Question

Ali is hiking on the hill, whose height is given by f(u,v)=n^2 e^((u+n)/(v+n)). Currently, he is positioned at point (3, 5). Find the direction at which he moves down the hills quickly. Take n =12

Answer 1

`<-144e^(0.88),7.47e^(0.88)>`

Explanation:

We are given that

`f(u,v)=n^2e^{(u+n)/(v+n)}`

Point=(3,5)

n=12

We have to find the direction at which he moves down the hills quickly.

`f(u,v)=144e^{(u+12)/(v+12)}`

`f_u(u,v)=144e^{(u+12)/(v+12)}`

`f_u(3,5)=144e^{(3+12)/(5+12)}`

`f_u(3,5)=144e^{(15)/(17)}=144e^(0.88)`

`f_v(u,v)=144e^{(u+12)/(v+12)}* (-(u+12)/((v+12)^2))`

`f_v(3,5)=144e^{(15)/(17)}(-(15)/((17)^2)`

`f_v(3,5)=-(2160)/(289)e^{(15)/(17)}=-7.47e^(0.88)`

`\Delta f(3,5)=<f_u(3,5),f_v(3,5)>`

`\Delta f(3,5)=<144e^(0.88),-7.47e^(0.88)>`

The direction at which he moves down the hills quickly=-
`\Delta f(3,5)`

The direction at which he moves down the hills quickly=
`<-144e^(0.88),7.47e^(0.88)>`