极值对称于反曲点
2016-04-18 13:46
本范例演示了函数极值点关于其反曲点(拐点)对称的数学性质。通过GeoGebra绘制函数图像,定位极值点与反曲点,并利用交点和正交线等几何构造工具,直观验证极值点关于反曲点的对称关系。适用于微积分与函数图像分析的教学与探索。
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.226.0" id="4b30963a-60c1-49ce-ad9b-9494ec839bb3" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1936" height="1056" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.10330578512396695" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1,1" size="400" window="610,245,700,550" />
<view id="512" toolbar="0 | 1 501 5 19 , 67 | 2 15 45 18 , 7 37 | 514 3 9 , 13 44 , 47 | 16 | 551 550 11 , 20 22 21 23 , 55 56 57 , 12 | 69 | 510 511 , 512 513 | 533 531 , 534 532 , 522 523 , 537 536 , 535 | 521 520 | 36 , 38 49 560 | 571 30 29 570 31 33 | 17 | 540 40 41 42 , 27 28 35 , 6 , 502" visible="false" inframe="false" stylebar="false" location="1,1,1" size="500" window="100,100,600,400" />
<view id="4" toolbar="0 || 2020 , 2021 , 2022 , 66 || 2001 , 2003 , 2002 , 2004 , 2005 || 2040 , 2041 , 2042 , 2044 , 2043" visible="false" inframe="false" stylebar="false" location="1,1" size="300" window="100,100,600,400" />
<view id="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 || 1008 1009 || 66 68 || 6" visible="false" inframe="false" stylebar="false" location="1,3" size="300" window="100,100,600,400" />
<view id="1" visible="true" inframe="false" stylebar="false" location="1" size="1704" window="100,100,600,400" />
<view id="2" visible="true" inframe="false" stylebar="false" location="3" size="200" window="100,100,250,400" />
<view id="16" visible="false" inframe="false" stylebar="false" location="1" size="150" window="50,50,500,500" />
<view id="32" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="64" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="70" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
</views>
<toolbar show="true" items="0 39 | 1 501 67 , 5 19 , 72 75 76 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 | 16 51 64 , 70 | 10 34 53 11 , 24 20 22 , 21 23 | 55 56 57 , 12 | 36 46 , 38 49 50 , 71 | 30 29 54 32 31 33 | 17 26 62 73 , 14 68 | 25 52 60 61 | 40 41 42 , 27 28 35 , 6" position="1" help="false" />
<input show="true" cmd="true" top="algebra" />
<dockBar show="true" east="true" />
</perspective>
</perspectives>
<labelingStyle val="0"/>
<font size="12"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="1704" height="889"/>
<coordSystem xZero="439.9999999999999" yZero="521.0" scale="50.0" yscale="50.000000000000014"/>
<evSettings axes="true" grid="true" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="0"/>
<bgColor r="255" g="255" b="255"/>
<axesColor r="0" g="0" b="0"/>
<gridColor r="192" g="192" b="192"/>
<lineStyle axes="1" grid="0"/>
<axis id="0" show="true" label="" unitLabel="" tickStyle="1" showNumbers="false"/>
<axis id="1" show="true" label="" unitLabel="" tickStyle="1" showNumbers="false"/>
</euclidianView>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0" spreadsheet="0"/>
<coordStyle val="0"/>
<angleFromInvTrig val="false"/>
</kernel>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<element type="numeric" label="a">
<value val="0.7"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="-5" max="5" absoluteScreenLocation="true" width="200.0" x="832.0" y="79.0" fixed="true" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.05" speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="b">
<value val="-3.25"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="-5" max="5" absoluteScreenLocation="true" width="200.0" x="832.0" y="129.0" fixed="true" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.05" speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="c">
<value val="2.65"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="-5" max="5" absoluteScreenLocation="true" width="200.0" x="832.0" y="179.0" fixed="true" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.05" speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="d">
<value val="3.35"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="-5" max="5" absoluteScreenLocation="true" width="200.0" x="832.0" y="229.0" fixed="true" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.05" speed="1" type="0" playing="false"/>
</element>
<expression label="f" exp="f(x) = (a * x^(3)) + (b * x^(2)) + (c * x) + d" />
<element type="function" label="f">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="204" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="15" typeHidden="1"/>
</element>
<expression label="e" exp="y = ((3 * a) * x^(2)) + ((2 * b) * x) + c" type="conic" />
