Cálculo iii

Universidade Federal do Maranh˜o a
CCET-Departamento de Matem´tica a ´ lculo Diferencial e Integral III Ca

Curso: Qu´ ımica

Professor: Lu´ Fernando ıs

3a Lista de Exerc´ ıcios - 2013/1 Nome: Matr´ ıcula: −

1. Determine as derivadas parciais (a) f (x, y) = 5x4 y 2 + xy 3 + 4 (b) z = cos(xy) (c) z = x3 + y 2 x2 + y 2
2 −y 2

(d) f (x, y) = e−x (f) z = xyexy

(e) z = x2 ln(1 + x2 + y 2 ) (g) f (x, y) = (4xy − 3y 3 )3 + 5x2 y (h) z = arctan( x ) y (i) g(x, y) = (x2 + y 2 ) ln(x2 + y 2 ) (j) g(x, y) = xy √ (k) f (x, y) = 3 x3 + y 2 + 3 x sen y (l) z = cos(x2 + y 2 ) 2. Considere a fun¸˜o z = ca ∂z ∂z xy 2 . Verifique que x +y = z. 2 + y2 x ∂x ∂y ∂z ∂z +y = z. ∂x ∂y

3. Considere a fun¸˜o dada por z = x sen( x ). Verifique que x ca y

  x + y4 ∂f ∂f , se (x, y) ̸= (0, 0) 4. Determine e sendo f (x) = x2 + y 2  ∂x ∂y 0, se (x, y) = (0, 0)

5. Calcule as derivadas parciais (a) f (x, y, z) = xex−y−z (b) f (x, y, z) = sen(x2 + y 2 + z 2 ) (c) w = x2 arctan( y ) z xyz (d) w = x+y+z (e) s = f (x, y, z, w), dada por s = xw ln(x2 + y 2 + z 2 + w2 ) 6. Seja f (x, y, z) = x2 x ∂f ∂f ∂f . Verifique que x +y +z = −f. 2 + z2 +y ∂x ∂y ∂z

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