Reference no: EM131006085

Question 1. A force F = 3i + 4j - 3k acts on a body at point A (3, 2, -1) and moves it to point B (4, -5, 2). Find the work done by the force on the body.

Question 2. A force F = 3i + 4j - 3k acts at point A (1, 5, -3) on a solid, which is free to turn about a fixed point O (3, 2, 1). Find the torque exerted by the force on the body.

Question 3.

1) Find parametric equations of the straight line that passes through the point A (1, 5, -3) and is orthogonal to the plane 2x + y - 3 z = 2.

2) At what point does this line intersect the plane?

Question 4.

1) Find the angle between the following planes: -x + 3y - 2 z = 1 and 2x + y - 2 z = 5.

2) Find the line of their intersection.

Question 5. Find the angle between the following lines:

(x-2)/2 = (y+1)/4 = z/-1  and (x +1)/-2 = (y+5)/3 = (z-3)/-1.

Question 6. Name and sketch each of the following surfaces:

1) z = x² + y², 2) x² + y² = z². Write the equation of each surface: a) in cylindrical coordinates, b) in spherical coordinates.

Question 7. Suppose the movement of an object is described by the position function
 x = 2t - sin 2t
 y = 1- cos 2t. Find the speed and acceleration of the object: 1) at time t; 2) at time t = Π/6.
 z = 2

Question 8. Suppose the movement of an object is described by the position function

x = cos³ t

y = sin³t. Find the distance travelled from time t = 0 to time t = Π/2.

z = 1

Question 9. The function f is defined by:f (x, y) = xy/(x² + y²  if x ≠ 0 or y ≠ 0,

                                                                           0          if x = y = 0.

1) Find the limit of f at point (0, 0) or explain why it does not exist.

2) Is the function f continuous at point (0, 0)?

Question 10. Draw each of the following two surfaces in Matlab. Attach Matlab code and picture. What is the name of the surface?

1) z = x² - y² , - 2 ≤ x ≤ 2, - 3 ≤ y ≤ 3 .

2) z = 2x² + 2 y². Change the equation to cylindrical coordinates (θ, ρ, z) and draw the surface in Matlab for 0 ≤ θ ≤ 2Π, 0 ≤ ρ ≤ 3.