題目
8. (10%) Let
F(x,y,z)=xy+z3+xyz−5. Suppose a function
f(x,y) is defined in a neighborhood of
(2,1) such that
F(x,y,f(x,y))=0. Estimate
f(1.97,1.04) by linear approximation.
解答
解法一
在 (2,1)，由 F(x,y,z)=0 得 z=1。且
Fx=2,Fy=4,Fz=5,fx=−52,fy=−54.
線性近似給出
f(1.97,1.04)≈1−52(−0.03)−54(0.04)=0.98. 