(a) Find the derivative of r(t) = (4 + t)i + tej + sin(5t) k. (b) Find the unit tangent vector at the point t = 0. Solution (a) According to the theorem that states if r(t) = (f(t), g(t), h(t)) = f(t)i + g(t)j + h(t) k, where f, g, and h are differentiable functions, then r'(t) = (f'(t), g'(t), h'(t)) = f'(t)i + g'(t)j + h'(t) we differentiate each component of r. r'(t) = (b) Since r(0) = T(0) = r'(0) Ir'(0)| and r'(0)j + 5k, the unit tangent vector at the point (4, 0, 0) is j + 5k