3. Consider the following initial-boundary value problem for the wave equation. Utt - Uxx = 0, x Є R, t> 0, u(x, 0) = e˜¼** x Є R, u₁(x, 0) = −x³ e˜¼×³¸ x Є R. (a) Find the solution to the above problem. (b) Sketch the solution you found and give a physical interpretation.

4. Find the parametric equation of the line through (1, 0, -2) and parallel to the line ⅓ (x – 4) = ½ (y) = z + 2.

1. Find the second derivative of the parametric equations x = a cos t and y = b sin t 2. Use second order derivative to derive the acceleration, a for an object that falls where its movement is described by s = V,t + gt? %3D

a2u satisfies the wave equation əx² -n a²u Verify that U(x, t) = e¬Vkt cos\ax %3D k at2

Which 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 + 1

Find the First and Second derivatives of y with respect to x from parametric equations. a). x=3 (t + 2) 2 ; y = 9t2-5

2. Find a set of parametric equations for a particle that oscillates on the line y = 6x – 5 between the points (1, 1) and (5, 25) starting at (1, 1). That is when t = 0 the particle is at the point (1, 1).

Find dy/dx of the following set of parametric equations 1. x = 3(t-2)2 ,  y = 9t2 + 4 2. x = 1/(t+1)2 ,  y = t2 + 3

Suppose that a particle follows the path r(t) = 2 sin(3t) i+4 cos(3t) j. Give an equation (in the form of a formula involving x and y set equal to 0) whose whose solutions consist of the path of the particle. = 0. (Answer in terms of x and y.) Determine the velocity vector of the particle when t = T : v(7) = (Answer in terms of t.) Determine the acceleration vector of the particle whent = t : a(7) = (Answer in terms of t.)