Finding the Equation of a Tangent Line (Improved)
2021-01-03 12:55
This GeoGebra example demonstrates the complete steps for finding the equation of a tangent line to a curve. By utilizing core commands such as Tangent, Point, Vector, and OrthogonalLine, it dynamically illustrates the construction process of a tangent line at a specific point on a function's graph and the derivation of its equation. This improved model combines rich text explanations with geometr
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.570.0" app="classic" platform="d" id="7743DA0B-3B4A-4730-A10A-A82A84EB3441" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1641" height="971" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.9244363193174894" orientation="1" />
</panes>
<views>
<view id="512" toolbar="0 | 1 501 5 19 , 67 | 2 15 45 18 , 7 37 | 514 3 9 , 13 44 , 47 | 16 51 | 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 , 538 | 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 || 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 || 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="true" location="3" size="1623" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="true" location="3" size="366" window="100,100,600,400" />
<view id="16" visible="false" inframe="false" stylebar="false" location="1" size="300" window="50,50,500,500" />
<view id="32" visible="false" inframe="false" stylebar="true" location="1" size="300" window="50,50,500,500" />
<view id="64" toolbar="0" visible="false" inframe="false" stylebar="false" location="1" size="480" window="50,50,500,500" />
<view id="4097" visible="false" inframe="false" stylebar="true" location="1" size="380" window="100,100,700,550" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="false" stylebar="true" location="1" size="900" window="50,50,500,500" />
</views>
<toolbar show="true" items="0 77 73 62 | 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 14 68 | 30 29 54 32 31 33 | 25 17 26 60 52 61 | 40 41 42 , 27 28 35 , 6" position="1" help="false" />
<input show="true" cmd="true" top="algebra" />
<dockBar show="false" east="false" />
</perspective>
</perspectives>
<labelingStyle val="1"/>
<font size="24"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="1623" height="729"/>
<coordSystem xZero="1151.071649147727" yZero="431.668058238636" scale="51.337357954545446" yscale="51.337357954545446"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="-1"/>
<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="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<tableview min="-2.0" max="2.0" step="1.0"/>
<scripting blocked="false" disabled="false"/>
<construction title="求切线方程" author="" date="">
<expression label="text1" exp=""过点P（1，-1）作圆x^{2}+y^{2}-2x-2y+1=0\\的切线，试求切线方程。""/>
<element type="text" label="text1">
<show object="true" label="true" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.0" size="0" style="0"/>
<startPoint x="-18.06277326618985" y="7.58426891523595" z="1.0"/>
</element>
<expression label="eq1" exp="x^(2) + y^(2) - (2 * x) - (2 * y) + 1 = 0" type="conic"/>
<element type="conic" label="eq1">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eigenvectors x0="1.0" y0="0.0" z0="1.0" x1="0.0" y1="1.0" z1="1.0"/>
<matrix A0="1.0" A1="1.0" A2="1.0" A3="0.0" A4="-1.0" A5="-1.0"/>
<eqnStyle style="user"/>
</element>
<expression label="C" exp="(1, 1)" type="point"/>
<element type="point" label="C">
<show object="true" label="true" ev="4"/>
<objColor r="21" g="101" b="192" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="1.0" y="1.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<expression label="P" exp="(1, -1)" type="point"/>
<element type="point" label="P">
<show object="true" label="true"/>
<objColor r="21" g="101" b="192" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-13" y="44"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="1.0" y="-1.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Tangent">
<input a0="P" a1="eq1"/>
<output a0="f" a1="g"/>
</command>
<element type="line" label="f">
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-1.5" y="0.8660254037844386" z="2.3660254037844384"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="explicit"/>
</element>
<element type="line" label="g">
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-1.5" y="-0.8660254037844386" z="0.6339745962155614"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="explicit"/>
</element>
<element type="boolean" label="a">
<value val="true"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="244" y="206"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<ggbscript onUpdate="b=false
c=false
"/>
<caption val="第一步，判断点与圆的位置关系"/>
</element>
<expression label="text2" exp=""∵\left| PC\right|=\sqrt{\left(1-1 \right) ^{2}+\left( -1-1\right) ^{2}}=2>r\\∴点P在圆外""/>
<element type="text" label="text2">
<show object="true" label="true" ev="40"/>
<condition showObject="a ≟ true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="false" sizeM="1.0" size="0" style="0"/>
<startPoint x="-16.355378304718954" y="3.407193254292162" z="1.0"/>
</element>
<element type="boolean" label="b">
<value val="true"/>
<show object="true" label="true"/>
<condition showObject="a ≟ true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="249" y="377"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<caption val="第二步，求圆心到切线的距离"/>
</element>
<expression label="text3" exp=""设切线方程为y+1=k\left( x-1\right)\\即 kx-y+\left( -1-k\right)\\圆心到切线的距离d=\frac{\left| k-1+\left( -1-k\right)\right| }{\sqrt{k^{2}+\left(-1\right) ^{2}}}=\frac{2}{\sqrt{k^{2}+1}}\\由于圆心到切线的距离等于半径，所以\\\frac{2}{\sqrt{k^{2}+1}}=1,得k=±\sqrt{3}""/>
<element type="text" label="text3">
<show object="true" label="true" ev="40"/>
<condition showObject="b ≟ true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.0" size="0" style="0"/>
<startPoint x="-16.23005614044797" y="-0.040461740658244816" z="1.0"/>
</element>
<command name="Point">
<input a0="xAxis"/>
<output a0="A"/>
</command>
<element type="point" label="A">
<show object="false" label="true" ev="4"/>
<objColor r="21" g="101" b="192" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-3.9342041973908093" y="0.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Point">
<input a0="xAxis"/>
<output a0="B"/>
</command>
<element type="point" label="B">
<show object="false" label="true"/>
