APTER 14 . CALCULUS OF VECTOR-VALUED FUNCTIONS b do happen fo sil favremi gne navo ter. Speed is the derivative of distance wars respect to j ds U(1) = dt " () velocity vector wor'() points in the object's speed. [provid 19. At "?)*0] and its magnitude v(t) = Ill is the object's speed If when the say that russ magnitudeength parametrizationbis b - a. . In the its veloc the are length of the curve that is traced over [a, b] is b - a 20. Wha "If r() is a parametrization of a curve C such that r (f) 7 0 for all f, the hyperbo * I (s ) = 1 (8 -' (5 ) ) is (2. 2 ) 21. fund are length parametrization of C, where t = 8 verse of the arc ley and flie 22. The helices has a helix h (a) Sh 14.3 EXERCISES helix. (c) The velocity vectors may point in opposite directions (b) Fi Preliminary Questions 23. L 1. At a given instant, a car on a roller coaster has velocity vector v = (25, -35, 10) (in miles per hour). What would at would the velocity vector be 3. Starting at the origin, a mosquito flies along a parabola with if the speed were doubled? What would it be if the car's direction were U(1) = 12. Let L(1) be the total distance traveled at time t. reversed but its speed remained unchanged? (a) How fast is L(t) changing at t = 2? (a) 2. Two cars travel in the same direction along the same roller coaster (at different times). Which of the following statements about their velocity (b) Is L(t) equal to the mosquito's distance from the origin? N co (b) vectors at a given point P on the roller coaster are true? (a) The velocity vectors are identical. 4. What is the length of the path traced by r(t) for 4