Gradients and Directional Derivatives

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• Questions?
• Directional Derivatives
  • Section 14.5
    • Derivative of f at point in direction of u, and some related formulas
      
    • Tangent line to curve: f_x(x₀y₀)(x-x₀) + f_y(x₀,y₀)(y-y₀) = 0
    • Derivative along path: d/dt f(r(t)) above
    • Variation in directional derivative with angle?
      • Angle is between direction vector and gradient
      • The closer direction is to gradient, the bigger the directional derivative
        
  • Tangible examples
    • Pick out directional derivatives and gradients for auditorium ceiling
  • Other examples
    • Just what was directional derivative for auditorium (z = √( 400 - (x-y)²/2 )) at your location in direction ⟨3,4⟩?
      
    • z = sinx cosy (the “egg carton” function from problem set 1)
      • Gradient (symbolically)?
      • grad z = ⟨ ∂z/∂x, ∂z/∂y ⟩
      • = ⟨ cosy cosx, -siny sinx ⟩
      • Look at this in muPad, looking towards xy plane, to see how gradient is orthogonal to level curves
        
    • f(x,y,z) = x² + y² + z²
      • To find rate at which f increases per unit distance moved from (1,0,0) in direction ⟨0,√2/2,√2/2⟩…
      • Find vector of partial derivatives (gradient)
      • Calculate dot product with ⟨0,√2/2,√2/2⟩
      • i.e., today’s ideas work in more dimensions exactly as you’d expect from how they work in 2
• Problem Set
  • Handout
• Next
  • Tangent planes
  • Read section 14.6

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