Problem 1E Problem 2E: Explain the geometric meaning of r(t). Problem 3E Problem 4E: Compute r(t) when r(t) = t10, 8t, cos t. Problem 5E: How do you find the indefinite integral of r(t) = f(t), g(t), h(t)? Problem 6E: How do you evaluate abr(t)dt? Problem 7E: Derivatives of vector-valued functions Differentiate the following functions. 7. r(t) = cos t, t2,... Problem 8E Problem 9E Problem 10E: Derivatives of vector-valued functions Differentiate the following functions. 10. r(t) = 4, 3 cos... Problem 11E Problem 12E: Derivatives of vector-valued functions Differentiate the following functions. 12. r(t) = tan t i +... Problem 13E Problem 14E Problem 15E Problem 16E Problem 17E Problem 18E Problem 19E Problem 20E Problem 21E Problem 22E Problem 23E Problem 24E Problem 25E Problem 26E Problem 27E Problem 28E Problem 29E Problem 30E Problem 31E: Derivative rules Let u(t)=2t3i+(t21)j8kandv(t)=eti+2etje2tk. Compute the derivative of the following... Problem 32E: Derivative rules Let u(t)=2t3i+(t21)j8kandv(t)=eti+2etje2tk. Compute the derivative of the following... Problem 33E: Derivative rules Let u(t)=2t3i+(t21)j8kandv(t)=eti+2etje2tk. Compute the derivative of the following... Problem 34E: Derivative rules Let u(t)=2t3i+(t21)j8kandv(t)=eti+2etje2tk. Compute the derivative of the following... Problem 35E: Derivative rules Let u(t)=2t3i+(t21)j8kandv(t)=eti+2etje2tk. Compute the derivative of the following... Problem 36E: Derivative rules Let u(t)=2t3i+(t21)j8kandv(t)=eti+2etje2tk. Compute the derivative of the following... Problem 37E: Derivative rules Compute the following derivatives. 37. ddt(t2(i+2j2tk)(eti+2etj3etk)) Problem 38E: Derivative rules Compute the following derivatives. 38. ddt((t3i2tj2k)(tit2jt3k)) Problem 39E: Derivative rules Compute the following derivatives. 39. ddt((3t2i+tj2t1k)(costi+sin2tj3tk)) Problem 40E: Derivative rules Compute the following derivatives. 40. ddt((t3i+6j2tk)(3ti12t2j6t2k)) Problem 41E: Higher-order derivatives Compute r(t) and r(t) for the following functions. 41. r(t) = t2 + 1, t +... Problem 42E Problem 43E: Higher-order derivatives Compute r(t) and r(t) for the following functions. 43. r(t) = cos 3t, sin... Problem 44E: Higher-order derivatives Compute r(t) and r(t) for the following functions. 44. r(t) = e4t, 2e4t +... Problem 45E: Higher-order derivatives Compute r(t) and r(t) for the following functions. 45. r(t)=t+4i+tt+1jet2k Problem 46E: Higher-order derivatives Compute r(t) and r(t) for the following functions. 46.... Problem 47E: Indefinite integrals Compute the indefinite integral of the following functions. 47. r(t) = t4 3t,... Problem 48E Problem 49E: Indefinite integrals Compute the indefinite integral of the following functions. 49. r(t) = 2 cos t,... Problem 50E: Indefinite integrals Compute the indefinite integral of the following functions. 50.... Problem 51E: Indefinite integrals Compute the indefinite integral of the following functions. 51.... Problem 52E: Indefinite integrals Compute the indefinite integral of the following functions. 52.... Problem 53E: Finding r from r Find the function r that satisfies the given conditions. 53. r(t) = et, sin t, sec2... Problem 54E Problem 55E Problem 56E: Finding r from r Find the function r that satisfies the given conditions. 56. r(t)=t,cost,4/t; r(1)... Problem 57E: Finding r from r Find the function r that satisfies the given conditions. 57. r(t) = e2t, 1 2et, 1 ... Problem 58E: Finding r from r Find the function r that satisfies the given conditions. 58.... Problem 59E: Definite integrals Evaluate the following definite integrals. 59. 11(i+tj+3t2k)dt Problem 60E: Definite integrals Evaluate the following definite integrals. 60. 14(6t2i+8t3j+9t2k)dt Problem 61E: Definite integrals Evaluate the following definite integrals. 61. 0ln2(eti+etcos(et)j)dt Problem 62E: Definite integrals Evaluate the following definite integrals. 62. 1/21(31+2ticsc2(2t)k)dt Problem 63E: Definite integrals Evaluate the following definite integrals. 63. (sinti+costj+2tk)dt Problem 64E: Definite integrals Evaluate the following definite integrals. 64. 0ln2(eti+2e2tj4etk)dt Problem 65E: Definite integrals Evaluate the following definite integrals. 65. 02tet(i+2jk)dt Problem 66E: Definite integrals Evaluate the following definite integrals. 66. 0/4(sec2ti2costjk)dt Problem 67E Problem 68E Problem 69E Problem 70E Problem 71E Problem 72E Problem 73E: Derivative rules Let u(t) = 1, t, t2, v(t) = t2, 2t, 1, and g(t)=2t. Compute the derivatives of the... Problem 74E Problem 75E: Derivative rules Let u(t) = 1, t, t2, v(t) = t2, 2t, 1, and g(t)=2t. Compute the derivatives of the... Problem 76E: Derivative rules Let u(t) = 1, t, t2, v(t) = t2, 2t, 1, and g(t)=2t. Compute the derivatives of the... Problem 77E: Derivative rules Let u(t) = 1, t, t2, v(t) = t2, 2t, 1, and g(t)=2t. Compute the derivatives of the... Problem 78E: Relationship between r and r 78. Consider the circle r(t) = a cos t, a sin t, for 0 t 2, where a... Problem 79E: Relationship between r and r 79. Consider the parabola r(t) = at2 + 1, t, for t , where a is a... Problem 80E Problem 81E: Relationship between r and r 81. Consider the helix r(t) = cos t, sin t, t, for t . Find all... Problem 82E: Relationship between r and r 82. Consider the ellipse r(t) = 2 cos t, 8 sin t, 0, for 0 t 2. Find... Problem 83E: Relationship between r and r 83. Give two families of curves in 3 for which r and r are parallel for... Problem 84E Problem 85E: Vectors r and r for lines a. If r(t) = at, bt, ct with a, b, c 0, 0, 0, show that the angle between... Problem 86E: Proof of Sum Rule By expressing u and v in terms of their components, prove that... Problem 87E: Proof of Product Rule By expressing u in terms of its components, prove that... Problem 88E Problem 89E: Cusps and noncusps a. Graph the curve r(t) = t3, t3. Show that r(0) = 0 and the curve does not have... Problem 90E: Motion on a sphere Prove that r describes a curve that lies on the surface of a sphere centered at... format_list_bulleted