31 plese VIIII8545. limX46. lim (te1181 + f²1 - 7² ³te-¹CHAPTER 13 Vector FunctionsXZA1³ + t2t³ - 1ZAtan-¹t,7-14 Sketch the curve with the given vector equation. Indicatewith an arrow the direction in which t increases.-N7. r(t) = (sin t, t)9. r(t) = (t, 2 t, 2t)11. r(t) = (3, t, 2 - t²)12. r(t) = 2 cos ti + 2 sin tj + k13. r(t) = t²i + tªj + tºk14. r(t)25nSimonian21-26 Match the parametric equations with the graphs(labeled I-VI). Give reasons for your choices.II1 - e-21= costi - cos tj + sin t k15-16 Draw the projections of the curve on the three coordinateplanes. Use these projections to help sketch the curve. B15. r(t) = (t, sin t, 2 cos t)16. r(t) = (t, t, t²)(d)SI17-20 Find a vector equation and parametric equations for theline segment that joins P to Q.17. P(2, 0, 0), Q(6, 2,-2)19. P(0, -1, 1), (1, 3, 4)→y1sin =)t sinMAt664016y8. r(t) = (t²-1, t)10. r(t) = (sin πt, t, cos mt)18. P(-1, 2, -2), Q(-3, 5, 1)20. P(a, b, c), Q(u, v, w)VIXKIVno bavioChaZAXellenZAyZAZĄ timil di bor!21. x = t cos t,22. x = cos f,y=t, z=tsint, t≥0y = sint, z = 1/(1+²)23. x = t, y = 1/(1 + f²), z = ²24. x = cos t,y = sin t, z = cos 2t25. x = cos 8t,26. x = cos²t, y = sin²t, z = ty = sin 8t, z=e08r, t≥0y = t sin t, zfact to help sketch the curve.27. Show that the curve with parametric equations x = t cost,=t lies on the cone z² = x² + y², and use this128. Show that the curve with parametric equations x =sin t,XIV10y = cost, zsin't is the curve of intersection of the surfacesz = x² and x² + y² = 1. Use this fact to help sketch the curve.dalags vid sir(t) = 2ti + e'j + e2¹ k.29. Find three different surfaces that contain the curve202280ur30. Find three different surfaces that contain the curver(t) = t²i + ln tj+ (1/t) k.FOREERtzwoda26 Sorlayd31. At what points does the curve r(t) = ti + (2t - 1²) k inter-sect the paraboloid z = x² + y²?32. At what points does the helix r(t) = (sin t, cos t, t) intersectthe sphere x² + y² + z² = 5?2& Tr33-37 Use a computer to graph the curve with the given vectorequation. Make sure you choose a parameter domain and view-points that reveal the true nature of the curve.33. r(t) = (cos t sin 2t, sin t sin 2t, cos 2t)34. r(t) = (te', e¹, t)35. r(t) = (sin 3t cos t, 1t, sin 3t sin t)39. Graph the curve with parametric equationsx = (1 + cos 16t) cos t36. r(t) = (cos(8 cos t) sin t, sin(8 cos t) sin t, cos t)) = (-37. r(t) = (cos 2t, cos 3t, cos 4t)38. Graph the curve with parametric equations x = sint,y = sin 2t, z = cos 4t. Explain its shape by graphing itsprojections onto the three coordinate planes.y = (1 + cos 16t) sin tF.Ez = 1 + cos 16t 2133X3 1.8equationsx = √1 -0.25 cos² 10t cos ty =√1 -0.25 cos² 10t sin tiz = 0.5 cos 10tExplain the appearance of the graph by showing that it lies onen bil Sela cone.40. Graph the curve with parametric

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We have to find that at what point curve r(t)=ti+(2t-t2)k intersect the paraboloid z=x2+y2

31. At what points does the curve r(t) = ti + (2t - 1²) k inter- sect the paraboloid z = x² + y²? 32. At what points does the helix r(t) = (sin t