# Math 221: Chapter 12 Exam Study Guide

1. Given two points, find a vector-valued function for the segment between them. Also write a set of parametric equations for the line segment.
2. Represent the curve as a vector-valued function. Two parts.
3. Find the limits. Two parts.
4. Given two vectors, find their derivatives, the derivative of the dot product, and the derivative of the cross product.
5. Find the indicated integral. Two parts.
6. 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.
7. Use the acceleration function and initial conditions to find the velocity and position functions.
8. Find a unit tangent vector at the indicated point.
9. Find the curvature and radius of curvature for an explicitly defined function.
10. Given a vector-valued position function, find the velocity, speed, and acceleration
11. Application problem involving path of a projectile.
12. Find the tangential and normal components of acceleration.
13. Find the length of the space curve.
14. Find the curvature at the indicated point for the space curve.
15. Graph a space curve and then find and graph the derivative at an indicated point.
16. Graph a plane curve and find the unit tangent and normal vectors and graph them.
17. 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.
18. 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

 # Pts 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Total 4 6 6 12 8 6 6 4 4 6 6 4 4 4 5 5 5 5 100