Find a unit vector U such that the rate of change of f in the direction of U at the given point is maximum.
point being (3,5,pi)
I know hos to do this when given a direction but since im asked to ind the unit vector first it kinda throws me off. do i just use my point? like make it a vector and divide by it's absolute value so i get a unit vector. would that be in the same direction of ∇f so i can get a max?
<3,5,pi>*(1/(106+pi^2)^(1/2)
You would just want to find the unit vecter like you had said in the first place.
To find a unit vector U such that the rate of change of f in the direction of U is maximum at the point (3, 5, π), you'll need to follow these steps:
1. Compute the gradient vector ∇f at the given point (3, 5, π).
The gradient vector ∇f represents the direction of steepest ascent of f at any given point in its domain. It is computed by taking the partial derivatives of f with respect to each variable and combining them into a vector.
2. Normalize the gradient vector to obtain a unit vector U.
Divide each component of the gradient vector by its magnitude (or absolute value), which will give you a vector with the same direction but with a magnitude of 1, making it a unit vector.
3. Verify that U and ∇f are in the same or opposite directions.
To determine if U and ∇f are in the same or opposite directions, compute the dot product between the two vectors. If the dot product is positive, it means they are in the same direction. If the dot product is negative, it means they are in opposite directions.
Note: If the dot product is zero, it means the rate of change will be zero, indicating a stationary point rather than a maximum.
If the dot product is positive, you have found the unit vector U such that the rate of change of f is maximum in that direction at the point (3, 5, π).