess, 0);
        </div>
    </div>

    <p class="question">
    </p>

    <div class="problem">
        <div class="graphie" id="fnplot">
            initAutoscaledGraph([XRANGE, YRANGE]);
            addMouseLayer();
            plot(FNX, XRANGE, {
                stroke: BLUE,
                strokeWidth: 3
            });
            graph.moved = false;
            // start the selection at the first zero of f'(x) which is
            // guranteed to be wrong but not give info about the right answer
            var startX = DDX_INTERVALS[0][1] - 0.5
            graph.slidingWindow = addRectGraph({
                x: startX,
                y: YRANGE[0],
                width: 1,
                height: YRANGE[1] - YRANGE[0],
                normalStyle: {
                    area: { "fill-opacity": 0.2 },
                    edges: { "stroke-width": 0 }
                },
                hoverStyle: {
                    area: { "fill-opacity": 0.3 }
                },
                fixed: {
                    points: [true, true, true, true],
                    edges: [true, true, true, true]
                },
                constraints: {
                    constrainX: false,
                    constrainY: true,
                    xmin: XRANGE[0],
                    xmax: XRANGE[1]
                },
                onMove: function() {
                    graph.moved = true;
                }
            });
        </div>
        <p class="graph-caption">
            <code class="hint_blue">f(x) = <var>POLYNOMIAL.text()</var></code>
        </p>
    </div>

    <div class="problems">
        <div id="inc-and-concave-up">
            <div class="vars" data-apply="prependVars">
                <var id="PREDICATE">function(x) { return DDX(x) &gt; 0 &amp;&amp; D2DX(x) &gt; 0; }</var>
            </div>
            <p class="question" data-apply="appendContents">
                A function <code>f(x)</code> is plotted below.
                Highlight an interval where <code>f^\prime(x) &gt; 0</code> and
                <code>f^{\prime\prime}(x) &gt; 0</code>.
            </p>
            <div class="hints">
                <p>
                    The first derivative, <code>f^\prime(x)</code>, is greater
                    than <code>0</code> wherever the function is increasing.
                </p>
                <div>
                    <p data-if="isSingular(DDX_INTERVALS_POS.length)">
                        The interval
                        where <code>f(x)</code> is increasing
                        is
                        <span class="hint_orange">highlighted</span> above.
                    </p><p data-else="">
                        The intervals
                        where <code>f(x)</code> is increasing
                        are
                        <span class="hint_orange">highlighted</span> above.
                    </p>
                    <div class="graphie" data-update="fnplot">
                        _.each(DDX_INTERVALS_POS, function(interval) {
                            plot(FNX, interval, {
                                stroke: ORANGE,
                                strokeWidth: 16,
                                opacity: 0.7
                            });
                        });
                    </div>
                </div>
                <p>
                    The second derivative, <code>f^{\prime\prime}(x)</code>, is greater
                    than <code>0</code> wherever the function is concave up.
                </p>
                <div>
                    <p data-if="isSingular(D2DX_INTERVALS_POS.length)">
                        The interval
                        where <code>f(x)</code> is concave up
                        is
                        <span class="hint_red">highlighted</span> above.
                    </p><p data-else="">
                        The intervals
                        where <code>f(x)</code> is concave up
                        are
                        <span class="hint_red">highlighted</span> above.
                    </p>
                    <div class="graphie" data-update="fnplot">
                        _.each(D2DX_INTERVALS_POS, function(interval) {
                            plot(FNX, interval, {
                                stroke: RED,
                                strokeWidth: 6,
                                opacity: 0.8
                            });
                        });
                    </div>
                </div>
                <div>
                    <p>
                        Select any part of the function that is highlighted for
                        both conditions.
                    </p>
                    <div class="graphie" data-update="fnplot">
                        graph.slidingWindow.moveTo(
                            (SOLUTION_INTERVAL[1] - SOLUTION_INTERVAL[0]) / 2 +
                            SOLUTION_INTERVAL[0] - 0.5, 0);
                    </div>
                </div>
            </div>
        </div>

