Vector Derivatives and Arc Length
                                2     3
                1.  Let r(t) = t i + t j.
                a) Compute, velocity, speed, unit tangent vector and acceleration.
                b) Write down the integral for arc length from t = 1 to t = 4. (Do not compute the integral.)
                Answer: a) Velocity = v = dr = h2t, 3t2i.
                               p              dt
                                    2     4
                Speed = |v| =     4t +9t .                                     
                                               v            2t           3t2
                Unit tangent vector = T = ds/dt =       √ 2       4, √ 2       4  .
                                                          4t +9t       4t +9t
                Acceleration = a = dv = h2,6ti.
                                 Z dt         Z
                                    4 ds        4 p
                                                      2     4
                b) Arc length =       dt dt =       4t +9t dt.
                                   1           1
                2.  Consider the parametric curve
                                               x(t) = 3t +1,    y(t) = 4t + 3.
                a. Compute, velocity, speed, unit tangent vector and acceleration.
                b. Compute the arc length of the trajectory from t = 0 to t = 2.
                Answer: a) Velocity = v = dr = h3, 4i.
                               √              dt
                Speed = |v| =    9+16=5.                    
                Unit tangent vector = T =      v   = 3, 4 .
                                             ds/dt      5 5
                Acceleration = a = dv = h0,0i.
                                 Z dt         Z
                                    2 ds        2
                b) Arc length =       dt dt =    5dt = 10.
                                   0           0
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       18.02SC Multivariable Calculus 
       Fall 2010 
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