Answer:
Rate of change of function in the direction of normal to the given surface at ( 3 , 4 , 3 ) is [tex]\frac{573}{\sqrt{985}}[/tex]
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
[tex](\bigtriangledown_x\:,\:\bigtriangledown_y\:,\:\bigtriangledown_z)[/tex]
[tex]\implies\:(2xy+y^2+0\:,\:x^2+2xy+z^2\:,\:0+0+2zy)[/tex]
[tex]\implies\:(2xy+y^2\:,\:x^2+2xy+z^2\:,\:2zy)[/tex]
Gradient at ( 3 , 4 , 3 ) [tex]=\:(2(3)(4)+(4)^2\:,\:(3)^2+2(3)(4)+(3)^2\:,\:2(4)(3))[/tex]
[tex]\implies\:(40\:,\:42\:,\:24)[/tex]
The Change in the directional derivative of f in given direction is,
[tex]\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}}[/tex]
[tex]=\frac{(12,9,12).(40,42,24)}{\sqrt{3940}}=\frac{480+378+288}{\sqrt{3940}}=\frac{1146}{2\sqrt{985}}=\frac{573}{\sqrt{985}}[/tex]
Therefore, Rate of change of function in the direction of normal to the given surface at ( 3 , 4 , 3 ) is [tex]\frac{573}{\sqrt{985}}[/tex]
