00.GEOMETRIE DIFERENTIALA_128p

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1.Sa se scrie prima forma fund pentru supraf definita implicit de ecuatia (Σ):x+y=z 3   ds  2 =(1+ 1/9z  4  )dx  2  +2/9z  4  dxdy +( 1 +1/9z  4  )dy  2  2. Sa se scrie prima forma fund pentru supraf definita de ecuatiile:   =   =   =    ( , )   2 ds 2 =(1+v) 2 du 2  + 2uvdudv +(1+u 2 )dv 2 3.Sa se scrie a doua forma fundame ntala pentru suprafata definite parametric de ecuatiile =   =   =   +    ( , )   2  n*d 2 r= -2(1+2u 2)-1/2 dudv +u 2 (1+2u 2 ) -1/2 dv 2 4.Fie suprafata definite parametric de ecuatiile =   =   =   +  ( , )   2  Sa se scrie ecuatiile planului tang ent si al normalei in punctual  M 0 (u=1, v=π) t: x+y+z=π n: x+1/1=y/1=z -1-π/1 7.Fie suprafata (S): x=ucosv, y=usinv, z=av Sa se afle elemental de arie pe suprafata ds=sqrt(u 2 +a 2 )dudv 11.Fie suprafata (Σ): x=u  2 +v  2  , y=u  2 -v  2  , z=uv  Sa se calculeze elemental de arc pe curba (γ  2  ):v=1   =   8 2 +1 1.Sa se scrie ecuatiile tangentei(T) si normalei(N) la curba (C) y=x 3 in pct A de abcisa  1  (T) 3x-y-2=0 (N) x+3y-4=0 2. ..y=sinx ..in pct A de abcisa 0 (T) x-y=0 (N) x+y=0 3. ..y=tgx..in pct A de abscisa π/4 (T) 4x-2y-π+2=0 (N) 4x-8y-π+8=0 4…x 3 +y 3 -3axy=0..in pct A(3a/2, 3a/2) (T) x+y-3a=0 (N)x-y=0 5…(x 2 +y 2 )x-ay 2 =0..in pct A(a/2,a/2) (T) 4x-2y-a=0 (N) 2x+4y=0 6…(x 2 +y 2 ) 2 -2a 2 (x 2 -y 2 )=0..in pct A(a√ 2, 0)…(T)  √ 2=0 (N) y=0 7…x 3 +3x 2 y-y 2 -2x+9=0..in pct A(2,-1) (T) x-7y-9=0 (N) 7x+y-13=0 8……x 3 +xy 2 +2x+y-3=0..in pct inters. axa Ox (T) 5x+y-5=0 (N) x-5y-1=0 9. ……y=lnx..in pct inters. A x=1 (T) x-y-1=0 (N) x+y-1=0 10… =   3 2 =   2 +1  ..in pct A(t=2) (T) 2x -5y+17=0 (N) 5x+2y-30=0 11………in pct A(t=1) (T) 2x-y+4=0 (N) x+2y-3=0 12…. = 1 +1 =   +1  ….. in pct A(t=1).(T) x+y-1=0 (N) x-y=0 13………………... in pct A(t=0).(T) x+y-1=0 (N) x-y-1=0 14…. =   3 =   3  in pct A(t=π/4) . (T) 2 +2 √ 2=0 (N) 4 4 ( 2 2 )√ 2=0 15. Sa se afle punctele singulare, sa se determine natural or sa se scrie i n cazul in care exista, ecuatiile tangentelor in aceste puncte pentru curba: 2 =   3 2 , 0 A(0, 0) =0, punct de intoarcere admite o tangent M=0 ecuatia tangentei : y=0 16. ..y 2 =x 2 (9-x 2 )…. A(0, 0) =36, nod ( 2 tangente) m=±3tg1 y=3xtg2 y=-3x 17…(x 2 +y 2 )x-ay 2 =0, a0  A(0, 0) =0, punct de in toarcere m=0 ec tang y=0 18…y 2 =x(x+1) 2 …A(-1,0) = - 4<0 punct izolat  Nu admite tangente 19….. 1 + 1 2 =   ….= - 1<0 punct izolat  Nu admite tangente 20. ….x 3 +y 3 -3axy=0…A(0,0) punct de intoarcer,e tgy=0 21…..xy 2 -x 2 -y 2 =0…..A(0,0) = - 4<0 punct izolat  Nu admite tangente 22. …x 4 -16x 2 +16y 2 =0…A(0,0) A(0, 0) =32*32 nod ( 2 tangente) m=±1 ec tang y=x y=-x 23. Sa se scrie ecuatia cercului osculatoare pentru curba y=sinx, A( 2 ,1)…(x-π/2) 2 +y 2 -1=0 24. =   ( ) =   (1  −)  .. …t=π (x-aπ) 2 +(y+2a) 2 -4a=0 25. Sa se calculeze raza de curbura a curbei y=x 3 -x 2 +2x-2 in pct de intersectie cu axa Ox .. R= 5√ 10 2 26. ….y2=8x… in A(9/8, 3) R=125/16  GEOMETRIE DIFERENTIALA

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