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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 MIT OpenCourseWare http://ocw.mit.edu 18.02SC Multivariable Calculus Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
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