Problems #24, 40, 41

In Exercises 21-24., give the equation of the line tangent to the graph of r (t) at the given t value. 21- -(1) (to + t, to - t) att = 1. 22-T(t) - (3 cost, sint) at t - 7/4 23.+ (t) = (3 cost, 3 sint, t) at t - 7 24. T (t) = (e , tant, t) at t = 0.In Exercises 37-40., solve the given initial value problems. Find 7 (t), given that r (t) - (t, sint) and r (0) = (2, 2). 38. Find r (t), given that r (t) - (1/ (t + 1), tant) and 7 (0) = (1, 2). 39. Find r (t). given that (t) - (t2, t. 1). " (0) = (1, 2, 3) and 7 (0) = (4, 5, 6). 40. Find r (t), given that r (t) = (cost, sint, e ). T' (0) = (0, 0,0) and T'(0) = (0, 0,0).In Exercises 41-46., find the arc length of r (t ) on the indicated interval. 41. 7(t) = (2 cost, 2 sint, 3t) on |0, 27]. 42. 7(t) = (5 cost, 3 sint, 4 sint) on |0, 27] 43. 7(t) - (+3, t2, +3) on [0, 1] 44-T(t) = (e-* cost, e * sint) on [0, 1] 45. 7(t ) = (t, 13 ) on [0, 3]. 46.7(t) = (t cost, t sint, t) on [0, 4x]