Problem 41E: Compute the derivatives of the vector-valued functions. 41. r(t)=t3i+3t2j+t36k Problem 42E: Compute the derivatives of the vector-valued functions. 42. r(t)=sin(t)i+cos(t)j+etk Problem 43E: Compute the derivatives of the vector-valued functions. 43. r(t)=eti+sin(3t)j+10tk . A sketch of the... Problem 44E: Compute the derivatives of the vector-valued functions. 44. r(t)=eti+2etj+k Problem 45E: Compute the derivatives of the vector-valued functions. 45. r(t)=i+j+k Problem 46E: Compute the derivatives of the vector-valued functions. 46. r(t)=teti+tln(t)j+sin(3t)k Problem 47E: Compute the derivatives of the vector-valued functions. 47. r(t)=1t+1i+arctan(t)j+lnt3k Problem 48E: Compute the derivatives of the vector-valued functions. 48. r(t)=tan(2t)i+sec(2t)j+sin2(t)k Problem 49E: Compute the derivatives of the vector-valued functions. 49. r(t)=3i+4sin(3t)j+tcos(t)k Problem 50E: Compute the derivatives of the vector-valued functions. 50. r(t)=t2i+te2tj5e4tk Problem 51E: For the following problems, find a tangent vector at the indicated value of t. 51.... Problem 52E: For the following problems, find a tangent vector at the indicated value of t. 52.... Problem 53E: For the following problems, find a tangent vector at the indicated value of t. 53.... Problem 54E: For the following problems, find a tangent vector at the indicated value of t. 54.... Problem 55E: Find the unit tangent vector for the following parameterized curves. 55.... Problem 56E: Find the unit tangent vector for the following parameterized curves. 56. r(t)=costi+sintj+sintk,0t2.... Problem 57E: Find the unit tangent vector for the following parameterized curves. 57.... Problem 58E: Find the unit tangent vector for the following parameterized curves. 58. r(t)=ti+3tj+t2k Let... Problem 59E: Find the following. 59. ddt[r(t2)] Problem 60E: Find the following. 60. ddt[t2.s(t)] Problem 61E: Find the following. 61. ddt[r(t).s(t)] Problem 62E: Compute the first, second, and third derivatives of r(t)=3ti+6ln(t)j+5e3tk . Problem 63E: Find r(t)r(t) for r(t)=3t5i+5tj+2t2k . Problem 64E: The acceleration function, initial velocity, and initial position of a particle are... Problem 65E: The position vector of a particle is r(t)=5sec(2t)i4tan(t)j+7t2k. Graph the position function and... Problem 66E: Find the velocity and the speed of a panicle with the position function r(t)=(2t1)(2t+1)i+ln(14t2)j... Problem 67E: Find the velocity function and show that v(t) is always orthogonal to r(t) . Problem 68E: Show that the speed of the particle is proportional to the angular velocity. Problem 69E: Evaluate ddt[u(t)u(t)] given u(t)=t2i2tj+k . Problem 70E: Find the antiderivative of r(t)=cos(2t)i2sintj+11+t2k that satisfies the initial condition... Problem 71E: Evaluate 03ti+t2jdt . Problem 72E: An object starts from nest at point P(1,2,0) and moves with an acceleration of a(t)=j+2k , where... Problem 73E: Show that if the speed 0f a particle traveling along a curve represented by a vector-valued function... Problem 74E: Given r(t)=ti+3tj+t2k and u(t)=4ti+t2j+t3k , find ddt(r(t)u(t)) . Problem 75E: Given r(t)=t+cost,tsint , find the velocity and the speed at any time. Problem 76E: Find the velocity vector for the function r(t)=et,et,0 . Problem 77E: Find the equation of the tangent line to the curve r(t)=et,et,0 at t=0 . Problem 78E: Describe and sketch the curve represented by the vector-valued function r(t)=6t,6tt2 . Problem 79E: Locate the highest point on the curve r(t)=6t,6tt2 and give the value of the function at this point. Problem 80E: The position vector for a particle is r(t)=ti+t2j+t3k . The graph is shown here: 80. Find the... Problem 81E: The position vector for a particle is r(t)=ti+t2j+t3k . The graph is shown here: 81. Find the speed... Problem 82E: The position vector for a particle is r(t)=ti+t2j+t3k . The graph is shown here: 82. Find the... Problem 83E: A particle travels along the path of a helix with the equation r(t)=cos(t)i+sin(t)j+tk . See the... Problem 84E: A particle travels along the path of a helix with the equation r(t)=cos(t)i+sin(t)j+tk . See the... Problem 85E: A particle travels along the path of a helix with the equation r(t)=cos(t)i+sin(t)j+tk . See the... Problem 86E: A particle travels along the path of a helix with the equation r(t)=cos(t)i+sin(t)j+tk . See the... Problem 87E: A particle travels along the path of an ellipse with the equation r(t)=costi+2sintj+0k . Find the... Problem 88E: A particle travels along the path of an ellipse with the equation r(t)=costi+2sintj+0k . Find the... Problem 89E: A particle travels along the path of an ellipse with the equation r(t)=costi+2sintj+0k . Find the... Problem 90E: Given the vector-valued function r(t)=tant,sect,0 (graph is shown here), find the following: 90.... Problem 91E: Given the vector-valued function r(t)=tant,sect,0 (graph is shown here), find the following: 91.... Problem 92E: Given the vector-valued function r(t)=tant,sect,0 (graph is shown here), find the following: 92.... Problem 93E: Find the minimum speed of a particle traveling along the curve r(t)=t+cost,tsintt[0,2] . Problem 94E: Given r(t)=ti+2sintj+2costk and u(t)=1ti+2sintj+2costk , find the following: 94. r(t)u(t) Problem 95E: Given r(t)=ti+2sintj+2costk and u(t)=1ti+2sintj+2costk , find the following: 95. ddt(r(t)u(t)) Problem 96E: Now, use the product rule for the derivative of the cross product of two vectors and show this... Problem 97E: Find the unit tangent vector T(t) for the following vector-valued functions. 97. r(t)=t,1t . The... Problem 98E: Find the unit tangent vector T(t) for the following vector-valued functions. 98. r(t)=tcost,tsint Problem 99E: Find the unit tangent vector T(t) for the following vector-valued functions. 99. r(t)=t+1,2t+1,2t+2 Problem 100E: Evaluate the following integrals: 100. ( e ti+sintj+ 1 2t1k) Problem 101E: Evaluate the following integrals: 101. 01r(t)dt , where r(t)=t3,1t+1,et format_list_bulleted
Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage