SPIRAT GOING IS A-NEHISI COATES. EGOING 748 CHAPTER 14 CALCULUS OF VECTOR-VALUED FUNCTIONS functions: 14. Match the space curves in Figure 7 with the following vector-valued 18. Describe the projections of the circle r(f) = (sint, 0.4 + cost) onto (a) ri(t) = (cos 2t, cost, sint) (c) 13(t) = (1, t, 1) (b) r2(1) = (t, cos 2t, sin 2t) the coordinate planes. In Exercises 19-22, the function r(t) traces a circle. Determine the radius, 15. Match the vector-valued functions (a)-(f) with the space curves center, and plane containing the circle. i) (vi) in Figure 9. 19. r(1) = (9 cost)i + (9 sint) (a) r(t) = (1+ 15, e0.08 cost, e0.08 sint 20. r(1) = 71 + (12 cost)j + (12 sint)k (b) r(t) = (cost, sint, sin 12t) 21. r(1) = (sin t, 0.4 + cost) (d) r(t) = (cost, sin' t, sin 2t) (c) r(1) = (1.1. 1+ 12 ) 22. r(1) = (6 + 3 sint, 9,4 + 3 cost) if) r(t) = (cost, sint, cost sin 12t) (e) r(t) = (1, 12, 21) 23. Consider the curve C given by r(t) = (cos(2t) sin t, sin(2t), cos(2t) cost) (a) Show that C lies on the sphere of radius 1 centered at the origin. b) Show that C intersects the x-axis, the y-axis, and the z-axis. 24. Show that the curve C that is parametriz r(1) = (12 - 1,1-212, 4-61) lies on a plane as follows: (i) (ii) (a) Show that the points on the curve at t = 0, 1, and 2 do not lie on a line, (iii) and find an equation of the plane that they determine. (b) Show that for all t, the points on C satisfy the equation of the plane in (a). 25. Let C be the curve given by r(t) = (t cost, t sint, t). a) Show that C lies on the cone x2 + y? = z. b) Sketch the cone and make a rough sketch of C on the cone. 26. CAS Use a computer algebra system to plot the projections onto (iv) (v ) the xy- and xz-planes of the curve r(1) = (t cost, t sint, t) in Exercise 25. (vi) In Exercises 27 and 28, let FIGURE 9 r(t) = (sint, cost, sint cos 2t) 16. Which of the following curves have the same projection onto the be a parametrization of the curve shown in Figure 11. xy-plane? (a) ri(1) = (t, 12, er) (b) 12(1) = (e', 12, 1) (c) 13(1) = (t, 12, cost) 17. Match the space curves (A)-(C) in Figure 10 with their projections (i) (iii) onto the xy-plane. FIGURE 11 (A) (B) 27. Find the points where r() intersects the xy-plane. 28. Show that the projection of r(t) onto the xz-plane is the curve z = x - 2x3 for - 1 5 x 1 29. Parametrization the intersection of the surfaces 12 - 22 = x-2, 12+ 2 2 = 9 (1i) (iji) where z 2 0 using t = y as the parameter. FIGURE 10 30. Find a parametrization of the entire intersection of the surfaces in Exercise 29 using trigonometric functions.31. Viviani's Curve C is the intersection of the surfaces (Figure 12) SHOTTOMU1 030 SECTION 14.1 Vector-Valued Functions 749 x2 + 12 = 2 2. y = 22 (a) Separately parametrization the two parts of C corresponding to In Exercises 36-38, two paths ri(t) and r2(1) intersect if there is a point P x 2 0 and x