<element type="conic" label="e">
<show object="true" label="true" ev="4"/>
<condition showObject="k"/>
<objColor r="51" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="7" type="0" typeHidden="1"/>
<eigenvectors x0="0.0" y0="1.0" z0="1.0" x1="-1.0" y1="0.0" z1="1.0"/>
<matrix A0="-2.0999999999999996" A1="0.0" A2="-2.65" A3="0.0" A4="3.25" A5="0.5"/>
<eqnStyle style="explicit"/>
</element>
<command name="Intersect">
<input a0="e" a1="xAxis" a2="2"/>
<output a0="A"/>
</command>
<element type="point" label="A">
<show object="true" label="false"/>
<condition showObject="m"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.48309100447104003" y="-0.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Intersect">
<input a0="e" a1="xAxis" a2="1"/>
<output a0="B"/>
</command>
<element type="point" label="B">
<show object="true" label="false"/>
<condition showObject="m"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="2.612147090767056" y="-0.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="OrthogonalLine">
<input a0="A" a1="xAxis"/>
<output a0="g"/>
</command>
<element type="line" label="g">
<show object="true" label="true" ev="4"/>
<condition showObject="m"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-1.0" y="0.0" z="0.48309100447104003"/>
<lineStyle thickness="2" type="15" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="OrthogonalLine">
<input a0="B" a1="xAxis"/>
<output a0="h"/>
</command>
<element type="line" label="h">
<show object="true" label="true" ev="4"/>
<condition showObject="m"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-1.0" y="0.0" z="2.612147090767056"/>
<lineStyle thickness="2" type="15" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="f" a1="g" a2="1"/>
<output a0="C"/>
</command>
<element type="point" label="C">
<show object="true" label="false"/>
<condition showObject="m"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.48309100447104003" y="3.950635779414596" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Intersect">
<input a0="f" a1="h" a2="1"/>
<output a0="D"/>
</command>
<element type="point" label="D">
<show object="true" label="false"/>
<condition showObject="m"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="2.612147090767056" y="0.5728714012354437" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<expression label="i" exp="y = ((6 * a) * x) + (2 * b)" type="line" />
<element type="line" label="i">
<show object="true" label="true" ev="4"/>
<condition showObject="l"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-4.199999999999999" y="1.0" z="6.5"/>
<lineStyle thickness="7" type="10" typeHidden="1"/>
<eqnStyle style="explicit"/>
</element>
<command name="Intersect">
<input a0="i" a1="xAxis"/>
<output a0="E"/>
</command>
<element type="point" label="E">
<show object="true" label="false"/>
<condition showObject="反曲點"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="6.5" y="-0.0" z="4.199999999999999"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="OrthogonalLine">
<input a0="E" a1="xAxis"/>
<output a0="j"/>
</command>
<element type="line" label="j">
<show object="true" label="true" ev="4"/>
<condition showObject="反曲點"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-4.199999999999999" y="0.0" z="6.5"/>
<lineStyle thickness="2" type="15" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="f" a1="j" a2="1"/>
<output a0="I"/>
</command>
<element type="point" label="I">
<show object="true" label="true" ev="4"/>
<condition showObject="反曲點"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="1" y="-7"/>
<labelMode val="3"/>
<coords x="1.547619047619048" y="2.2617535903250183" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
<caption val="反曲點"/>
</element>
<element type="boolean" label="k">
<value val="true"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="844" y="274"/>
<labelMode val="4"/>
<checkbox fixed="true"/>
<caption val="一階導數"/>
</element>
<element type="boolean" label="l">
<value val="true"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="844" y="324"/>
<labelMode val="4"/>
<checkbox fixed="true"/>
<caption val="二階導數"/>
</element>
<element type="boolean" label="反曲點">
<value val="true"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="944" y="324"/>
<labelMode val="4"/>
<checkbox fixed="true"/>
</element>
<element type="boolean" label="m">
<value val="true"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="944" y="274"/>
<labelMode val="4"/>
<checkbox fixed="true"/>
<caption val="極大值與極小值"/>
</element>
</construction>
</geogebra>