784 CHAPTER 14 CALCULUS OF VECTOR-VALUED FUNCTIONS 4. Use the decomposition of acceleration into tangential and normal com- 6. What ponents to explain the following statement: If the speed is constant, then around the acceleration and velocity vectors are orthogonal. 7. Two 5. If a particle travels along a straight line, then the acceleration and ve- both of locity vectors are (choose the correct description): (choose (a) orthogonal. (b) parallel. (a) aT Exercises 1. Use the table to calculate the difference quotients "(1 + 7) -r(1) for In Ex h = -0.2, -0.1, 0.1, 0.2. Then estimate the velocity and speed at t = 1. positic 15. a r(0.8) (1.557, 2.459, -1.970) r(0.9) (1.559, 2.634, -1.740) 16. r(1) (1.540, 2.841, -1.443) r(1.1) (1.499, 3.078, -1.035) 17. r(1.2) (1.435, 3.342, -0.428) 18. 2. Draw the vectors r(2 + h) - r(2) and "(2 + h) - r(2) for h = 0.5 for 19. the path in Figure 10. Draw v(2) (using a rough estimate for its length). h . tial loca r(2.5) 20. spe we syd bodhpesb er hing 21 cel at 22 in W In Exercises 3-6, calculate the velocity and acceleration vectors and the speed at the time indicated. 3. r(t) = (13, 1 - t, 412 ), t = 1( )( 4 r(t) = etj- cos(2t)k, t = 0 5. r(0) = (sin 0, cos 0, cos 30), 0 = 6. r(s) = + $2 ' S = 2 7. Find a(t) for a particle moving around a circle of radius 8 cm at a con- stant speed of v = 4 cm/s (see Example 4). Draw the path, and on it, draw the acceleration vector at t = 4. 8. Sketch the path r(t) = (1 - 12, 1 - 1) for -2