L=,dd +(I)  1Courier New1Symbol New1Courier New1Courier NewTimes New RomanTimes New Roman1Courier New1Courier New1Courier New1Courier NewQPTQPTQPTQPT?/ <```@@@?~~~~~ԱޱޱԱ޿޿ph (((Pdd{ d< p {2 䘉?Uq;?$!@~=xk+@r,%odN -@???9>@by, @@tQy[T1@̰@2 Dq@033333??@^ߛOw?HzG???ut I@x@@?@ @@@?@@?@?@?@ffffff4нTM@@8h$`@@/2??h!@$$$Uy@r3܀@eـa#˖r3܀@tz˖@1<S@???@ 0L¹H䘼pXo 2ucos(t) 3usin(t)u^2$x = 2ucos(t); y = 3usin(t); z = u^2O0O52???$$$|aS D$k??4333333@@,\(\@4333333@̄@,\(\@???4333333@@,\(\@lӾ׾˶PYV=abf(a,b)(x,y,z) = (a,b,f(a,b))42???$$$??@5Gztffffff@GzG@????1d??sF|]xXLԙ@0$a00aseg (a,0,0) to (a,b,0)b042???$$$??ΈQ333333@Q@@????1d??d(tTصԵ䦻l䇱b0b0aseg (0,b,0) to (a,b,0)b042???$$$??\ ףp= @\ ףp= @Mh$W@???1d??ѵ~䒵ХpSf0JTٻ+ab0aseg (a,b,0) to (a,b,f(a,b))bf(a,b)@ɨ'@X-02??h!@$$$@?@ffffffSۗ@tffffff@ffffff@2Tv@???tU;'p"8++Ȧ8Шatf(a,t)x = a; y = t; z = f(a,t)m@@, NY@~02??h!@$$$@?)> ףp=Gz)> ףp=@ű ̣@??? d:uԪl?@ttbf(t,b)x = t; y = b; z = f(t,b)Lr@=\g@>͵42???$$$??@333333@@tffffff@@)\(@???1d??DX5 0ܳxhLuaXabf(a,b)a+1+seg (a,b,f(a,b)) to (a+1,b,f(a,b)+fx(a,b))bf(a,b)+fx(a,b)@{@I:sf42???$$$??@333333@(b/@tffffff@@@???1d??Գѳźhزp2xabf(a,b)a+seg (a,b,f(a,b)) to (a,b+1,f(a,b)+fy(a,b))b+1f(a,b)+fy(a,b)@᷌@} 42???$$$??֣p= ׋@ t@ @(\(@)\(@N贁N@l(\@???1d?? a( K^4}ʽtabf(a,b) a+fx(a,b)3seg (a,b,f(a,b)) to (a+fx(a,b),b+fy(a,b),f(a,b)-1) b+fy(a,b) f(a,b)-1@ϓh$@oюM>; 2???$$$@@QU @ZC*u#ן?` %@bg;V@OУ1@|/4+y^2/9F h8  D pJ@b 6@h%x@Times New RomanLime: <1,0,fx(a,b)>^2/9YA!ITYLS.}sL\@Z$p@Z{(F@Times New Roman<Aqua: (normal to tangent plane)N[cҵ@o/@_uH@Times New RomanOlive: tangent plane ,-1)>\SSSe3`8Njc`k@w(@Times New RomanAnimate: a to change x; b to change y; c to rotate u c to )َy@^;1\jTimes New RomandYellow: ertical plane containing uy; c to rotate u \.Zbu.M9@z _@Times New Roman Black curve: surface intersect vertical planee u H YurHw?"ݠ`T @܏WW@Times New Roman Black segment in xy-plane: unit vector ulaneSSI=(I>jnE[V0X_@}@Times New Roman Fuchsia: tangent vector to black curveuxp<=ʟs@zdy#@X =@Times New Romanԇ This file authored by John Ganci of Richland CC, Dallas TX<[Bf@S} l@*~@Times New Romanx vertical plane containing uanci of Richland CC, Dallas TXS$\,L)֌KM9@z _@Times New Romanx = 2ucos(t); y = 3usin(t); z = u^23@((׭qr@1Courier New(x,y,z) = (a,b,f(a,b))2/9)3:+P!@޸Eo>Ͽ@1Courier Newseg (a,0,0) to (a,b,0)3D6@Nb HIsd.@1Courier Newl seg (0,b,0) to (a,b,0)347KR@fi|eMe@1Courier New$!seg (a,b,0) to (a,b,f(a,b))2/9)347KR@fiWD@1Courier New!x = a; y = t; z = f(a,t)^2/9?347KR@fiȓC@1Courier New"x = t; y = b; z = f(t,b)^2/9 /3Ql]>M@ĵ%:0xP@1Courier NewL#seg (a,b,f(a,b)) to (a+1,b,f(a,b)+fx(a,b))3Ql]>M@ĵ%:\m}@1Courier New$seg (a,b,f(a,b)) to (a,b+1,f(a,b)+fy(a,b)) 3Ql]>M@ĵ%:@"r@1Courier New$ seg (a,b,f(a,b)) to (a+fx(a,b),b+fy(a,b),f(a,b)-1)3Ql]>M@ĵ%:JlZ@1Courier Newt% plane{[fx(a,b),fy(a,b),-1];(a,b,a^2/4+b^2/9)}3Գ84\@&@=ˑE†@1Courier New,& z = f(x,y)^2/93?l[@6qS?1Courier New& seg (a,b,0) to (a+cos(c),b+sin(c),0)?3c6e@`ȝ ll'?1Courier New' plane{[-sin(c),cos(c),0];(a+cos(c),b+sin(c),0)}3+@z1iQ5%51Courier NewT(x = a+t*cos(c); y = b+t*sin(c); z = f(a+t*cos(c),b+t*sin(c))3J@p@P:H `6[?1Courier New )seg (0,0,0) to (a,b,f(a,b))3*:H@}?y+灟1Courier New)seg (a,b,f(a,b)) to (a+cos(c),b+sin(c),f(a,b)+fx(a,b)*cos(c)3 -i@{f>v:!1Courier New