微積分方向導數一題

May 2021 2 5 Report

微積分方向導數一題

f(x,y,z)=(xyz)^1/2 (3,2,6) v=<-1,-2,2>

Find the directional derivative of the function at the given point in the direction of the vector v.

Update:

不好意思,補充一題

Find the directional derivative of f(x,y,z)=xy+yz+zx at P(1,-1,3) in the direction of Q(2,4,5)

1. f(x,y,z)=√(xyz), grad(f)=0.5<yz, zx, xy>/√(xyz)

(x,y,z)=(3,2,6), grad(f)=0.5<12, 18, 6>/6=<1, 3/2, 1/2>

v=<-1,-2,2>, |v|=3, unit vector u=<-1,-2,2>/3

directional derivative=grad(f)‧u=<1, 3/2, 1/2>‧<-1,-2,2>/3= -1

2. f(x,y,z)=xy+yz+zx, grad(f)=<y+z, x+z, x+y>

(x,y,z)=(1,-1,3), grad(f)=<2, 4, 0>

vector PQ=<1, 5, 2>, unit vector u=<1, 5, 2>/√30

directional derivative=grad(f)‧u=<2,4,0>‧<1,5,2>/√30= 22/√30