A Trigonometric Derivative Problem

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• Misc
  • Hour exam
    • Monday, Oct. 6
    • Material since start of semester (e.g., limits, limit definition of derivative, differentiation rules including product, quotient, trig derivatives)
    • About 7 short-answer questions, e.g., like problem sets
    • Whole class period
    • Open book, notes, computer; closed person
  • Useful fact for trig derivatives problem set
    • Antiderivatives of 1/x are of the form lnx + C
• Questions?
  • Limits from right or left?
    • Limits computed based only on how function behaves for x values slightly smaller than c and approaching it (limit from left), or slightly larger and approaching (limit from right)
      
    • For example...
      • lim_x→0-(1/x) = -∞
      • lim_x→0+(1/x) = ∞
      • lim_x→0(1/x) undefined
  • Homework question 4
    • Derive derivative of secx
      • i.e., show why d( secx ) / dx = secx tanx
    • secx = 1 / cosx (quotient rule!)
    • d( secx ) / dx
    • = sinx / cos²x
    • = ( sinx / cosx ) ( 1 / cosx )
    • = tanx secx
• Trigonometric Derivatives
  • d( sinx / (x + secx) ) / dx
    • Quotient rule!
      • u(x) = sinx, u′(x) = cosx
      • v(x) = x + secx, v′(x) = 1 + secx tanx
    • d( sinx / (x + secx) ) / dx
    • = ( (x + secx)cosx - sinx(1 + secx tanx) ) / (x + secx)²
    • = ( (x cosx + 1) - (sinx + sinx secx tanx) ) / (x + secx)²  (sec = 1/cos)
    • = ( (x cosx + 1) - (sinx + sin²x / cos²x ) ) / (x + secx)²
    • = ( x cosx + 1 - sinx - tan²x ) / (x + secx)²
  • Higher order derivatives
    • Find d²y / dx² if y = sinx - cosx
      • = d( dy/dx ) / dx
      • = d( cosx + sinx )/ dx
      • = -sinx + cosx
    • Do derivatives of sinx ever reach 0 like derivatives of polynomials do?
      • No
  • Antiderivatives
    • Find an antiderivative of -cosx
      • = -sinx (+ C, in general)
    • Antiderivative of -sinx
      • = - (antiderivative of sinx)
      • = - ( -cosx)
      • = cosx
      • Or, simpler
        • = cosx
• Silo problem
  
  • How fast does sun move?
    • 2π radians / day
    • = π/12 radians / hour
  • dL/dΘ = -30 csc²Θ feet/radian
  • = -30 csc²Θ (π/12) feet/hour
  • = -30 csc²(πt/12) (π/12) feet/hour
• Next
  • The Chain Rule
  • Section 3.6

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