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1. Folosind denit ia s a se calculeze derivatele part iale f

x(x0, y0), f

y(x0, y0)n punctele indicate (x0, y0), pentru urm atoarele funct ii f : D R2 R,unde D este domeniul maxim de derivabilitate :1. f(x, y) = x3y + x, (x0, y0) = (1, 1);2. f(x, y) = ln(x2+ y3), (x0, y0) = (1, 1);3. f(x, y) = e3x+y+ ex+3y, (x0, y0) = (1, 2);4. f(x, y) = 3

x5y2, (x0, y0) = (1, 1);5. f(x, y) = arctan xy, (x0, y0) = (2, 1);6. f(x, y) =

xy + xy, (x0, y0) = (2, 1).Rezolvare:1. f

x(1, 1) = limx1f(x, 1) f(1, 1)x 1 = limx1x3+ x 2x 1 = 4,f

y(1, 1) = limy1f(1, y) f(1, 1)y 1 = limy1y 1y 1 = 12. f

x(1, 1) = limx1f(x, 1) f(1, 1)x + 1 = limx1ln(x2+ 1) ln 2x + 1 == limx12xx2+ 1 = 1f

y(1, 1) = limy1f(1, y) f(1, 1)y 1 = limy1ln(1 + y3) ln 2y 1 == limy13y21 + y3 = 322. S a se studieze existent a derivatelor part iale, continuitatea si diferent ia-bilitatea n origine pentru funct iile f : R2R, denite prin:1. f(x, y) =

3xyx2+ y2, dac a (x, y) = (0, 0)0, dac a (x, y) = (0, 0);2. f(x, y) =

xy

x2+ y2, dac a (x, y) = (0, 0)0, dac a (x, y) = (0, 0);3. f(x, y) =

xyx2y2x2+ y2, dac a (x, y) = (0, 0)0, dac a (x, y) = (0, 0);14. f(x, y) =

(x2+ y2) cos 1x2+ y2, dac a (x, y) = (0, 0)0, dac a (x, y) = (0, 0).Rezolvare:1. f

x(0, 0) = limx0f(x, 0) f(0, 0)x = 0,f

y(0, 0) = limy0f(0, y) f(0, 0)y = 0.Deoarece limx0,y=mxf(x, y) = limx03mx2(1 + m2)x2 = 3m1 + m2, m R, rezult ac a f nu este continu a n origine.Dac a f ar diferent iabila n origine, atunci ar rezulta c a f este continua norigine, ceea ce este absurd. Prin urmare, f nu este continu a si diferent iabilan origine, dar admite derivate part iale n origine.2. f

x(0, 0) = 0, f

y(0, 0) = 0.Deoarece |f(x, y| = |xy|

x2+ y2 |xy||x| = |y|, avem lim(x,y)(0,0)f(x, y) = 0 =f(0, 0), deci funct ia f este continu a n origine.Dac a f ar diferent iabila n origine, atunci ar existadf(0, 0) = f

x(0, 0) + f

y(0, 0) = 0 dx + 0 dy = 0si lim(x,y)(0,0)f(x, y) f(0, 0) df(0, 0)(x, y)

x2+ y2= lim(x,y)(0,0)xyx2+ y2 = 0. Dar,cum limx0,y=mxxyx2+ y2 = m1 + m2, m R, limita de mai sus nu exist a. Asadarf nu este diferent iabila n origine.3. S a se determine derivatele part iale de ordinul ntai si diferent iala deordinul nt ai pentru urm atoarele funct ii f : D R2 R, unde D estedomeniul maxim de diferent iabilitate:1. f(x, y) =

x2+ y2;2. f(x, y) = 2x2y 3xy + x2y + 2;3. f(x, y) = ln xy;4. f(x, y) = ln yx;5. f(x, y) = ln yyln xx;6. f(x, y) = ln xy;27. f(x, y) = x ln y ey2+x;8. f(x, y) = xexy;9. f(x, y) = ex2y;10. f(x, y) = xy;11. f(x, y) = ex2+y2sin2x;12. f(x, y) = arctan xy;13. f(x, y) = arctan

x2+ y2;14. f(x, y) = x2arctan(x2+ y2);15. f(x, y) = arcsin yx, x > 0;16. f(x, y) =

