758 CHAPTER 14 CALCULUS OF VECTOR-VALUED FUNCTIONS . If R', (t) = R'2(t), then Ri(t) = R2(f) + c for some constant vector c. . The Fundamental Theorem of Calculus for vector-valued functions: If r() is contin- yous , then crowes bent bred ow srouthoo tedint ard - if R' (t) = r(t), then r(t) dt = R(b) - R(a), d [ r(s ) ds = *(t ) 14.2 EXERCISES Preliminary Questions 1. State the three forms of the Product Rule for vector-valued functions. 5. The derivative of a vector-valued function is the slope of the tangent In Questions 2-6, indicate whether the statement is true or false, and if it is false, provide a correct statement. line, just as in the scalar case. 2. The derivative of a vector-valued function is defined as the limit of a 6. The derivative of the cross product is the cross product of the difference quotient, just as in the scalar-valued case. derivatives. component. 3. The integral of a vector-valued function is obtained by integrating each 7. State whether the following derivatives of vector-valued functions ri(t) 4. The terms "velocity vector" and "tangent vector" for a path r(t) mean and r2(t) are scalars or vectors: the same thing. (a) -ri(t) ( b ) (ri(t) . r2(t) ) (@) (ri(1) x 12(1)) Exercises In Exercises 1-6, evaluate the limit.() ()5 - 16() 18. Determine the values of t between 0 and 2x such that the tangent vec- tor to the cycloid r(t) = (t - sint, 1 - cost) is a unit vector. 1. lim (12 , 41, - not hne \\ v2. jim sin 2ti + costj + tan 4/k In Exercises 19-22, evaluate the derivative by using the appropriate Prod- uct Rule, where 3. lim e2ti + In(t + 1)j + 4k e' - 1, 41) 4. lim 1 1 ri(1) = (12, 13, 1), 12 (1) = (@31, e2t, et ) 19. ~ ( ri(t) . 12 (t ) ) 20 . (14ri(t ) ) h -+ 0 5. Evaluate lim "( +h) -r( for r(t) = (t-1, sint, 4). 21. 7 (ri(t) x 12(1)) 1 0 t 6. Evaluate lim -for r(t) = (sint, 1 - cost, -2t). In Exercises 7-12, compute the derivative. wayomni Line .noin 22. a (r(t) . ri(t)) , assuming that 7. r(t) = (1, 12, 13) 8. r(1) = (2+2, 121, 2-1-2) I ( 2 ) = (2, 1, 0) , I' (2) = (1, 4, 3) 9. r(s) = (el-s, 1 - s, In(1 - s)) In Exercises 23 and 24, let 10 . b(1 ) = (@ 31-4,@6-1, (1+ 1)-1 ) 11. c(1) =1-li-e2k Just SetluM mmtoo? ~ ri(t) = (12, 1, 21), r2(t) = (1, 2, et ) 12. a(0) = (cos 30)i + (sin20)j + (tan 0)k 23 . Compute -ri(t ) . r2(1) in two ways : 13. Calculate r'(t) and r"(t) for r(t) = (t, 12,13). (a) Calculate ri(t) . r2(t) and differentiate. 14. Sketch the curve parametrization) for -1 St 1. (b) Use the Dot Product Rule. Compute the tangent vector at t = 1 and add it to the sketch. 24. Compute _ri(t) x r2(t) _in two ways: 15. Sketch the curve parametrization2) together with its tan- (a) Calculate ri(t) x r2(t) and differentiate. gent vector at f = 1. Then do the same for r2(t) = (13, to). (b) Use the Cross Product Rule. 16. Sketch the cycloid r(t) = (t - sint, 1 - cost) together with its tangent In Exercises 25-28, evaluate -,r(8(t)) using the Chain Rule. vectors at t = ] and 17. Determine the value of t between 0 and 2xt such that the tangent vector 1 sib ballen at ( 25. r(t) = (12, 1-1), 8(t) = e' to the cycloid r(t) = (t - sint, 1 - cost) is parallel to ( 3, 1). our26. r(1) = (12, 13 ), 8 (1 ) = sint