4. Consider the wave equation for a vibrating rectangular membrane 0 < y < H. 0 < x < L, J²u J²u J²u მt2 მე2 + მყ2 subject to the initial condition ди u(x, y, 0) = 0, and (x, y, 0) = f(x, y). Ət Solve the initial value problem if ди ди Ju ди (0, y, t) = 0, (L, y, t) = 0, (x, 0,t) = 0, (x,H,t) = 0. მე მე მყ მყ Note: You may assume without derivation that product solutions u(x, y, t) = (x,y)h(t) satisfy dh = -Ac²h dt and the two-dimensional eigenvalue problem V2 + X = 0, and you may use results of the two-dimensional eigenvalue problem.

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).

11.6: Problem 8 Find the directional derivative of f(x, y, z) point (1, 2, 3) in the direction of a vector making an angle of 2 with V f(1, 2, 3). = zx + y“ at the

5C. Under suitable assumptions derive one dimensional wave equation.

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.)

9.) A particle moves in a straight line according to the law of motion: s=cos(x). a. Find the acceleration of the particle when the velocity is zero in the domain [0, 2m). b. Find jerk of the particle when the acceleration is zero in the domain [0, 2n).

2. If a runner traveled a distance of d(h) = V16h4 in time t(h) = 3h4 – 3h3 -5h2, what is the runner's speed? s(h) = 3-3h-5 O s(h) = 12h8 – 12h5 – 20h4 O s(h) = 34 3h-5 O s(h) = 8h° - 12h5 - 10h Black Boy SG.pdf Open file P Type here to search E D F G H C

3. Solve the radial wave in R3 with initral data gler) =4-r², Yer)=0 equation = Au U, =

; (x.v) +(0,0) 0; (x.y) = (0,0) Given the hunclion aboue, fund the directional denvatue along