Find r'(t), r(t), and r'(t,) for the given value of to: Then sketch the space curve represented by the vector-valued function, and sketch the vectors r(t,) and r'(t,). r(t) = ti + tj + k, to = 2 r'(t) = r(t,) = r'(t,) = -2 2- y -2. -4, -6, 4 -8 -8 r" C y 4. -4 r -2 -2
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- Find a vector equation for the tangent line to the curve of intersection of the cylinders x2+y2=25 and y2+z2=20 at the point (3, 4, 2).If an object travels in the xy-plane along the curve traced out by the vector function r(t) = (t 3/2,-t) for t ≥ 0, then the total distance traveled by the object from t = 0 to t = 4 isSuppose that r(u)r(u) is a vector valued function of uu, and suppose dr/du(3)=(0.5933,−0.3068,0.4641).drdu(3)=(0.5933,−0.3068,0.4641). A particle moves in three dimensional space along the reparametrized curve r(u), where u=2t+t^3. What is the speed of the particle at time t=1?
- give the position vectors of particles moving alongvarious curves in the xy-plane. In each case, find the particle’s velocityand acceleration vectors at the stated times, and sketch them asvectors on the curve. Motion on the parabola y = x2 + 1r(t) = ti + (t2 + 1)j; t = -1, 0, and 1use the parameter x=t to find a vector-valued function for the space curve represented by the intersection of the surfices x2 + z2=4 and x - y=0Find a vector function that represents the curve of intersection of the surface 4x+2y-8z^2=16 and the cylinder of radius 3 wrapped around the y-axis.