Let u(x, t) be the solution of the one-dimensional wave equation with speed c> 0 Show that Utt = c²uxx u(0, t) = 0 = u(L, t), 0 < x 0 t>0 u(x, 0) = f(x) u₁(x, 0) = 0. u(x, t) = ½ [F(x − ct) + F(x + ct)] where F(x) is the odd periodic extension of f(x).
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- Find 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 =Show that t, e^t, and sin(t) are linearly independent.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).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.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 =
- The directional derivatives of fAx, y,2) = x'y+ 4y°z+ 3xz? at (3,3,3) in the direction of = 3i + 6ị + 6k 1s4. Determine the directional derivatives of +V3 sin" xy $(x, y) =tan at the point (1,1) in the direction of the vector 31-2j .Suppose a particle, whose initial position is (1, 0, 0), moves with velocity given by v(t) = (-1, cos(t), - sin(t)). Compute the vector-valued function that represents the particle's position at any time t = [0, 2π].
- A particle traveling in a straight line is located at the point (1, -1, 2) and has speed 2 at time t = 0. The particle moves toward the point (3, 0, 3) with constant acceleration 2i + j + k. Find its position vector r(t) at time t.2. (4 points) Compute the directional derivative of f at the given point in the direction of the given vector: f(x, y) = In(3+ 2a? + y?), P(2,1), ū = (-3, 4)Q1:- Find the domain of the following vector functions:- (a) f (t) = (cos t)i – Ln(t)j + vt – 2k (b) f (t) = Ln|t – 1|i + e'j + vtk Q2:- Find the domain and the range of the following equations:- 1 -1 (1)W (2)W = sin x y (3)W x²+y2 ху 1 (4)W (5)W = /x² + y2 + z² (6) W = x – y x²+y2+z² (7)W = Ln(x² + y²) (8) W = xy (9) W = 4x² + 9y²