These were selected from the first midterm given by another instructor.

1.

A particle starts at the origin with initial velocity i+2 j+k. Its acceleration at any later time t is given by
a = sin(t) i+cos(t) j+2e^t k. Find its velocity and position for all t > 0.

2.

Let f(x,y,z) = x²y+y^1/2+z.

(a)

Find the directional derivative of f at the point (2,1,0) in the direction towards the point (3,3,2).

(b)

Find the direction of maximum rate of increase of f at the point (2,1,0) and find the maximum rate of increase of f.

(c)

Find the equation of the tangent plane to the level surface f(x,y,z) = 5 at the point (2,1,0) for the function f defined above.

3.

A particle travels in the plane so that its position vector is given by r(t) = 3e^t i+(t²+4t) j for all real numbers t.

(a)

Find the velocity, v, the speed, v, and the acceleration a for all times t.

(b)

Find the velocity, speed, and acceleration at the point (3,0), that is, when t = 0.

(c)

Find the unit tangent vector (T), unit normal vector (N), the curvature (K), and the tangential and normal components of acceleration (a_T and a_N), all at the point (3,0) (as above in (b)).

4.

Find the critical points of the function f(x,y) = x³-3xy+y³ and use the second derivative test to determine whether each is a relative maximum, a relative minimum, or a saddle point.

5.

Find the maximum and the minimum values of the function f(x,y) = x+y+1 on the closed bounded region whose interior is defined by the inequality x²-2xy+2y² < 6.