Tangent Planes and Linearization

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• Misc
  • Colloquium
    • “Perpetual Pest Control in the Face of Resistance Evolution?”
    • Dr. John Ringland, University at Buffalo
    • Thursday, Oct. 29, 4:00 PM
    • Newton 202
    • Extra credit
  • Hour exam 2 next Monday (Nov. 2)
    • Material since 1st hour exam (e.g., curvature, multivariable functions, partial derivatives, gradient, directional derivatives, tangency, linearization, extreme values, etc.)
    • Rules and format otherwise similar to 1st hour exam (esp. open reference materials)
• Questions?
  • Plan for question 4, PS 7?
    • Find directions of fastest change
    • Plug those directions into directional derivative
      • Directional derivative is a dot product
        
  • Why are there two directions of fastest change?
    • Perpendicular to level curves/contours, with one upward and one downward
      
  • Why is directional derivative a dot product of gradient and direction vector?
    
  • Vertical bar notation for evaluating expression at a point?
    
• Tangent Planes and Linearization
  • Section 14.6
    • Tangent plane in space
    • Straightforward equations
      • f_x(p₀) (x-x₀) + f_y(p₀) (y-y₀) + fz(p₀) ( z - z₀) = 0
        • Tangent to f(x,y,z) = c at p₀
      • z = g(x,y) → g(x,y) - z = 0
        • g_x(p₀) (x-x₀) + g_y(p₀) (y-y₀) - ( z - z₀) = 0
    • Linearization?
      • Example: linearization of z = y² + e^xy at (0,2)
        
• Next
  • Extreme Values
  • Read section 14.7

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