find the direction in the xy-plane in which the following functions have zero ch

Question

find the direction in the xy-plane in which the following functions have zero change at the given point. i need help for number 60,f(x,y)=x^2-4y^2-8, p(4,1,4)

Gradient of a composite function Consider the function F(x, y, z) = exyz. Write F as a composite function f degree g, where f is a function of one variable and g is a function of three variables. Relate Delta F to Delta g. Directions of zero change Find the directions in the xy-plane in which the following functions have zero change at the given point. Express the directions in terms of vectors.

Explanation / Answer

The directional change for a function can be calculated by finding the directional derivative at a point in a given direction which can be found by finding the gradient of the function at the point and then finding the dot product of the gradient with the given direction vector.... so the gradient of given f(x) will be del(f(x,y))=(partial derivative w.r.t x)i+(partial derivative w.r.t y)

so gradient =(2x)i-(8y)j

thus at the point (1,2,4)we have gradient =2i-16j

Let the required direction be xi+yj in the xy plane

so according to question (2i-16j).(xi+yj)=0

=> 2x-16y=0

=>x=8y

Thus the vector will be xi+yj=8yi+yj

where i and j are unit vectors

Removing the common y we have 8i+j as the required answer because all other vectors will be parallel to the answer vector for integral y values

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