Question

Let f(x, y, z) = x^2 + 2y^2 - 3z^2. (a) Find the gradient nabla f. (b) Find the directional derivative of f in the direction of u = (1, 2, 3) at the point P_0 (1, 1, 1). Observe that u is not a unit vector. (c) Find the directions in which the function increases and decreases most rapidly at the point P_0 (1, 1, 1).

Answer 1

`grad(f(x,y,z))=(\partial f)/(\partial x)\widehat{i}+(\partial f)/(\partial y)\widehat{j}+(\partial f)/(\partial z)\widehat{k}\\\\`

Applying values we get

`grad(f(x,y,z))=2x\widehat{i}+4y\widehat{j}-6z\widehat{k}`

b) The directional derivative in the direction of

`\overrightarrow{u}=\widehat{i}+2\widehat{j}+3\widehat{k}` is given by

`\overrightarrow{\triangledown f}.\frac{\overrightarrow{u}}u`

Applying values we get the directional derivative equals

`(2x\widehat{i}+4y\widehat{j}-6z\widehat{k}).(\widehat{i}+2\widehat{j}+3\widehat{k})* (1)/(√(14))\\\\=(2x+8y-18z)/(√(14))`

Thus value at
`P_(o)=(1,1,1)=-2.13`

c)

The direction of rate of maximum increase at
`P_(o)=(1,1,1)` is given by

`\triangledown \overrightarrow{f}(1,1,1)=2\widehat{i}+4\widehat{k}-6\widehat{k}`