# Math 221: Chapter 12 Exam Study Guide

- Given two points, find a vector-valued function for the segment between them. Also write a set of parametric equations for the line segment.
- Represent the curve as a vector-valued function. Two parts.
- Find the limits. Two parts.
- Given two vectors, find their derivatives, the derivative of the dot product, and the derivative of the cross product.
- Find the indicated integral. Two parts.
- Given a two dimensional vector-valued function and its graph, draw arrows to indicate the orientation, draw a vector for
**r** and **r**' at a particular point.
- Use the acceleration function and initial conditions to find the velocity and position functions.
- Find a unit tangent vector at the indicated point.
- Find the curvature and radius of curvature for an explicitly defined function.
- Given a vector-valued position function, find the velocity, speed, and acceleration
- Application problem involving path of a projectile.
- Find the tangential and normal components of acceleration.
- Find the length of the space curve.
- Find the curvature at the indicated point for the space curve.
- Graph a space curve and then find and graph the derivative at an indicated point.
- Graph a plane curve and find the unit tangent and normal vectors and graph them.
- Graph a plane curve that is defined parametrically. Find the curvature and radius of curvature at the indicated point. Find the equation for and draw the osculating circle.
- Find the unit tangent, normal, and binormal vectors at the indicated point. Graph the space curve and the three vectors that form the TNB frame.

## Notes

- Questions 1-9 are to be worked without a computer.
- Questions 10-14 are to be worked using Maxima.
- Questions 15-18 are to be worked using Maxima and Winplot. The graphs should be saved and emailed to the instructor.
- You may copy the formulas from the summary table on page 877 onto a sheet of paper and use those during the exam.
- You may find the Introduction to Maxima for Multivariate Calculus to be useful.
- Andre found this site: Maxima
Book

## Points per problem

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

Pts |
4 |
6 |
6 |
12 |
8 |
6 |
6 |
4 |
4 |
6 |
6 |
4 |
4 |
4 |
5 |
5 |
5 |
5 |
100 |