<objColor r="21" g="101" b="192" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="8.629745832353393" y="0.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Vector">
<input a0="A" a1="B"/>
<output a0="u"/>
</command>
<element type="vector" label="u">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="12.563950029744202" y="0.0" z="0.0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<coordStyle style="cartesian"/>
<startPoint exp="A"/>
</element>
<command name="Point">
<input a0="yAxis"/>
<output a0="D"/>
</command>
<element type="point" label="D">
<show object="false" label="true" ev="4"/>
<objColor r="21" g="101" b="192" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="0.0" y="-2.6671014484733573" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Point">
<input a0="yAxis"/>
<output a0="E"/>
</command>
<element type="point" label="E">
<show object="false" label="true"/>
<objColor r="21" g="101" b="192" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="0.0" y="5.591991284257709" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Vector">
<input a0="D" a1="E"/>
<output a0="v"/>
</command>
<element type="vector" label="v">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.0" y="8.259092732731066" z="0.0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<coordStyle style="cartesian"/>
<startPoint exp="D"/>
</element>
<expression label="text4" exp=""O""/>
<element type="text" label="text4">
<show object="true" label="true" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<startPoint x="-0.5643385304981783" y="-0.5828492729964136" z="1.0"/>
</element>
<expression label="text5" exp=""x""/>
<element type="text" label="text5">
<show object="true" label="true" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<startPoint x="8.220686994175676" y="-0.6607652421731218" z="1.0"/>
</element>
<expression label="text6" exp=""y""/>
<element type="text" label="text6">
<show object="true" label="true" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<startPoint x="-0.5253805459098241" y="5.046579500020752" z="1.0"/>
</element>
<expression label="text7" exp=""1""/>
<element type="text" label="text7">
<show object="true" label="true" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<startPoint x="0.9745018607418094" y="-0.3685803577604661" z="1.0"/>
</element>
<expression label="text8" exp=""1""/>
<element type="text" label="text8">
<show object="true" label="true" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<startPoint x="-0.3695486075564076" y="1.014428095126104" z="1.0"/>
</element>
<expression label="text9" exp=""-1""/>
<element type="text" label="text9">
<show object="true" label="true" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<startPoint x="-0.5643385304981782" y="-1.167219041821725" z="1.0"/>
</element>
<expression label="text10" exp=""-1""/>
<element type="text" label="text10">
<show object="true" label="true" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<startPoint x="-1.129229307029313" y="-0.5438912884080594" z="1.0"/>
</element>
<command name="Point">
<input a0="g"/>
<output a0="F"/>
</command>
<element type="point" label="F">
<show object="false" label="true" ev="4"/>
<objColor r="21" g="101" b="192" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-2.012465138935603" y="4.217742676666502" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Point">
<input a0="g"/>
<output a0="G"/>
</command>
<element type="point" label="G">
<show object="false" label="true"/>
<objColor r="21" g="101" b="192" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="1.7277003989602926" y="-2.260414063687369" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Point">
<input a0="f"/>
<output a0="H"/>
</command>
<element type="point" label="H">
<show object="false" label="true" ev="4"/>
<objColor r="21" g="101" b="192" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="0.30335952734039584" y="-2.206616693255232" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Point">
<input a0="f"/>
<output a0="I"/>
</command>
<element type="point" label="I">
<show object="false" label="true"/>
<objColor r="21" g="101" b="192" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="3.8128301636750788" y="3.871964756547518" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="F" a1="G"/>
<output a0="h"/>
</command>
<element type="segment" label="h">
<show object="true" label="false" ev="4"/>
<condition showObject="c"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="6.478156740353871" y="3.740165537895896" z="-2.737991202457975"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="explicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="H" a1="I"/>
<output a0="i"/>
</command>
<element type="segment" label="i">
<show object="true" label="false" ev="4"/>
<condition showObject="c"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-6.0785814498027495" y="3.509470636334683" z="9.588052086137433"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="explicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="boolean" label="c">
<value val="true"/>
<show object="true" label="true"/>
<condition showObject="a ≟ true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="664" y="207"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<caption val="切线"/>
</element>
<command name="OrthogonalLine">
<input a0="C" a1="i"/>
<output a0="j"/>
</command>
<element type="line" label="j">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-3.509470636334683" y="-6.0785814498027495" z="9.588052086137433"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="j" a1="i"/>
<output a0="M"/>
</command>
<element type="point" label="M">
<show object="true" label="true" ev="4"/>
<condition showObject="c ≟ true"/>
<objColor r="97" g="97" b="97" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="9" y="10"/>
<labelMode val="0"/>
<coords x="91.93074280648437" y="24.632768294590733" z="49.265536589181465"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="C" a1="M"/>
<output a0="k"/>
</command>
<element type="segment" label="k">
<show object="true" label="false" ev="4"/>
<condition showObject="c ≟ true"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.5" y="0.8660254037844386" z="-1.3660254037844386"/>
<lineStyle thickness="5" type="10" typeHidden="1" opacity="127"/>
<eqnStyle style="explicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
</construction>
</geogebra>