Directional Derivatives and the Gradient 2. Captain Ralph is flying his rocket t

Question

Directional Derivatives and the Gradient

2. Captain Ralph is flying his rocket through a cloud of poisonous and corrosive gas in the atmosphere of a distant planet. The equation of the density of this gas at a point 2) is known to be: 42 f(r, y, z) a) At the point (1,1,0), calculate the rate that the density changes in the direction of the vector b) At the point (1,1,0), find a vector that points in the direction that Captain Ralph should fly if he wishes to decrease the density of the poisonous gas surrounding his rocket as quickly as possible

Explanation / Answer

2)

a)given f(x,y,z)=e-x^2 -y^2 -4z^2

gradient f=<-2xe-x^2 -y^2 -4z^2,-2ye-x^2 -y^2 -4z^2,-8ze-x^2 -y^2 -4z^2>

at point (1,1,0)

gradient f=<-2*1e-1^2 -1^2 -4*0^2,-2*1*e-1^2 -1^2 -4*0^2,-8*0*e-1^2 -1^2 -4*0^2>

gradient f=<-2e-2,-2e-2,0>

gradient f=(-2e-2)i+(-2e-2)j+0k

given u =(1/2)(1i+0j+1k)

|u|=1

rate of change of density in the direction of vector u is =f.u/|u|

rate of change of density in the direction of vector u is =((-2e-2)i+(-2e-2)j+0k).(1/2)(1i+0j+1k)/1

rate of change of density in the direction of vector u is =(1/2)((-2e-2*1)+(-2e-2*0)+(0*1))

rate of change of density in the direction of vector u is =(1/2)((-2e-2)+0+0)

rate of change of density in the direction of vector u is =-(2)e-2

b)vector that points in the direction that captain ralph should fly if he wishes to decrease the density of the poisnous gas surrounding his rocket as quickly as possible =-f/f

=-[(-2e-2)i+(-2e-2)j+0k]/[(-2e-2)2+(-2e-2)2+02]

=-[(-2e-2)i+(-2e-2)j+0k]/((2e-2)2)

=-[(-1/2)i+(-1/2)j+0k]

=(1/2)i+(1/2)j+0k

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