Find an equation of the tangent plane to the given parametric surface at the specified point. r(u, v) = uv i + u sin(v) j + v cos(u) k; u = 0, v = t
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- (a) Find the exact area of the surface obtained by rotating the curve y = e^x about thex-axis over the interval 0 ≤ x ≤ 1.(b) Determine the length of the parametric curve given by the following set ofparametric equations.x = 3 cos t − cos 3t, y = 3 sin t − sin 3t, 0 ≤ t ≤ πYou may assume that the curve traces out exactly once for the given range of t.Find parametric equations for the tangent line to the curve x=cos(t), y=sin(t), z=t at the point(cos(5pi/6),sin(5pi/6),5pi/6). Use u as the parameter.x(u) = y(u) = z(u) = (The line should be parametrized so that it passes through the given point at u=0).(a) By eliminating the parameter, sketch the trajectory over the time interval 0 ≤ t ≤1 of the particle whose parametric equations of motion are x = cos (πt), y =sin(πt)(b) Indicate the direction of motion on your sketch.(c) Make a table of x-and y-coordinates of the particle at times t = 0, 0.25, 0.5, 0.75, 1.(d) Mark the position of the particle on the curve at the times in part (c), and label those positions with the values of t.
- 40) Find an equation of the tangent plane to the parametric surface at the given point. If you have software that graphs parametric surfaces, use a computer to graph the surface and the tangent plane. x = u^2, y = v^2, z = uv; u = 1, v = 1find the moment of inertia with respect to the origin: 1. the area of the curve r^2 =a^2 cos 2 θThe tangent line to a smooth curve r(t)=f(t)i+g(t)j+h(t)k at t=t0 is the line that passes through the point (f(t0), g(t0), h(t0)) parallel to v(t0), the curve's velocity vector at t0. find the parametric equation for the line that is tangent to the given curve at the given parametric value t=t0. r(t)= (sin t)i + (cos t)j + (sin 2t)k, t0= pi/2 . x=?
- Part 1. Find the curvature κ(t)κ(t) of the curve r(t)=(−2sint)i+(−2sint)j+(2cost)k Part 2. Find parametric equations for the tangent line at the point(cos(−4π/6),sin(−4π/6),−4π/6)(cos(−4π6) on the curve x=cos(t), y=sin(t), z=(t) x(t)= y(t)= z(t)=Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. x = e−8t cos(8t), y = e−8t sin(8t), z = e−8t; (1, 0, 1)Find the equation for the tangent plane to the surface at the indicated point. (Hint: Solve for z in terms of x and y.) z = e6x2 + 8y2, P(0, 0, 1)
- (b) Determine the length of the parametric curve given by the following set of parametric equations. x = 3 cos t − cos 3t, y = 3 sin t − sin 3t, 0 ≤ t ≤ πYou may assume that the curve traces out exactly once for the given range of t.Eliminate the parameter t. Then use the rectangular equation to sketch the plane curve represented by the given parametric equation x = 1 + 3 cos t, y = 2 + 3 sin t; 0≤ t < 2π. Use arrows to show the orientation of the curve corresponding to increasing values of t. If an interval for t is not specified, assume that - ∞ < t < ∞.