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.