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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![Let u(x, t) be the solution of the one-dimensional wave equation with speed c > 0
Utt = c²uxx
u(0, t) = 0 = u(L, t),
u(x, 0) = f(x)
0 < x < L,
t> 0
t> 0
u₁(x, 0) = 0.
Show that
u(x,t)
=
[F(x-
[F(x − ct) + F(x + ct)]
where F(x) is the odd periodic extension of f(x).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff6cd132a-964f-4a31-af3d-1b1c2cf60729%2F7ab79c86-8f50-4cca-a87f-31948d7f9f16%2F6wyg02p_processed.jpeg&w=3840&q=75)
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- 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).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 -? .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.The directional derivatives of fAx, y,2) = x'y+ 4y°z+ 3xz? at (3,3,3) in the direction of = 3i + 6ị + 6k 1s
- 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 =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 .10. Given the curve C defined by the vector-valued function F(t) = a) find an equation of the line that is tangent to this curve at the point (1,1,0). (You can give the parametric equations of the line.) b) Find the length of the curve C from the point (1,1,0) to the point. (1,-1,5) (in the direction of increasing t). c) Compute the curvature of the curve C at the point (1,1,0). 11. Determine whether the lines L₁ given by x=t, y=1+2t, z=2+ 3t, (with -∞ < t < oo) and L₂ given by x = 3 - 4s, y = 3+2s, z = 2-s, (with -∞ < s <∞o) are parallel, intersect or are skew lines. 12. Consider the function f(x, y) = { 27² if (x, y) = (0,0) 1 if (x, y) = (0,0) a) Does lim(z,y)+(0,0) f(x, y) exist? Justify your answer. b) Is f continuous at (0,0)? Justify your answer. af c) Compute of (0,0), (0,0), if they exist there. (Hint: You will need to use the limit definition of the partial derivative.)
- 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)