Question

what is the answer for number 4

1. Let r(t) = -i-e2t j + (t? + 2t)k be the position of a particle moving in space. a. Find the particle's velocity, speed and direction at t = 0. Write the velocity as a product of speed and direction at this time. b. Find the parametric equation of the line tangent to the path of the particle at t = 0. 2. Find the integrals: a. S (tezi - 3sin(2t)j + ick) at b. SGhi - 3sin(2t)j + 1+ k) 3. A projectile is launched from a height of 64 feet with initial velocity of 96 feet per second and initial angle of 30°. Determine the projectile's: a. maximum height, b. flight time and c. range 4. Let f(x, y, z) = In(x2+ya+z2 - 4) for this problem. a. Evaluate f(0, -1, 2) and f(2,3,-1) b. Describe the domain of f. C. Describe the level surface f(x,y,z) = ln(5) 5. Determine the following limits: lim Sxy-5y x2+x2-222 b lim - C. lim X-2-2 - (xyz)--(1,-1,2) 22 (xy,z)-(2,3,2) xy-zy (x.2)-(4,2) VX - VZ+2 a 6. Show the following limit does not exist: lim - (xy)-(0,0) x+y 7. a. Find the first partial derivatives of f(x,y,z) = xcos?(xz) + ye*y - yln(yz) b. Find the second partial derivatives of g(x,y) = xsin(xy) c. Show f(x, y, z) = 2x3 - 3x(y2 + z) satisfies the LaPlace equation fxx + fyy + fzz = 0 8. a. Let w = f(x,y,z), x = g(p,q), y = h(p, q), z = k(p, q).p = u(t),q = v(t). Draw a dependency diagram for and the use it to write a Chain Rule for b. Let z = 3e* sin(y), x = vln(u),y = uv. Use a Chain Rule to find evaluated at (u, v) = (1,3) 9. Assume the equation x3z-3xy2 + 2yz= 0 defines z as a differentiable function of x and y. Find d then evaluate these at (x,y,z) = (2,-1,3). 9. Assume the edz and az and then eva

Answer 1

f(x,y,z)=In (x²+y? +z–4) f(0,-1,2)=ln (oP+(-1) +22–4]=In 1=0 |(2,3,-1)=ln (22+(3)2+(-1)2–4]=In 10 (6) Domain off Function f is defined for all values of x,y,z for which (x²+ya+z2-4)>0 x?+y+z<>4 D={x,y,z\ 2°+y+z>4} (c) f(x,y,z)=In 5 +In (x²+y²+z2 – 4)=ln 5 +x+y+z2–4=5 +x+y?+z=9 +x+y +z+=32 Level surfaces is sphere with centre at the origin and radius=3