Prove that the functions (a)
are solutions of the wave equation with the specified initial-boundary conditions:
Want to see the full answer?
Check out a sample textbook solutionChapter 8 Solutions
Numerical Analysis
- 4. ) Find the directional derivative of the function f(x, y) = sin²(xy) at the point P (2,7) in the direction of the unit vector ū = −²¹+] √5arrow_forwardUse Theorem 13.9 to find the directional derivative of the function at P in the direction of the unit vector u = cos ei + sin ej. f(x, y) = v P(3, 0), 0 = - 7 4x + y' 1 2/3 2arrow_forwardA particle's position vector is given by: F(t) = R(1+ cos(wot + q cos wot))& + R sin(wnt + q cos wot)ŷ (= What is the particle's maximum speed? If it helps, you can assume that R, wo, and q are all positive numbers, and that q is very small.arrow_forward
- 2. Suppose that a motion described by the vector valued function (i.e. position function) r(t) has velocity given by r'(t) = v(t) = (5 cos t, 5 sin t, –2) and that r(0.) = (2,1, 1). Find the formula for the position function r(t).arrow_forward8) 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).arrow_forwardWhich of the following are parametric equations for the entire line y = x + 1? Choose all that apply. x(t) = cos(t), y(t) = cos(t) + 1 Ox(t)=t+2, y(t) = t + 3 x(t)=t, y(t) = t + 1 Ox(t)=t+1, y(t) = t Ox(t) = t1, y(t) = t x(t) =tan(t), y(t) = tan(t) + 1 x(t) = t², y(t) = ² + 1 Ox(t) = t³, y(t) = t³ + 1arrow_forward
- 01. Find the directional derivative of p = x²z + 2xy? + yz? at (1, 2,-1) in the direction of vector (2i + 3j – 4k)arrow_forwardFind the principal unit norma l r(t) = 6 cos ti + 6 sin tj + vector to the curve at the specified value of the parameter t = 3 π /4.arrow_forward6. Find the position vector 7(t) velocity vector v (t),acceleration a (t), and the speed for the motion of a particle described with parametric equations: a = 3 sin(2t), the distance that the particle travels from t = 0, to t = r. y = 3 cos(2t), z = 2t – 1. Findarrow_forward
- Let r(t) = (sin(2t), 3t, cos(2t)), t E [-n, 7] be the position vector of a particle at time t. (a) Show that the velocity and acceleration vectors are always perpendicular. (b) Is there any time t for which r(t) and the velocity vector are perpendicular? Is so, find all such values of t.arrow_forwardThe acceleration vector for the spacecraft Dolphin 163 is given by d(t) = (-2 cos(t), 0,–2 sin(t)). It is also known that the velocity and position at t = 0 are ü(0) = (0, V5, 2) and r(0) = (3, 0,0 ). Assume distances are measured in kilometers (km) and time is measured in seconds (s). (a) Find the position function F(t) for the spacecraft. (b) Find the function for the speed of the spacecraft and the speed when t = 0. (c) Compute the curvature of the trajectory when t = 0. (d) At time t = T seconds the spacecraft launches a probe in a direction opposite of N, the unit normal vector to r. If the probe travels along a straight line in the direction it was launched from the spacecraft for 5 km and then stops, what is its resting coordinate?arrow_forwardFind parametric equations for the tangent line at the point(cos(−4π/6),sin(−4π/6),−4π/6) on the curve x=cos(t), y=sin(t), z=(t) x(t)= y(t)= z(t)=arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage