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- Suppose C is the curve from (0, 0) to (2, 0) to (2, 3) to (0, 3) to (0, 0). Find the work done by the vector fieldF(x, y) = <x^(3)−2y^(2), x + cos(√y)> on a particle moving along C.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 1Find 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).
- Consider the curve y = 10 + 4x − x^(2) at (x, y) = (3, 13). Find a vector, v, that has length 4 and is parallel to the tangent line to y = 10 + 4x−x^(2) at x = 3.Find the parametrization of two different curves from the point (2,4) to (3,9). Compute the work done of the vector field F=〈2xy,x2+2〉over the two curves found in part (a).Suppose 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?
- Der. 3D 19 0 Find the unit tangent vector at the point with the givenvalue of the parameter t.Suppose z is given implicitly by the equation ln (yz2) + x3z = 1 in a neighborhood of the point P (1, 1), in which z = 1. The value of the directional derivative of z at P in the direction of the vector w = (1, −1), corresponds to:8.1 If C is the curve given by r(t)=(1+5sint)i+(1+3sin2t)j+(1+3sin3t)kr(t)=(1+5sint)i+(1+3sin2t)j+(1+3sin3t)k, 0≤t≤π20≤t≤π2 and F is the radial vector field F(x,y,z)=xi+yj+zkF(x,y,z)=xi+yj+zk, compute the work done by F on a particle moving along C.
- Suppose that z is an implicit function of x and y in a neighborhood of the point P = (0, −3, 1) of the surface S of equation: exz + yz + 2 = 0 An equation for the tangent line to the surface S at the point P, in the direction of the vector w = (3, −2), corresponds to:Consider I = ∫CF⋅dr, where (img17) is a conservative vector field and curve C is parameterized by:α (t): = ((2 − cos (5t)) cost, (2 − cos (5t)) synt, sin (5t)) with 0≤t≤π. We have that the value of I is equal to: (img18)2.10 : (a) Express the vector fieldH = xy*2zax + x*2yzay+xyz*2a2in cylindrical and spherical coordinates. (b) In both cylindrical and spherical coordinates, determine H at (3, —4, 5).