Math Problem Statement

The uploaded image contains mathematical problems, including tasks to calculate gradients, directional derivatives, extrema of functions, and indefinite integrals.

Solution

The uploaded image contains mathematical problems, including tasks to calculate gradients, directional derivatives, extrema of functions, and indefinite integrals.

Please let me know which problem you would like to focus on! For instance:

  • Gradient and directional derivative problems (e.g., #95–100).
  • Extremum analysis of functions (e.g., #101–105).
  • Indefinite integral evaluation (e.g., #106–109).

Feel free to specify, and I will solve and explain in detail!

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Math Problem Analysis

Mathematical Concepts

Calculus
Directional Derivatives
Gradient Calculation
Extremum Analysis
Indefinite Integration

Formulas

Gradient of a function ∇f = (∂f/∂x, ∂f/∂y)
Directional derivative formula D_uf = ∇f ⋅ u
Critical points using ∇f = 0 for extrema
Integration techniques: substitution, partial fractions

Theorems

Chain Rule for differentiation
Fermat's theorem on stationary points
Fundamental Theorem of Calculus

Suitable Grade Level

Undergraduate Mathematics