1 x2y2;17. f(x, y) = ln(x +

x2+ y2);18. f(x, y) = x2y2x2+ y2;19. f(x, y) = x23xy + 2y2+ 3x 4y + 2;20. f(x, y) = sin(2x + 3y);21. f(x, y) = x3+ y3+ 3xy;22. f(x, y) = (x a)(y a)(x b)(y b), unde a, b R;23. f(x, y) = xy2exy;24. f(x, y) = x4y3(1 x2+ y);25. f(x, y) = cos x + cos y + sin(x + y);26. f(x, y) = sin x sin y sin(x + y);27. f(x, y) = ln 3

x2+ y2+ arctan xy;28. f(x, y) = lnm n

x2+ y, unde m N, n N, n 2;29. f(x, y) = (x2y2) sin yx;330. f(x, y) = sin2(x2y + xy3);31. f(x, y) = xy + yx;32. f(x, y) = xy3x4+ y2;33. f(x, y) = sin xx2+ y2;34. f(x, y) = (x2+ y2) cos 1

x2+ y2;35. f(x, y) = 1x + xy + 1y.4. S a se determine derivatele part iale de ordinul ntai si diferent iala deordinul nt ai pentru urm atoarele funct ii f : D R3 R, unde D estedomeniul maxim de diferent iabilitate:1. f(x, y, z) = (x + y) sin yz;2. f(x, y, z) = x2+ y2+ xyz;3. f(x, y, z) = 3x2+ 2y2+ 3z2+ 2xy 3yz;4. f(x, y, z) = x3y2z + 10xy 2z3+ 4;5. f(x, y, z) =

x2+ y2+ z2;6. f(x, y, z) = x

y2+ z2;7. f(x, y, z) = arctan xyz ;8. f(x, y, z) = arctan z

x2+ y2;9. f(x, y, z) = ln zxy;10. f(x, y, z) = ln xyyzzx;11. f(x, y, z) = ln yx2+ z2;412. f(x, y, z) = zexy2+ yex+z;13. f(x, y, z) = xy+ yz+ zx;14. f(x, y, z) = exyz3cos(x + y);15. f(x, y, z) = xy yz + zx;16. f(x, y, z) = (x + a)(y + b)(z + c), unde a, b, c R;17. f(x, y, z) = zexy;18. f(x, y, z) = zex2+y2+ xyez;19. f(x, y, z) = sin x cos y sin(x + z);20. f(x, y, z) = eyx(z2x2);21. f(x, y, z) = xy2z3(6 2x + 3y 4z);22. f(x, y, z) = xzx2y2x2+ z2;23. f(x, y, z) = (x y)n(y z)m, unde n, m N;24. f(x, y, z) = y2x + zy 2z;25. f(x, y, z) = cos2(ax + by + cz), unde a, b, c R;5. Sa se determine derivatele part iale de ordinul al doilea si diferent ialade ordinul al doilea n punctele indicate (x0, y0) pentru funct iile f : D R2R, unde D este domeniul maxim de diferent iabilitate:1. f(x, y) =

x2+ y2, (x0, y0) = (1, 1);2. f(x, y) = exy, (x0, y0) = (1, 0);3. f(x, y) = ln x2y, (x0, y0) = (3, 1);4. f(x, y) = ln xy2, (x0, y0) = (1, 2);5. f(x, y) = arctan(x2+ y2), (x0, y0) = (1, 1);6. f(x, y) = (1 + x)m(1 + y)n, m, n 2, (x0, y0) = (0, 0);57. f(x, y) = excos y, (x0, y0) = (0, );8. f(x, y) = x + yx y, (x0, y0) = (2, 1);9. f(x, y) = x3+ y3x2y2, (x0, y0) = (1, 1);10. f(x, y) = ex2y2, (x0, y0) = (1, 1).Rezolvare:1. f

x(x, y) = x

x2+ y2, f

y(x, y) = y

x2+ y2,f

x2(x, y) = y2(x2+ y2)

x2+ y2, f

xy(x, y) = xy(x2+ y2)

x2+ y2,f

y2(x, y) = x2(x2+ y2)

x2+ y2,f

x2(1, 1) = f

y2(1, 1) = f

xy(1, 1) = 122,d2f(1, 1) = 122(dx2+ 2dxdy + dy2) = 122(dx + dy)22. f

x(x, y) = yexy, f

y(x, y) = xexy, f

x2(x, y) = y2exy, f

y2(x, y) =x2exy, f

xy(x, y) = (1 + xy)exy, f

x2(1, 0) = 0, f

y2(1, 0) = 1, f

xy(1, 0) =1, d2f(1, 0) = 2dxdy + dy26. S a se determine derivatele part iale de ordinul al doilea si diferent ialade ordinul al doilea n punctele indicate (x0, y0, z0) pentru funct iile f : D R3R, unde D este domeniul maxim de diferent iabilitate:1. f(x, y, z) = ln(1 + x + y + z), (x0, y0, z0) = (1, 0, 1);2. f(x, y, z) = 1

x2+ y2+ z2, (x0, y0, z0) = (1, 1, 1);3. f(x, y, z) = xeyz+ yezx+ zexy, (x0, y0, z0) = (1, 1, 1);4. f(x, y, z) = x6yz4+x4y3yz2+6x +2yz 2, (x0, y0, z0) = (1, 2, 1);5. f(x, y, z) = zexyz, (x0, y0, z0) = (1, 1, 1);6. f(x, y, z) = x arctan yz, (x0, y0, z0) = (2, 1, 1);7. f(x, y, z) = sin2(x y z), (x0, y0, z0) = (, 2, 3);68. f(x, y, z) = x + zxy + z, (x0, y0, z0) = (1, 1, 2);9. f(x, y, z) =

