Answer:
a)
b)
c)
Step-by-step explanation:
The given function is . Recall the following:
- (The gradient of f is defined as the vector whose components are the partial derivatives of the function with respect to each of its variables)
- Given a direction vector v, that is a vector that is unitary, the rate of change of the function f in the direction v is given by
- Recall that given two vectors a and b, the dot product between them is given by
- REcall that given a vector x, then
where theta is the angle between both vectors and ||a|| is the norm of the vector a
- Given a vector of components (x,y) its norm is given by .
a)Let us calculate first the gradient of f and the calculate it at the given point. We will omit the inner steps of derivation, so you must check that the gradient of f is given by
. Since at P we have x=9, and y=81 the desired gradient is
and so the norm of the gradient at P is
.
b) We want an unitary vector v for the gradient of f, so we take the gradient and divide it by its norm (i.e )
Hence, the rate of change is given by
c). We are given that . We consider a vector a that is unitary, hence, the rate of change of f in the direction of vector a is given by