Geometric Meaning of Derivatives and Tangent Line Construction
2017-04-21 06:10
This applet demonstrates the geometric meaning of derivatives on function graphs, illustrating tangent line construction for parabolas and piecewise functions. Users interact with buttons and points to observe tangent changes, understanding the relationship between derivatives and slopes. It offers a visual exploration of calculus and function properties.
Loading the GeoGebra board and drawing, please wait…
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.317.0" id="64575a18-7bb2-4f97-8f70-813ab83bf870" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1464" height="969" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.5655737704918032" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1,1" size="400" window="348,402,764,559" />
<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="true" 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="true" location="1,1" size="300" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="false" location="3,3" size="295" window="100,100,250,400" />
<view id="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 || 1008 1009 || 66 68 || 6" visible="false" inframe="false" stylebar="true" location="1,3" size="300" window="100,100,600,400" />
<view id="1" visible="true" inframe="false" stylebar="false" location="3" size="828" window="100,100,600,400" />
<view id="16" visible="true" inframe="false" stylebar="false" location="1" size="609" 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" toolbar="0" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" 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 | 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 | 100006 100002 100001" position="1" help="true" />
<input show="true" cmd="true" top="true" />
<dockBar show="false" east="true" />
</perspective>
</perspectives>
<labelingStyle val="0"/>
<font size="28"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="828" height="684"/>
<coordSystem xZero="106.91036108585624" yZero="553.7796632292791" scale="69.25560809286783" yscale="69.25559040467633"/>
<evSettings axes="true" grid="false" 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="x" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="true" label="y" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<euclidianView>
<viewNumber viewNo="2"/>
<size width="609" height="684"/>
<coordSystem xZero="196.30000000000015" yZero="284.0000000000066" scale="50.00000000000003" yscale="50.00000000000109"/>
<evSettings axes="false" grid="false" 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="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>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<element type="button" label="按钮1">
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="1.0"/>
<layer val="0"/>
<labelOffset x="145" y="84"/>
<labelMode val="0"/>
<fixed val="true"/>
<isShape val="false"/>
<auxiliary val="true"/>
<ggbscript val="n=n-1 SetValue[a,if[n<1,0,1]] SetValue[b,if[n<2,0,1]] SetValue[c,if[n<3,0,1]] SetValue[d,if[n<4,0,1]] StartAnimation[a,false] StartAnimation[b,false] StartAnimation[c,false] StartAnimation[d,false]"/>
<font serif="false" sizeM="2.0" size="28" style="0"/>
<file name="/geogebra/gui/images/go-previous.png"/>
</element>
<element type="button" label="按钮2">
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="1.0"/>
<layer val="0"/>
<labelOffset x="245" y="84"/>
<labelMode val="0"/>
<fixed val="true"/>
<isShape val="false"/>
<auxiliary val="true"/>
<ggbscript val="SetValue[a,if[n<1,0,1]] SetValue[b,if[n<2,0,1]] SetValue[c,if[n<3,0,1]] SetValue[d,if[n<4,0,1]] StartAnimation[a,if[n==0,true,false]] StartAnimation[b,if[n==1,true,false]] StartAnimation[c,if[n==2,true,false]] StartAnimation[d,if[n==3,true,false]] n=n+1"/>
<font serif="false" sizeM="2.0" size="28" style="0"/>
<file name="/geogebra/gui/images/go-next.png"/>
</element>
<element type="button" label="按钮3">
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="1.0"/>
<layer val="0"/>
<labelOffset x="345" y="84"/>
<labelMode val="0"/>
<fixed val="true"/>
<isShape val="false"/>
<auxiliary val="true"/>
<ggbscript val="n=0 a=0 b=0 c=0 d=0 StartAnimation[a,false] StartAnimation[b,false] StartAnimation[c,false] StartAnimation[d,false]"/>