9 x2y2z2, (x0, y0, z0) = (2, 1, 1);10. f(x, y, z) = 3

x zy z, (x0, y0, z0) = (2, 0, 1) .Rezolvare:1. f

x(x, y, z) = f

y(x, y, z) = f

z(x, y, z) = 11 + x + y + zf

x2(x, y, z) = f

y2(x, y, z) = f

z2(x, y, z) = 1(1 + x + y + z)2f

xy(x, y, z) = f

xz(x, y, z) = f

yz(x, y, z) = 1(1 + x + y + z)2d2f(1, 0, 1) = 19(dx2+dy2+dz2+2dxdy +2dxdz +2dydz) = 19(dx+dy +dz)2.7. Sa se determine derivatele part iale de ordinul al doilea si diferent iala deordinul al doilea ntr-un punct curent (x, y) pentru funct iile f : D R2R,unde D este domeniul maxim de diferent iabilitate:1. f(x, y) = x3y22xy3+ 3x26y + 7;2. f(x, y) = x4+ y42x2+ 5xy 3y6;3. f(x, y) = 2x yx + 3y;4. f(x, y) = exy2;5. f(x, y) = yexy;6. f(x, y) = (x2+ y2)ex+y;7. f(x, y) = ln(x2+ y3);8. f(x, y) = sin xy;9. f(x, y) = arctan xy;10. f(x, y) = yx.7Rezolvare:1. f

x(x, y) = 3x2y22y3+ 6x, f

y(x, y) = 2x3y 6xy26,f

x2(x, y) = 6xy2+ 6, f

y2(x, y) = 2x312xy, f

xy(x, y) = 6x2y 6y2,d2f(x, y) = 6(xy2+ 1)dx2+ 12(x2y y2)dxdy + 2(x36xy)dy2.8. S a se determine derivatele part iale de ordinul al doilea si diferent ialade ordinul al doilea ntr-un punct curent (x, y, z) pentru funct iile f : D R3R, unde D este domeniul maxim de diferent iabilitate:1. f(x, y, z) = x2+ y2+ z2xyz ;2. f(x, y, z) = x + zy + z;3. f(x, y, z) = xy2z3(6 x + 2y z);4. f(x, y, z) = z2xey+ yez+ zex;5. f(x, y, z) = (x + z)eyx;6. f(x, y, z) = yx2+ z;7. f(x, y, z) = ln(x + y + z);8. f(x, y, z) = ln xyz;9. f(x, y, z) = cos(x + 2y + 3z);10. f(x, y, z) = x sin y + y sin z + z sin x.Rezolvare:1. f

x(x, y, z) = 1yz yx2z zx2y, f

y(x, y, z) = xy2z + 1xz zxy2,f

z(x, y, z) = xyz2 yxz2 + 1xy, f

x2(x, y, z) = 2(y2+ z2)x3yz ,f

y2(x, y, z) = 2(x2+ z2)xy3z , f

z2(x, y, z) = 2(x2+ y2)xyz3 ,f

xy(x, y, z) = x2y2+ z2x2y2z , f

xz(x, y, z) = x2+ y2z2x2yz2f

yz(x, y, z) = x2y2z2xy2z2 ,8d2f(x, y, z) = 2(y2+ z2)x3yz dx2+ 2(x2+ z2)xy3z dy2+ 2(x2+ y2)xyz3 dz2++ 2(x2y2+ z2)x2y2z dxdy + 2(x2+ y2z2)x2yz2 dxdz + 2(x2y2z2)xy2z2 dydz9. Sa se scrie matricea Jacobi si sa se determine jacobianul pentruurm atoarele funct ii:1. f : (0, ) [0, 2] R2, f(r, t) = (r cos t, r sin t);2. f : (0, ) [0, 2] [2, 2] R3,f(r, , ) = (r cos cos , r sin cos , r sin );3. f : R2R2, f(x, y) = (xexy, x3y +x2xy +y +3), n punctul (1,1).Rezolvare:1. Fie f1(r, t) = r cos t si f2(r, t) = r sin t.Atunci Jf(r, t) =

(f

1)r(r, t) (f

1)t(r, t)(f

2)r(r, t) (f

2)t(r, t)

=

cos t r sin tsin t r cos t

,si detJf(r, t) =

cos t r sin tsin t r cos t

= r.10. Sa se scrie matricea Jacobi pentru urmatoarele funct ii:1. f : R2R3, f(x, y) = (xy, x + y, x