00020001010008main.ACT0002020009deriv.EAC010000004ca6  deriv.EAC main.ACT )- @ `E ('j[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; #U `@Gs# .y ! DvIKr  AkL'C uxyqq H" u  XBtp qdP9P !6 uf D5b0E Y@Y  cB 6@ gff ')<Y`"` # $T<gT@Y` D5b ufP Y!6@  qdP9p XBt<YBFgAs8P6C!u,T(\Y a E7s\ ihU ihg0R&A4 "yaT eW30 ( $f<6$TlgT슜$wiO>H K `G6! shBBRc %C`/ !&`1WG G6YcxP  $ 1WG@1!&0 6 0 $BRc shB<w!  gAst6C!upT( a E7 ihUihg0R&A4"yaT eW3ǎ ($f<6TlgT@L$wdiO5@ /  MAx6 `50GpC &  / sin(x)BUNYy &1WGSZ N.@N 3"` ,z@6!XFsf4z \.H \r rz"@.1H2PH  iv[Step 1.-Insert a Geometry strip. "6-Select the coordinate system.[-Under 'Draw', s4function'-Type in sin(x)@Step 2'2NMLConstruct->Tangen o a curve:-Tappoi"and%'-EdiGAnim->Add a ->Go onc/3>6fonSinj tabl?YP 'slope' of&lEa^Ď֌L This shouldwoV`s.[Step 4.[TNow drag the x-column andslopeinto'graph, we should see@sketch [,of. derivativeSine function.[R-Dyˆl a circle. Z Conjecture:m 6-Let's guessequonD is x^2+y^2=1Q;M8' back 0do,wos almost4 identicalJ(-Therefore, if we use x=sin(t) and y=cos ,ar" ing the equation of [>2+!R!>=1,s this suggestsjatoderivqveqe x) should be \\x).[Remark. Symbolicallycan compute   f(x+h)-)h0) directly or by(follow6step# approaches:R?Edefine )x)Rdone. f(x+h)-) hR"sin-+-x:T8 tExpand((l)tucos^;+,B,[ Notice that ) can be written as [ sinxcos!h)-1 +&(x)98Y7? but Rs g0R Jy@1[We conclude tha f(x+h)-)؎9Yfoo Exercise 1.A'Use the similiar procesure described in$Exampl0o show8at [  ddx cos(x)=-sin graphically.[?\GTry your answer hereΈ$(m ve `@Gs$ .i DvKr  A kL'C uxy qv/ ̎ g2L*(a) Use the similar procesure described in# Example 1 tomonstrat@at[ddx (tan(x))=sec&2.x.[8 Solution:ROM define secJ=1/cosSR"done[R-RepeatStepshrough 4sketch graphs of y= ants rivative  toge0r.k V-We drag <back5nase{y satisfy a parabolicoequa %  Conjecture.[W-We c< that tan(x) and its derivative satisfying 1+x^2=y, where x=6t6y a$isUeEofb t), i.e. y=secC2). (DragHequation back toe?geometry strip below)e do see two parabolas amost identical.)[-Noticz1+'. This suggesfz*Htan(x) should be sec2.\ Try your answer hereΈ<@F Os `@Gs( .a>JDvKr  A kL'C uxy qqH  !Ј %?^^(8 C @ qJ6I&52H  % `I'P` Y12b 6!%@$Y0%& 4P R$ CGwTr6% wHStx%wFx %5ChFAwC"p(! $ < Tl```Y,,0, D \ t2H6I&8 @ q8( ($("" U(QfAV % p B$h0'@$@Ge00HP<#%r`a`T)$@VV` )w$A) wdsTWR8lUgRl8Tf5 $% `BTeCC !U tan(x)BN VAVAlZ2.HN3Ȗr"$@$cV Y)2$#$ S.H \r .@ 21H!2H 7h(!u[,:(b) Explore the parametic equation [x(t),y]=[tansecC2K t] by using `1+ ;6=!UjX.[Hint: See below.\A bonusrЈt4N<%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<   <110;869`9504+69s302603508+ 0.23566295 -3.163265306 1.000234898& 0.0216936$ 2.959183673716870N -0.187557678'2.102041q79636597: 439349283:9020391682M 0.807058922.34877 42748Ԕ454104488142857148471794 86I1729^n552.779K788 7.209791582z7& 6.128774554 -37.05282775 1.53061224 24.89217447% 618.7353451.326;K 4.134898556] 6.58813768 8l(o 2.E92846 -4.795647H -0.9183673469 1."12547 -2.15570754n9 7142857143932351926147460689 510204081Άr 14594154401276$ N40487574833145177p0288 -0.1029190617 0408163 1.00522884%944696 0.3F22449748757482\33148321o5p8 145941544 64132276387142853235192J754347879183673%64712547 2.155889785$ 1.89H 29284 4.79620373}3265$13D55 6.59156905324| 24.892174~ 619.5059083 1.734693878 -6.128774554% 37.064184977 1.375516 2.779857789[ 7.2107961652.142$14 -1.84717949%868960732.767427484542_7m55102040 1.20391682 0.807116026ۆ8׈91.36584355289591836701687032p187583.16!530ވ0002348 -0.02166864333 3.367346939 -1'6035082 :2356'067957142891000`9% s 5042766658 775510204q 241136256q9123q97959183^ 1.49487694^6609695^ 4.29824566193326844.38%1351715889.3151I54.U -8%28477d 68.6278759pޓb15767 -143.11487695 3.525320086- 1.91634641[.