Question

Find the rate of change of f(x,y,z)=xyzf(x,y,z)=xyz in the direction normal to the surface yx2+xy2+yz2=120yx2+xy2+yz2=120 at (3,4,3)(3,4,3). (Use symbolic notation and fractions where needed.) Rate of change =

Answer 1

Answer:

Rate of change of function in the direction of normal to the given surface at ( 3 , 4 , 3 )  is $\frac{573}{\sqrt{985}}$

Step-by-step explanation:

Given:

Function, f( x , y , z ) = xyz

Equation of surface, yx² + xy² + yz² = 120

To find: Rate of change of function in the direction of normal to the given surface at ( 3 , 4 , 3 )  

The Gradient of the normal to the surface

$(\bigtriangledown_x\:,\:\bigtriangledown_y\:,\:\bigtriangledown_z)$

$\implies\:(2xy+y^2+0\:,\:x^2+2xy+z^2\:,\:0+0+2zy)$

$\implies\:(2xy+y^2\:,\:x^2+2xy+z^2\:,\:2zy)$

Gradient at ( 3 , 4 , 3 ) $=\:(2(3)(4)+(4)^2\:,\:(3)^2+2(3)(4)+(3)^2\:,\:2(4)(3))$

$\implies\:(40\:,\:42\:,\:24)$

The Change in the directional derivative of f in given direction is,

$\bigtriangledown f_{(3,4,3)}.\frac{(40,42,24)}{\sqrt{40^2+42^2+24^2}}=(yz,xz,xy)_{(3,4,3)}.\frac{(40,42,24)}{\sqrt{1600+1764+576}}=((4)(3),(3)(3),(3)(4)).\frac{(40,42,24)}{\sqrt{3940}}$

$=\frac{(12,9,12).(40,42,24)}{\sqrt{3940}}=\frac{480+378+288}{\sqrt{3940}}=\frac{1146}{2\sqrt{985}}=\frac{573}{\sqrt{985}}$

Therefore, Rate of change of function in the direction of normal to the given surface at ( 3 , 4 , 3 )  is $\frac{573}{\sqrt{985}}$