Math Problem Statement

Find the directional derivative of f(x,y,z)=z3−x2y at the point (-2, 5, 2) in the direction of the vector v=⟨−3,−1,2⟩.

Solution

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Math Problem Analysis

Mathematical Concepts

Directional Derivatives
Gradient Vectors
Multivariable Calculus
Vector Normalization
Dot Product

Formulas

Gradient of f: ∇f(x, y, z) = (∂f/∂x, ∂f/∂y, ∂f/∂z)
Directional derivative: D_𝑣 f = ∇f ⋅ u
Vector normalization: u = v / |v|
Dot product: A ⋅ B = A₁B₁ + A₂B₂ + A₃B₃

Theorems

Gradient Theorem
Properties of Dot Product

Suitable Grade Level

Grades 11-12 or Early College