\6!Look at column 4 agains11(5;N SpreadsheetSk2#Y2@  yY6p,`sP8wU 66sFQlP@W(W!l2e0Ӣg0uQ3!@)4i8wlluQzsF`S$_a $HPQ lYl@0 @6IQQ Q6Ql)z_rW56sQ`l4@l6gll6Ww)pDm28RzY7yY6p@ x21RS `1vtP1Rw6;QQqX@$VaIHvlQ$6%lhi`5 4hp2l6Y 9pBt)qyIwWx`tU#HDp66HULQ0i(Fkdq%G 625&lYATWHQR(QQ)?S`??y""6wwBtQ 9l6Yhp2آ4`5 hil$6%lIHvP)$VaDQqX_1Rw`@vtPRS 1; S29D@9C ppo}oVAXo 1f#0o96ff Q CcQ 1V)5f)6fj GUvxlYl94K %6EAHpr X 6pRwPQsSEQzlev6yVGVWT6t`hA'aY1EvaF QH2A2'c0f6uCGXxP@ybspeV6PY0_QpdIQ6!`sOEB'`& 98RXHh6hd3Y)5cFQYl'fe^ 6h f iT`93&0` 1QaEP'Y6CHvĈ54d6`z!  ##ª#M XF# i (`_ a `# 4# F ddd| # ;^ &z!  `#J#FizS `Pu w$B thfY:!-8@@[$;Partial list of column 1 (x) and4 (3/# 2)[J-5S 3.38081324 -4.795918367x 11.94755958%#38 8.256652373K3879102 2.971647236 -4.183673469 -1.7118492633.97959+7&11119460 3.775510204 -0.7351580931'571428' 4584401848t3N 229683130 3.163265306' 021688565 -2.0444601a& 0.4069418215&0 0.67036103qT3877 1.64644J 4k%5ˇ92% GZ 2.593582  [ sin(x)/cos\Έ) A `@Gs . D}vKr  A 'LpC uxyqq*@ Չ yY6p sP8wU7`6sF00#@HW(W`6sF02e0`l;g0xuQ0j@4i8w0S0 uQ` 8$PGU!%fR70,G#3`qI&`pP5RYXDHY)h10VTYD` ! p64 FDU0 Y5*  [!Full list of M1:=column 1 (x) ['#and M24 ((3/02)=Tsec=ax.Re-5 -4.7959183673& -4.387755102 1836734693.9795977045714289J3L 1632653062.]u^L2.p089–L4L -1.#%1.߆K1.12245&1K"4898 -0./10.s4;&  0.306122449 0.510204081671428539183673469RD-5 -W 28211277T5882351)524 66666683R 2193877551S 500000}20-}2631548 50 7692309294|2} 119047618233331$58909091)52721078 166666634- 1286600214347821) 306122479911122S31887755S 125000000} 97789154|83 2142857158S 1Q00793~+7884979454545952 249M10544196f83333315L50 Q483351221 526315774- 7142858+ 10000007- 31887755W625.W 306464V V497849293 769230724 k·) S[dRj` M2:= 3.380813245i1.94M95{ -8.256652373 -2.971647236 1.71184926& 1.111194607 -0.7351580931458440>8( 2296831309 02168856540.s4a13' 4069418215 0.670361038 1.n64644$1.553092 2.593582q 6.045715 4.6209 4.0117399/07880202 /308786692 559606345 316042347tW10238371< 0.1024092146 0.3160723212 0.559647014% 86703949588 1.30888012R1220717c361a13Y20099-?510830061868%118194%39773J346783z02578n35489m319377305852442365223693487943L48536G2113,13033A6&480^ 817694 443324349862512252988669319323085447153474754697`8595765135582198011138974p644660-1020588^41059$3889426969511I8725690078J333659:25494I953751100095201453&21"6m6001156807- 494426 48291471639970 16013891:509:^6401062008H897KW240613e43086V20g7W79Raugment(M1,M2)R"5 3.3808132458$P; 4.795918367yY6pE11.94755958GU=EC35DsP 8.256652373%fRo4.3877551028wU  -" 2.971647236G#`'4.1836734696sFN1.711849263qI&0u3u959S7p1.111194607`p3.04@ 0.73515809315 3.571428W(W'9O4584401848XDO8399: 2296831309)h10 -" 3.1632653062e0`&' 0.0216885654VT& O2.959183673g0OF1844460139D`F2.755102041uQF4069418215!I2.E08U @F 0.670361034p64 2.3469387764i8w 1.018646445FDPe>E142857143(WE5 11892U0 1.93877551uQ" 2.593582885Y5P-'1.7346H8sFE6.045715WW E530612245S$l24.85662096HVb `326Q2e0a 4.0117399399% 4898$H%2.078802028. 0.918363 6sFu u308769480iG| 0.7142853(W0 -"(8669769429fiB(5102040816 @`P5596063459Y`cEpx 0.306122449$I3160423476#GYǒ3 383712987Gf 4092146@Ԛ 723212#! a @` " 0.559647014Ydp 0.7142853(W08670394958gU>9183673469 6sF{ 1.308880120[!Remark: We want to show that 'sec(x)4x(=tan#!. 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