<font serif="false" sizeM="2.0" size="28" style="0"/>
<file name="/geogebra/gui/images/nav_skipback.png"/>
</element>
<expression label ="a" exp="0"/>
<element type="numeric" label="a">
<value val="0.0"/>
<symbolic val="true" />
<show object="false" label="true" ev="3"/>
<objColor r="0" g="153" b="255" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="1" absoluteScreenLocation="true" width="250.0" x="77.0" y="167.0" fixed="true" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.01" speed="5" type="3" playing="false"/>
</element>
<expression label ="b" exp="0"/>
<element type="numeric" label="b">
<value val="0.0"/>
<symbolic val="true" />
<show object="false" label="true" ev="3"/>
<objColor r="0" g="153" b="255" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="1" absoluteScreenLocation="true" width="250.0" x="81.0" y="218.0" fixed="true" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.01" speed="5" type="3" playing="false"/>
</element>
<expression label ="c" exp="0"/>
<element type="numeric" label="c">
<value val="0.0"/>
<symbolic val="true" />
<show object="false" label="true" ev="3"/>
<objColor r="0" g="153" b="255" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="1" absoluteScreenLocation="true" width="250.0" x="71.0" y="265.0" fixed="true" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.01" speed="5" type="3" playing="false"/>
</element>
<expression label ="d" exp="0"/>
<element type="numeric" label="d">
<value val="0.0"/>
<symbolic val="true" />
<show object="false" label="true" ev="3"/>
<objColor r="0" g="153" b="255" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="1" absoluteScreenLocation="true" width="250.0" x="88.0" y="307.0" fixed="true" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.01" speed="5" type="3" playing="false"/>
</element>
<expression label ="n" exp="0"/>
<element type="numeric" label="n">
<value val="0.0"/>
<symbolic val="true" />
<show object="false" label="true" ev="3"/>
<objColor r="0" g="153" b="255" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="4" absoluteScreenLocation="true" width="250.0" x="94.0" y="361.0" fixed="true" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="3" playing="false"/>
</element>
<command name="If">
<input a0="n ≟ 1" a1="a" a2="If[n ≟ 2, 1 + b, If[n ≟ 3, 2 + c, If[n ≟ 4, 3 + d, 0]]]"/>
<output a0="m"/>
</command>
<element type="numeric" label="m">
<value val="0.0"/>
<symbolic val="true" />
<objColor r="0" g="153" b="255" alpha="0.10000000149011612"/>
</element>
<expression label ="f" exp="f(x) = (1 / 20 * x^(2.5))"/>
<element type="function" label="f">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="153" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<lineStyle thickness="13" type="0" typeHidden="1"/>
</element>
<element type="point" label="A">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="204" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-3" y="59"/>
<labelMode val="3"/>
<fixed val="true"/>
<isShape val="false"/>
<coords x="3.0" y="0.0" z="1.0"/>
<pointSize val="7"/>
<pointStyle val="0"/>
<caption val="$\large{x_0}$"/>
</element>
<expression label="B" exp="(x(A), f(x(A)))" type="point" />
<element type="point" label="B">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="204" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-31" y="0"/>
<labelMode val="3"/>
<fixed val="true"/>
<isShape val="false"/>
<coords x="3.0" y="0.7794228634059949" z="1.0"/>
<pointSize val="7"/>
<pointStyle val="0"/>
<caption val="$\large{P}$"/>
</element>
<command name="Tangent">
<input a0="B" a1="f"/>
<output a0="g"/>
</command>
<element type="line" label="g">
<show object="true" label="false" ev="4"/>
<condition showObject="e"/>
<objColor r="255" g="102" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isShape val="false"/>
<coords x="-0.649519052838329" y="1.0" z="1.169134295108992"/>
<lineStyle thickness="9" type="0" typeHidden="1"/>
<eqnStyle style="explicit"/>
</element>
<expression label="C" exp="(x(A) + 4 - m, 0)" type="point" />
<element type="point" label="C">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="204" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="1" y="14"/>
<labelMode val="3"/>
<fixed val="true"/>