        <div id="inc-and-concave-down">
            <div class="vars" data-apply="prependVars">
                <var id="PREDICATE">function(x) { return DDX(x) &gt; 0 &amp;&amp; D2DX(x) &lt; 0; }</var>
            </div>
            <p class="question" data-apply="appendContents">
                A function <code>f(x)</code> is plotted below.
                Highlight an interval where <code>f^\prime(x) &gt; 0</code> and
                <code>f^{\prime\prime}(x) &lt; 0</code>.
            </p>
            <div class="hints">
                <p>
                    The first derivative, <code>f^\prime(x)</code>, is greater
                    than <code>0</code> wherever the function is increasing.
                </p>
                <div>
                    <p data-if="isSingular(DDX_INTERVALS_POS.length)">
                        The interval
                        where <code>f(x)</code> is increasing
                        is
                        <span class="hint_orange">highlighted</span> above.
                    </p><p data-else="">
                        The intervals
                        where <code>f(x)</code> is increasing
                        are
                        <span class="hint_orange">highlighted</span> above.
                    </p>
                    <div class="graphie" data-update="fnplot">
                        _.each(DDX_INTERVALS_POS, function(interval) {
                            plot(FNX, interval, {
                                stroke: ORANGE,
                                strokeWidth: 16,
                                opacity: 0.7
                            });
                        });
                    </div>
                </div>
                <p>
                    The second derivative, <code>f^{\prime\prime}(x)</code>, is less
                    than <code>0</code> wherever the function is concave down.
                </p>
                <div>
                    <p data-if="isSingular(D2DX_INTERVALS_NEG.length)">
                        The interval
                        where <code>f(x)</code> is concave down
                        is
                        <span class="hint_red">highlighted</span> above.
                    </p><p data-else="">
                        The intervals
                        where <code>f(x)</code> is concave down
                        are
                        <span class="hint_red">highlighted</span> above.
                    </p>
                    <div class="graphie" data-update="fnplot">
                        _.each(D2DX_INTERVALS_NEG, function(interval) {
                            plot(FNX, interval, {
                                stroke: RED,
                                strokeWidth: 6,
                                opacity: 0.8
                            });
                        });
                    </div>
                </div>
                <div>
                    <p>
                        Select any part of the function that is highlighted for
                        both conditions.
                    </p>
                    <div class="graphie" data-update="fnplot">
                        graph.slidingWindow.moveTo(
                            (SOLUTION_INTERVAL[1] - SOLUTION_INTERVAL[0]) / 2 +
                            SOLUTION_INTERVAL[0] - 0.5, 0);
                    </div>
                </div>
            </div>
        </div>

        <div id="dec-and-concave-up">
            <div class="vars" data-apply="prependVars">
                <var id="PREDICATE">function(x) { return DDX(x) &lt; 0 &amp;&amp; D2DX(x) &gt; 0; }</var>
            </div>
            <p class="question" data-apply="appendContents">
                A function <code>f(x)</code> is plotted below.
                Highlight an interval where <code>f^\prime(x) &lt; 0</code> and
                <code>f^{\prime\prime}(x) &gt; 0</code>.
            </p>
            <div class="hints">
                <p>
                    The first derivative, <code>f^\prime(x)</code>, is less
                    than <code>0</code> wherever the function is decreasing.
                </p>
                <div>
                    <p data-if="isSingular(DDX_INTERVALS_NEG.length)">
                        The interval
                        where <code>f(x)</code> is decreasing
                        is
                        <span class="hint_orange">highlighted</span> above.
                    </p><p data-else="">
                        The intervals
                        where <code>f(x)</code> is decreasing
                        are
                        <span class="hint_orange">highlighted</span> above.
                    </p>
                    <div class="graphie" data-update="fnplot">
                        _.each(DDX_INTERVALS_NEG, function(interval) {
                            plot(FNX, interval, {
                                stroke: ORANGE,
                                strokeWidth: 16,
                                opacity: 0.7
                            });
                        });
                    </div>
                </div>
                <p>
                    The second derivative, <code>f^{\prime\prime}(x)</code>, is greater
                    than <code>0</code> wherever the function is concave up.
                </p>
                <div>
                    <p data-if="isSingular(D2DX_INTERVALS_POS.length)">
                        The interval
                        where <code>f(x)</code> is concave up
                        is
                        <span class="hint_red">highlighted</span> above.
                    </p><p data-else="">
                        The intervals
                        where <code>f(x)</code> is concave up
                        are
                        <span class="hint_red">highlighted</span> above.
                    </p>
                    <div class="graphie" data-update="fnplot">
                        _.each(D2DX_INTERVALS_POS, function(interval) {
                            plot(FNX, interval, {
                                stroke: RED,
                                strokeWidth: 6,
                                opacity: 0.8
                            });
                        });
                    </div>
                </div>
                <div>
                    <p>
                        Select any part of the function that is highlighted for
                        both conditions.
                    </p>
                    <div class="graphie" data-update="fnplot">
                        graph.slidingWindow.moveTo(
                            (SOLUTION_INTERVAL[1] - SOLUTION_INTERVAL[0]) / 2 +
                            SOLUTION_INTERVAL[0] - 0.5, 0);
                    </div>
                </div>
            </div>
        </div>

