Partial Derivatives

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• Misc
  • PRISM Mentoring
    • Wednesday, March 9, 3:30
    • South 328
    • Advice from math majors re course selection, summer experiences, etc.
• Exam Solution
  • Gives general ideas if not concrete details for solving each problem.
  • See https://docs.google.com/document/d/1jNb-mOjDSHtkqmQveOhjwm1pzfXct8q2zIyDmKrMiK4/edit
  • Comments, feedback, etc. welcome
• Questions?
  • Curvature of parabola question?
    • Could do direct calculation of k
    • Or use formula for curvature of twice-differentiable function in exercise 5a
• Limits of Multivariable Functions
  • lim_{(x,y)→(0,0)}( 3x³ / (x³+y) )
    
• Partial Derivatives
  • Section 14.3
    • Definition of partial derivative
      
    • Notations
    • Can apply to functions with more than 2 variables
    • Treat all variables other than one you differentitate with respect to as constants
      • e.g., in ∂f(x,y)/∂x treat y as constant
    • Function can have partial derivatives without being continuous
    • 2nd order partial derivatives are derivatives of derivatives
      • Notations
        • ∂²f/∂x² or f_xx
        • Similar for other variables
        • ∂²f/(∂x∂y) differentiates wrt y first also f_yx
    • Mixed derivative theorem?
      • For suitably continuous functions, order of differentiation for second derivatives doesn’t matter
      • ∂f/(∂x∂y) = ∂f/(∂y∂x)
    • Increment theorem?
      • For a function with continuous derivatives, the change in a function’s value near (x₀,y₀) is the change in x and/or y times the corresponding derivative plus a correction factor that diminishes with diminishing change in x, y
        
    • Differentiability implies continuity?
      • Definition of “differentiable” says function satisfies increment theorem
      • This requires more than just that the partial derivatives exist
  • Examples
    • Find first partial derivatives of z = xy² - y sinx
      
    • Find slopes of auditorium function (z = √( 400 - (x-y)²/2 ) ) in x and y directions at (x,y) = (0,10)
      
      • Derivative with respect to y is similar, except with opposite sign
      • Slope in y direction is thus just the negative of slope in x direction
• Next
  • The chain rule for partial derivatives
  • Read section 14.4

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