<isShape val="false"/>
<coords x="7.0" y="0.0" z="1.0"/>
<pointSize val="7"/>
<pointStyle val="0"/>
<caption val="$\large{x_n}$"/>
</element>
<expression label="D" exp="(x(C), f(x(C)))" type="point" />
<element type="point" label="D">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="204" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-61" y="14"/>
<labelMode val="3"/>
<fixed val="true"/>
<isShape val="false"/>
<coords x="7.0" y="6.482090712108247" z="1.0"/>
<pointSize val="7"/>
<pointStyle val="0"/>
<caption val="$\large{P_n}$"/>
</element>
<command name="Line">
<input a0="B" a1="D"/>
<output a0="h"/>
</command>
<element type="line" label="h">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="0" b="51" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isShape val="false"/>
<coords x="-5.702667848702252" y="4.0" z="13.99031209248278"/>
<lineStyle thickness="9" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="Segment">
<input a0="B" a1="A"/>
<output a0="i"/>
</command>
<element type="segment" label="i">
<show object="true" label="false" ev="4"/>
<objColor r="96" g="96" b="96" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isShape val="false"/>
<coords x="0.7794228634059949" y="0.0" z="-2.3382685902179845"/>
<lineStyle thickness="9" type="15" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="D" a1="C"/>
<output a0="j"/>
</command>
<element type="segment" label="j">
<show object="true" label="false" ev="4"/>
<objColor r="96" g="96" b="96" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isShape val="false"/>
<coords x="6.482090712108247" y="0.0" z="-45.37463498475773"/>
<lineStyle thickness="9" type="15" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Point">
<input a0="g"/>
<output a0="E"/>
</command>
<element type="point" label="E">
<show object="true" label="true" ev="4"/>
<condition showObject="e"/>
<objColor r="0" g="204" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-1" y="59"/>
<labelMode val="3"/>
<fixed val="true"/>
<isShape val="false"/>
<coords x="7.987657994797936" y="4.01900176006867" z="1.0"/>
<pointSize val="7"/>
<pointStyle val="0"/>
<caption val="$\large{T}$"/>
</element>
<expression label="文本2" exp=""显示曲线在x=x_0处的切线""/>
<element type="text" label="文本2">
<show object="true" label="true" ev="43"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.0" size="0" style="0"/>
<absoluteScreenLocation x="72" y="276"/>
</element>
<expression label="文本3" exp=""k_{PT}=" + (LaTeX[Slope[g]]) + """ />
<element type="text" label="文本3">
<show object="true" label="true" ev="43"/>
<condition showObject="e"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.0" size="0" style="0"/>
<absoluteScreenLocation x="115" y="344"/>
</element>
<expression label="文本4" exp=""显示结论""/>
<element type="text" label="文本4">
<show object="true" label="true" ev="43"/>
<condition showObject="e"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<font serif="true" sizeM="1.1" size="3" style="0"/>
<absoluteScreenLocation x="75" y="458"/>
</element>
<expression label="文本5" exp=""f'(x_0)=\lim_{ Δx \to 0 }\frac{f(x_0+Δx)-f(x_0)}{Δx}=k_{PT}""/>
<element type="text" label="文本5">
<show object="true" label="true" ev="43"/>
<condition showObject="k"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.0" size="0" style="0"/>
<absoluteScreenLocation x="49" y="489"/>
</element>
<element type="boolean" label="e">
<value val="false"/>
<show object="true" label="false" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="40" y="279"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<ggbscript onUpdate="k=false"/>
</element>
<expression label ="k" exp="false"/>
<element type="boolean" label="k">
<value val="false"/>
<show object="true" label="false" ev="3"/>
<condition showObject="e"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="40" y="429"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
</element>
<expression label="文本1" exp=""k_{PP_n}=\frac{f(x_n)-f(x_0)}{x_n-x_0}" + (LaTeX[If[m ≟ 4, ">0.65", ("=" * Slope[h])]]) + """ />
<element type="text" label="文本1">
<show object="true" label="true" ev="11"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.0" size="0" style="0"/>
<absoluteScreenLocation x="103" y="142"/>
</element>
</construction>
</geogebra>