Limit Definition of the Derivative, Part 2

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• Misc
  • Sign up to grade problem set 2
  • Some things that counted as “more than expected” in problem set 1
    • R features that are helpful but not strictly needed, e.g, variables, plots
    • Solutions that work well but that I hadn’t thought of
    • Extra analysis of problems, e.g., graphs, double-check δ in proof
• Questions?
  • Problem set question 1 (derivative of t - 1/t)
    • f′(x) = lim_h→0( f(x+h) - f(x) ) / h
    • = lim_h→0( ((x+h) - 1/(x+h)) - (x-1/x) ) / h
    • (Rest of solution proceeds like this…)
      
• Derivatives as Limits: Examples
  • f′(x) = lim_h→0( f(x+h) - f(x) ) / h
  • f(x) = 3x
    • lim_h→0( f(x+h) - f(x) ) / h
    • = lim_h→0( 3(x+h) - 3x ) / h
    • = lim_h→0( 3x/h + 3h/h - 3x/h )
    • = lim_h→0( 3h/h )
    • = lim_h→0( 3 )
    • = 3
    • Derivative as slope of function—slope is constant, so is derivative, regardless of what ε and δ intervals you consider
      
  • f(x) = x²
    • lim_h→0( f(x+h) - f(x) ) / h
    • = lim_h→0( (x+h)² - x² ) / h
    • = lim_h→0( (x²+2xh+h²) - x² ) / h
    • = lim_h→0( 2xh+h² ) / h
    • = lim_h→0 h( 2x+h ) / h
    • = lim_h→0( 2x+h )
    • = 2x
  • f(x) = 1/x³
    
    
    
  • f(x) = √(2x)
    
    
• Basic Differentiation Rules
  • Hand out problem set
• Next
  • “Powers, Multiples, Sums, and Differences” part of section 3.3
  • Middle of page 118 - bottom of 121

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