# Sample Practice Exam - M261

This is just a SAMPLE; not all topics can be on the final exam, and just because a topic is not on this sample final does not mean that it will not be included on the real final.

Department of Mathematics
4/28/2000 5:47PM

1.
(50 points, 10 points each part.) Consider the vector function which describes a particle moving along a curve in space.
a.
Find the velocity v and the acceleration a at time t=0.
b.
Find the speed of the particle at time t=0.
c.
Write a parametric form of the tangent line to the curve at the point (1,-1,0).
d.
Compute the work done in moving along the curve above from the point (1,-1,0) to the point (e,-e,e) in the force field .
e.
Set up, but do not evaluate, the integral that computes the distance the particle travels from time t=0 to time t = 1.

2.
(10 points) Use the multivariable chain rule to find and , where (There is no need to simplify your answer.)

3.
(40 points; 10 points each part) The temperature at a point (x,y,z) in space is with some appropriate choice of units.
a.
Compute the gradient vector , at any point (x,y,z).
b.
Find the rate of change (i.e., the directional derivative) of the temperature at the point P = (1,0,2) in the direction towards the point (2,1,2).
c.
Specify a vector in whose direction the temperature increases the fastest at P.
d.
What is this maximum rate of increase at P?

4.
(10 points) Find if z is implicitly defined by the equation xy2z3 + x3y2z = x + y + z. (Your answer should be in the form .)

5.
(20 points) Find and classify all the critical points of the function

6.
(15 points) Set up completely, but do not evaluate, the double integral in polar coordinates that computes the mass of the plane lamina D in the first quadrant enclosed by the curves y=0,y= x, and the circle x2 + y2 = 9; use the density function Sketch the plane region D as well.

7.
(15 points) Set up completely, but do not evaluate, the triple integral in spherical coordinates that computes the yz-moment of the solid H, where H is the part of the sphere of radius 2, centered at the origin,lying in the first octant; use the density function . (Note that the yz-moment is the integral that is the numerator" of the formula computing the x-coordinate of the center of mass.)

8.
(20 points; 10 points each part.) Consider the closed surface S, where S consists of the part of the paraboloid z=1-(x2 + y2) lying above the xy-plane, and the disk in the xy-plane.
a.
Draw a careful sketch of the surface S.

b.
Let F(x,y,z) be the vector field Use the divergence theorem to set up a triple integral that computes the flux integral using outward pointing normal vectors to the surface.

Use cylindrical coordinates for your final answer. Do not evaluate the triple integral.

9.
(20 points; 10 points each part.) Let C be the boundary curve of the part of the surface S, given by the equation z=1-x2-y2, lying in the first octant. Note that the curve C lies entirely on the surface S.
a.
Draw a careful sketch of C and the surface. Draw a sample outward pointing normal vector to S. Indicate with arrows a direction of travel around C that positively orients C with respect to this outward pointing normal vector.
b.
Let be a vector field. Use Stokes' theorem to compute the line integral .