The Limit Definition of the Derivative

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• Misc
  • 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?
  • Where is the math learning center?
    • 3rd floor of South Hall, along hallway towards Fraser
      
• Derivatives as Limits: Examples
  • f′(x) = lim_h→0( f(x+h) - f(x) ) / h
  • f(x) = 1
    • lim_h→0( f(x+h) - f(x) ) / h
    • = lim_h→0( 1 - 1 ) / h
    • = lim_h→00 / h
    • = lim_h→0 0
    • = 0
    • Constant function vs one that varies
      
  • f(x) = 3x
    • lim_h→0( f(x+h) - f(x) ) / h
    • = lim_h→0( 3(x+h) - 3x ) / h
    • = lim_h→0( 3x+3h - 3x ) / h
    • = lim_h→0 3h / h
    • = lim_h→0 3
    • = 3
  • 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( 2x + h )
    • = 2x
  • f(x) = 1/x³
    • lim_h→0( f(x+h) - f(x) ) / h
    • = lim_h→0( 1/(x+h)³ - 1/x³ ) / h
    • = lim_h→0( x³/x³(x+h)³ - (x+h)³/x³(x+h)³) / h
    • Or, in a more (or at least differently) readable form...
      
    • Mini-Assignment: try to finish simplifying this. It should work out to -3/x⁴
• Next
  • Finish examples
    • Note mini-assignment above
  • Basic differentiation rules

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