        <div id="dec-and-concave-down">
            <div class="vars" data-apply="prependVars">
                <var id="PREDICATE">function(x) { return DDX(x) &lt; 0 &amp;&amp; D2DX(x) &lt; 0; }</var>
            </div>
            <p class="question" data-apply="appendContents">
                A function <code>f(x)</code> is plotted below.
                Highlight an interval where <code>f^\prime(x) &lt; 0</code> and
                <code>f^{\prime\prime}(x) &lt; 0</code>.
            </p>
            <div class="hints">
                <p>
                    The first derivative, <code>f^\prime(x)</code>, is less
                    than <code>0</code> wherever the function is decreasing.
                </p>
                <div>
                    <p data-if="isSingular(DDX_INTERVALS_NEG.length)">
                        The interval
                        where <code>f(x)</code> is decreasing
                        is
                        <span class="hint_orange">highlighted</span> above.
                    </p><p data-else="">
                        The intervals
                        where <code>f(x)</code> is decreasing
                        are
                        <span class="hint_orange">highlighted</span> above.
                    </p>
                    <div class="graphie" data-update="fnplot">
                        _.each(DDX_INTERVALS_NEG, function(interval) {
                            plot(FNX, interval, {
                                stroke: ORANGE,
                                strokeWidth: 16,
                                opacity: 0.7
                            });
                        });
                    </div>
                </div>
                <p>
                    The second derivative, <code>f^{\prime\prime}(x)</code>, is less
                    than <code>0</code> wherever the function is concave down.
                </p>
                <div>
                    <p data-if="isSingular(D2DX_INTERVALS_NEG.length)">
                        The interval
                        where <code>f(x)</code> is concave down
                        is
                        <span class="hint_red">highlighted</span> above.
                    </p><p data-else="">
                        The intervals
                        where <code>f(x)</code> is concave down
                        are
                        <span class="hint_red">highlighted</span> above.
                    </p>
                    <div class="graphie" data-update="fnplot">
                        _.each(D2DX_INTERVALS_NEG, function(interval) {
                            plot(FNX, interval, {
                                stroke: RED,
                                strokeWidth: 6,
                                opacity: 0.8
                            });
                        });
                    </div>
                </div>
                <div>
                    <p>
                        Select any part of the function that is highlighted for
                        both conditions.
                    </p>
                    <div class="graphie" data-update="fnplot">
                        graph.slidingWindow.moveTo(
                            (SOLUTION_INTERVAL[1] - SOLUTION_INTERVAL[0]) / 2 +
                            SOLUTION_INTERVAL[0] - 0.5, 0);
                    </div>
                </div>
            </div>
        </div>

    </div>

</div>
</body>
</html>
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