## Math 221 - Final Exam Study Guide

- Find parametric equations for the line passing through the given points.
- Given two vectors, find: the sum, scalar product, norm, dot product, cross product, projection,
angle between the vectors, unit vector.
- Name the quadric surfaces. Seven parts.
- Sketch the graph of the vector valued function and indicate the direction of increasing t.
- Find the arc length of the vector valued function.
- Find the unit tangent vector
**T** and the unit normal vector **N** for the given value of *t*.
- Given the position function of a particle moving in the plane, find the velocity, acceleration,
and speed at an arbitrary time
*t*. Then sketch the path of the particle together with the velocity
and acceleration vectors at the indicated time *t*.
- Find the multi-variable limits, if they exist.
- Find the second partial derivatives of a function.
- Find dz/dt using the chain rule.
- Find the gradient of
*f* at the indicated point.
- Locate all relative maxima, relative minima, and saddle points.
- Evaluate the iterated integral. Show all work.
- Evaluate the double integral over the region.
- Evaluate the iterated integral by converting into polar coordinates.
- Use cylindrical coordinates find the volume of the solid. Set up the integral only, do not
evaluate (Derive was unable to evaluate it, even in approximate mode, so don't waste time on
it).
- Change variables to evaluate an integral. The change of variables is not given to you.
- Find div
**F** and curl **F**.
- Show that
**F** is conservative and find a potential function for it.
- Evaluate the line integrals. Two parts.
- Evaluate the closed line integral. Green's theorem would be extremely useful.
- Find a unit vector in the direction in which
*f* most increases most rapidly at P; and find the rate
of change of *f* at P in that direction.

### Notes:

- Problems were selected by looking at previous exams.
- None of the problems are directly from the previous exams, but are similar. In some cases,
the numbers were changed, in other cases, problems similar to those were selected from the
text.
- With the exception of the last problem, the order of the problems on the final is in the order
covered on the previous exams.

# |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
19 |
20 |
21 |
22 |
Total |

Pts |
6 |
24 |
14 |
6 |
6 |
10 |
12 |
9 |
12 |
6 |
6 |
10 |
10 |
6 |
6 |
6 |
8 |
10 |
8 |
10 |
8 |
7 |
200 |