Math Problem Statement
Find the direction in which the maximum rate of change occurs for the function f(x,y)=2xsin(xy) at the point (5,3). Give your answer as a unit vector.
Solution
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Math Problem Analysis
Mathematical Concepts
Multivariable Calculus
Gradient
Partial Derivatives
Trigonometric Functions
Formulas
Gradient: ∇f(x, y) = (∂f/∂x, ∂f/∂y)
Partial derivatives of f(x, y) = 2x sin(xy): ∂f/∂x = 2 sin(xy) + 2xy cos(xy), ∂f/∂y = 2x^2 cos(xy)
Magnitude of the gradient vector: |∇f(5,3)| = √((∂f/∂x)^2 + (∂f/∂y)^2)
Theorems
The Gradient Theorem: The gradient vector points in the direction of the steepest ascent.
Suitable Grade Level
University Level - Calculus III or Multivariable Calculus
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