The Product and Quotient Differentiation Rules

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• Misc
  • “Introduction to R” handout (thanks to Prof. Feissner in Bio)
    • curve function
      

      > curve( x^2 - 3*x, -5, 5 )
      > curve( x * sin(x), -5, 5, add=TRUE )
      > curve( sqrt(x) - x, -5, 5, add=TRUE, col="red" )
      Warning message:
      In sqrt(x) : NaNs produced

      
      
  • Date for hour exam 1 could move as much as 1 week early
    • Test will be limits, limit definition of derivative, basic differentiation rules, product & quotient, trig derivatives
    • Moving a whole week is probably extreme, but some move (e.g., to Monday, Oct. 6) might be good
    • (Mostly) up to you
    • Let me know whether you’d like to move the exam or not
• Questions?
• “Real-World Math Bounty”
  • A mathematican walks into a bar and orders a pint of beer. Just as the bartender is about to pour it another mathematician comes in and orders a half pint of beer. Then another mathematician comes in and orders a quarter pint, a fourth comes in and orders an eighth of a pint, and so forth. Finally, in frustration, the bartender shouts at the by now huge crowd of mathematicians “you mathematicians just don’t know your limits!” and hands the whole group 2 pints.
  • Questions: What’s the reference to limits about, and why 2 pints?
  • Answers: If you add up all the beers ordered by the first mathematician (1 pint), the first two (1 + 1/2 = 1 1/2 pints), the first three (1 + 1/2 + 1/4 = 1 3/4), and so forth, the limit that the sums approach is 2.
• Product and Quotient Rules
  • Second half of section 3.3
    • Product rule
      • d( uv )/dx = v du/dv + u dv/dx
    • Quotient rule
      • d( u/v )/dx = (v du/dx - u dv/dx) / v²
    • Higher-order derivatives
      • Derivatives of derivatives of derivatives etc.
        • v = ds/dt, a = dv/dt
        • Thus a = d²s/dt² = s′′(t)
  • Another real-world example
    • 2 weights hung over a pulley, but one weight is leaking and so getting lighter.
      
    • Acceleration = a(t) = g ( (2.4 - 0.17t) - 1.3 ) / (2.4 - 0.17t) + 1.3 )
      • = g (1.1 - 0.17t) / (3.7 - 0.17t)
    • Find “jerk” = a′(t)
      • = g d( (1.1 - 0.17t) / (3.7 - 0.17t) )/dx
        • u = 1.1 - 0.17t
        • v = 3.7 - 0.17t
      • So a′(t) = g ( (v du/dx - u dv/dx)/v² )
      • = g ( -0.17(3.7 - 0.17t) + 0.17(1.1 - 0.17t) ) / (3.7 - 0.17t)²
      • = g (-0.442) / (3.7 - 0.17t)²
      • = -0.442g / (3.7 - 0.17t)²
• Next
  • More examples of product and quotient rules

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