Math Problem Statement
Find the directional derivative of 0 = 682+92722372 at the point (1,-1,1) in the direction of @ =î+§+k.
Solution
To find the directional derivative of the function at the point in the direction of the vector , let us solve this step by step:
1. General Formula for the Directional Derivative
The directional derivative of in the direction of a unit vector is given by: Where:
- is the gradient vector of .
- is the unit vector in the direction of .
2. Step-by-Step Solution
Step 1: Calculate the gradient
The function is a constant function. The gradient of a constant function is:
Step 2: Normalize the direction vector
The magnitude of is: The unit vector is:
Step 3: Compute the directional derivative
The directional derivative is given by: Substituting and , we get:
Final Answer:
The directional derivative of the given function at the point in the direction of is:
Would you like further clarification or details?
Related Questions:
- What is the meaning of a zero directional derivative in a given direction?
- How does the gradient relate to the steepest direction of change?
- What happens to the directional derivative if the vector is not normalized?
- How would the directional derivative change if were not constant?
- Can the directional derivative ever exceed the magnitude of the gradient?
Tip:
The gradient vector always points in the direction of the steepest ascent, and its magnitude represents the rate of change in that direction!
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Math Problem Analysis
Mathematical Concepts
Directional Derivative
Gradient Vector
Vector Normalization
Formulas
Directional derivative formula: D_u f(x, y, z) = ∇f · u
Gradient: ∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z)
Unit vector: u = v / ||v||
Theorems
Properties of Constant Functions
Suitable Grade Level
Grades 11-12
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