Calc 3 directional derivative question

In summary, a directional derivative is a measure of the rate of change of a function in a specific direction, denoted by ∇f. It is calculated using the formula ∇f · &vec;u, where ∇f is the gradient vector and · represents the dot product. It can be negative if the function is decreasing in the direction of the unit vector. The directional derivative has many real-world applications, such as in optimization problems and predicting system behavior. It is related to the gradient vector as it is essentially the projection of the gradient onto a specific direction, representing the maximum rate of change in that direction.
  • #1
meadow
19
0
The question asks:
Find the directional derivative of f (x, y, z) = z ln (x/y) at (1, 1, 2) toward the point (2, 2, 1).

What I did was find the distance between the two points to be the directional vector (i+j-k) and then I took the norm of the direction vector. so my unit vector = 1/sqrt(3) * u; then I found the gradient. From there, I found the scalar product of my unit vector and the gradient to get 0. Did I approach this problem right? Does that answer seem correct to you?
 
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  • #2
Seems all right to me :smile:
 

Related to Calc 3 directional derivative question

1. What is a directional derivative in Calc 3?

A directional derivative is a measure of the rate of change of a function in a specific direction. It represents the slope of the function along a given direction and is denoted by the symbol ∇f.

2. How is a directional derivative calculated?

The directional derivative of a function f in the direction of a unit vector &vec;u is calculated using the formula ∇f · &vec;u, where ∇f is the gradient vector of f and · represents the dot product.

3. Can a directional derivative be negative?

Yes, a directional derivative can be negative if the function is decreasing in the direction of the unit vector. This means that the function is decreasing at a faster rate in that direction compared to the overall rate of decrease.

4. What is the significance of the directional derivative in real-world applications?

The directional derivative is used in many fields of science and engineering to understand the rate of change of a function in a specific direction. It is especially useful in optimization problems, such as finding the steepest descent or ascent of a function, and in predicting the behavior of systems in different directions.

5. How does the directional derivative relate to the gradient vector?

The directional derivative is essentially the projection of the gradient vector onto a specific direction. This means that the directional derivative is the maximum rate of change of the function in that direction, and it occurs when the direction of the unit vector aligns with the direction of the gradient vector.

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