sec 4.7: Problem 3 Solve the wave equation with fixed endpoints and the given initial displacement and velocity. Utt = 4uxx, 0 < x < 7, t> 0 t>0 u(0,t)=0, u(7,t) = 0, u(x, 0) ut(x, 0) = = (0, 0 < x < 1/1 ½ < x < 7 [8, 금 5x(7x), 0 < x < 7 We solve the PDE using the series solution u(x,t) = sin (2) ( 7 2nπ 2nπ -bn sin t + cn cos 2nπ 7 7 ·t)). n=1 The coefficients b and c are Fourier coefficients we have computed many times: bn dx Cn = dx help (formulas) help (numbers)

ii. Find parametric equations for the Line through (7, 5) and (-5, 7) 7. Calculate dy/dx at the point indicated: f(0) = (7tan 0, cos O), 0=a/4

At time t=0, a particle is located at the point (3,9,4). It travels in a straight line to the point (7,8,6), has speed 6 at (3,9,4) and constant acceleration 4i-j+2k. Find an equation for the position vector r(t) of the particle at time t -O+¹+* The equation for the position vector r(t) of the particle at time t is r(t) = (Type exact answers, using radicals as needed.)

1. An object moves in space with acceleration a(t) = 12t²i+6tj+2k. When t 0, the object is at (1, 0, -1) with initial velocity v(0) = 2i - 3k. Find the position vector of the object, r(t).