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- 2. Find the directional derivative of 9 = yx? +x?z + 3xyz in the direction of vector n = Î + 2ŷ -? .8) Find the position vector r(t) for a particle with acceleration a(t) = (5t, 5 sin t, cos 6t), initial velocity (0) = (3, -3, 1) and initial position (0) = (5, 0, -2).A particle is moving with velocity V(t) = ( pi cos (pi t), 3t2+ 1) m/s for 0 ≤ t ≤ 10 seconds. Given that the position of the particle at time t = 2s is r(2) = (3, -2), the position vector of the particle at t is?
- Let u(t) = 5t°i+ 2-9)j-8k and v(t) = e'i+9 e-j- e k. Compute the derivative of the following function. u(t) • v(t) Select the correct choice below and fill in the answer box(es) to complete your choice. The derivative is the vector-valued function (i+ ( i+ (k. O B. The derivative is the scalar functionThe directional derivatives of fAx, y,2) = x'y+ 4y°z+ 3xz? at (3,3,3) in the direction of = 3i + 6ị + 6k 1sShow that the function Z = sin(wct)sin(wx) satisfies the wave equation
- Sketch the curve whose vector equation is Solution r(t) = 6 cos(t) i + 6 sin(t) j + 3tk. The parametric equations for this curve are X = I y = 6 sin(t), z = Since x² + y² = + 36. sin²(t) = The point (x, y, z) lies directly above the point (x, y, 0), which moves counterclockwise around the circle x² + y2 = in the xy-plane. (The projection of the curve onto the xy-plane has vector equation r(t) = (6 cos(t), 6 sin(t), 0). See this example.) Since z = 3t, the curve spirals upward around the cylinder as t increases. The curve, shown in the figure below, is called a helix. ZA (6, 0, 0) (0, 6, 37) I the curve must lie on the circular cylinder x² + y² =At time t = 0, a particle is located at the point (1, 2, 3). It travels in a straight line to the point (4, 1, 4), has speed 2 at (1, 2, 3) and constant acceleration 3i - j + k. Find an equation for the posi-tion vector r(t) of the particle at time t.4. Determine the directional derivatives of +V3 sin" xy $(x, y) =tan at the point (1,1) in the direction of the vector 31-2j .
- Let c (1) = e*i + 3 sin(t)j + t'k and e,(t) = e-8'i + 2 cos(t)j – 81'k. (Enter your solution as a single vector using the vector form (*,*,*). Use symbolic notation and fractions where needed.) d -[c (t) + c2(t)] = dtparticle moves along a path with velocity (t) = sin(t) i + t³j + etk. ts) Find its acceleration. Suppose the particle's initial position is P(2,1,-1). Find the particle'sFind the velocity and acceleration vectors in terms of u, and ug- de r=a cos 20 and dt = 5t, where a is a constant (- 10at sin 20 ) u, + ( 5at cos 20 ) ue y = - a cos (20) • (4 + 5t)) u, + (5a( cos (20) – 4t sin (20)) ue a =