00020001010008main.ACT0002020009deriv.EAC0100000014d4  deriv.EAC main.ACT )- $kE ('I[Z&About D`atives of sin(x) and ta .\6 AuthorM{Professor Wei-Chi Yang Radford University  , VA 241424U.S.A.Ce- l: wyang@rQ.eduJURLww./.Ώ Objec[W(1) We will demonstrate onay of 'sketching' the graphs derivativesin(x)  and ta.['(2tmake useOequati, FTt)T2H+!cos!"~=1 x1+SS=tecS@, to mo% respec!ly.[ Example 1. [shaSddxdsin(x)=cos graphically.\! Tap hereΈ5; # `F(x .  DvIKr  AkL'C uxy qv [Step 1.-Insert a Geometry strip. "-Select the coordinate systemH-Under 'Draw', s4function' -Type in 22F-Under Draw, select Construct->Tangent to a curve.[-Tap the poi"and%'-EdiGAnimate->Add a ion->Go onc/UStep 3>6-SfonsSinj tabl?Y? 'slope' of&lEa^Ď֌L This shouldwoV`s4TN7drag.x-column1JoU graph, we}eensketch, of the graph derivativeSine function.[R-Drag7y-column and slopeintoZ, we seel a circle. Z[ Conjecture:m 6-Let's guessequonD is x^2+y^2=1Q?M8' back dowo s almost identicalJ-Therefore, if use x=sin(t)y=cos ar"in[h>w2+!cos(t)2>=1, and this suggestsat!e derivative of sin(x) should be M\x).[] Remark.[u!Symbolically, we can compute   f(x+h)-)h0) directly or by ugfollowstep# approaches:Rdefine )=Rӌdone.01qӏ-x:Z) tExp((f(x+h)- )h )Rcos$xsin;+x,B,W-B+h[u Notice that () can be written as [{e@*-1 +f(x)8 but R  cos h-10R JsinI@1[We conclude tha f(x+h)-)Ґxo Exercise 1.AU#the similiar procesure described in$Example to show[O ddxk(x)=-)(x) graphically\Try your answer here  q `F(x .c Dv$Kr  A kL'xC uxy v [̎ Example 2.z,L(a) Use the similar procesure described in#A1 tomonstrat@at[Wddxh (tan(x))=sec}2x Solution:R?M define secJ=1/cosSRdone[ R-Repeat the Steps 1 rough 4 to sketch &graphs of y=tan(x) and its derivative f togeUr.k V-We drag <backunaseey satisfy a parabolico equation.[ ConjectureWcюving 1+x^2=y, where xty TVis the derivative. (Dragequon 1+x^2=y back to3geometry strip below, wL0o #see two parabolas amost identical.)[.-Noticezof}Qtan(t)_2;=!sec!"'. Thsuggestatjh x) should be `x`.\Tyour answer hereΉΊ ! `F(y 4S x Dvr  A "L' C uxy c p( [b,:(b) Explore the parametic equation [x(t),y]=[tansecC2K t] by using `1+ ;6=!UjX.[Hint: See below.\A bonusrЉI^ʋx4N<%FinaForm$NGraph2D 3, LISTSYS8@4< Modify lP<STATCALC d< \x S:equence,xSheetO | olveEq`wr(Up(tupFLG14(<Lis{\DPicViewWind_osvev.dxt_ Hy2( ( <O )  i1T`lx ̒؆ !, "@"#TF $h,%|8 &D !'P(\  )`hd*t+N-.012Ȇ3Ԇ45EF H I@ J^Kh L@MNX'Odņ PpnQr|RST]j } ^_`ab ĒІ͆܆Ά2/ב@XؑT ّh ڑ|$ 6ۑ0 FinancialFormat  +! system]list^]=_`a "b~u @ @x# ! a/ seq_hb NewFoldeF 0 M<   <Tangent to a curve.[-Tap the poi"and%'-EdiGAnimate->Add a ion->Go onc/UStep 3>6-SfonsSinj tabl?Y? 'slope' of&lEa^Ď֌L This shouldwoV`s4TN7drag.x-column1JoU graph, we}eensketch, of the graph derivativeSine function.[R-Drag7y-column and slopeintoZ, we seel a circle. Z[ Conjecture:m 6-Let's guessequonD is x^2+y^2=1Q?M8' back dowo s almost identicalJ-Therefore, if use x=sin(t)y=cos ar"in[h>w2+!cos(t)2>=1, and this suggestsat!e derivative of sin(x) should be M\x).[] Remark.[u!Symbolically, we can compute   f(x+h)-)h0) directly or by ugfollowstep# approaches:Rdefine )=Rӌdone.01qӏ-x:Z) tExp((f(x+h)- )h )Rcos$xsin;+x,B,W-B+h[u Notice that () can be written as [{e@*-1 +f(x)8 but R  cos h-10R JsinI@1[We conclude tha f(x+h)-)Ґxo Exercise 1.AU#the similiar procesure described in$Example to show[O ddxk(x)=-)(x) graphically\Try your answer here  q `F(x .c Dv$Kr  A kL'xC uxy v [̎ Example 2.z,L(a) Use the similar procesure described in#A1 tomonstrat@at[Wddxh (tan(x))=sec}2x Solution:R?M define secJ=1/cosSRdone[ R-Repeat the Steps 1 rough 4 to sketch &graphs of y=tan(x) and its derivative f togeUr.k V-We drag <backunaseey satisfy a parabolico equation.[ ConjectureWcюving 1+x^2=y, where xty TVis the derivative. (Dragequon 1+x^2=y back to3geometry strip below, wL0o #see two parabolas amost identical.)[.-Noticezof}Qtan(t)_2;=!sec!"'. 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