Vector-Valued Functions, Part 2

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• Questions?
• Vector-Valued Functions
  • Problem Set
  • Examples
    • Expanding helix along positive y axis
      
      • ⟨ t cos(t), t, tsin(t) ⟩
      • Look at this in muPad: with both negative and positive t values, it’s sort of like a double-conical spring
        
      • Make more turns of the helix by scaling arguments to sin and cos up
      • And get muPad to plot it as smooth curves via the Mesh = option to plot::Curve3d (default is 121)
        
    • Derivatives of angled parabola and helix
      • Parabola = ⟨ t, t, 1-t² ⟩
      • Derivative of r(t) is vector of derivatives of components
      • Derivative of parabola = ⟨ 1, 1, -2t ⟩
      • Derivative of helix: ⟨ cost - t sint, 1, sint + t cost ⟩
        • Use product rule, other differentiation rules, on component functions exactly as you always have
    • lim_t→1 ⟨ t²-1, 1/(t²-1), (t²-1)/(t-1) ⟩
      • Is this function continuous at t = 1? t = 0?
        
    • Derivative of ⟨ t, t², 1 ⟩ ⋅ ⟨ sin t, cos t, √t ⟩
      • d/dt ( r(t) ⋅ f(t) ) = dr/dt ⋅ f(t) + r(t) ⋅ df/dt
      • So derivative in this question is ⟨1,2t,0⟩ ⋅ ⟨ sin(t), cos(t), √t ⟩ + ⟨t,t²,1⟩ ⋅ ⟨ cos(t), -sin(t), 1/(2√t) ⟩
      • = ( sin(t) + 2t cos(t) + 0 ) + ( tcos(t) - t²sin(t) + 1/(2√t) )
      • = sin(t) ( 1 - t² ) + 3t cos(t) + 1/(2√t)
    • d/dt( u(t) × v(t) ) = du/dt × v(t) + u(t) × dv/dt
    • Moon’s orbit around Earth can be (very) approximately described by r(t) = ⟨ 385000 cos( πt/14), 385000 sin(πt/14) ⟩ with distances in kilometers and times in days
      • r(t) = 385000 ⟨ cosx, sinx ⟩ where x = πt/14
      • What is the moon’s velocity?
        • v(t) = dr/dt
        • = 385000 d/dx⟨cos,sinx⟩ dx/dt
        • = 385000 ⟨ -sin(πt/14), cos(πt/14) ⟩ π/14
      • What is its speed?
        • Speed = | v(t) |
• Next
  • Integrals of vector-valued functions and some applications
  • Read section 13.2

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