Question

高等数学:第2题的(1)(2)(3)(4)(5)怎么做，需要写清楚怎么计算的

Answer 1

(1) f(x+y, x-y) = x^2+y^2 = (1/2)[(x+y)^2 +(x-y)^2]
则 f(x, y) =(1/2)(x^2+y^2)
(2) ⊿z = (2.02^2)(-1.01)^2 - (2^2)(-1)^2 = 0.16241604
dz = (2xy^2)dx+(2yx^2)dy = 2xy(ydx+xdy)
= 2*2(-1)[-0.02 -0.02) = 0.16
(3) dz = 2xdx-2ydy , z = x^2 - y^2 + C
z(1, 1) = 2, 则 C = 2, z = x^2 - y^2 +2
(4) 旋转曲面方程 3x^2+2y^2+3z^2 = 12
法向量 (3x, 2y, 3z),
在点 (0, √3, √2) , 法向量 (0, 2√3, 3√2)
单位法向量 (0, 2/√10, 3/√15)
(5) z = 2x^2+ax+xy^2+2y,
z'<x> = 4x+a+y^2,
z'<y> = 2xy+2
P(1, -1) 是极值点，则 4+a+1= 0， a = -5
z''<xx> = 4 > 0, 则极值点为极小值点.