# Math 221: Chapter 13 Study Guide

1. Sketch the graph of the 2D vector valued function and show the direction of increasing t.
2. Find the limit of the vector valued function.
3. Find the derivative of the vector valued function.
4. Find parametric equations of the line tangent to the graph of the vector valued function at the given point.
5. Evaluate the integral.
6. Calculate a derivative using the chain rule.
7. Find the derivative of a dot product and a cross product using the product rules for dot and cross products. Do not find the dot or cross product and then take the derivative.
8. A 2D curve is given. Draw the unit tangent and unit normal vectors.
9. A 3D curve is given. Identify the T, N, and B vectors (they are drawn, you just need to identify which one is which).
10. Find parametric equations for the curve using arc length s as parameter.
11. Find the unit tangent vector T, the unit normal vector N, and the binormal vector B, for the given value of t.
12. Find the curvature and radius of curvature for a vector-valued function at the indicated point.
13. Find the scalar tangential and normal components of acceleration at the indicated time t.
14. Convert the rectangular equation into cylindrical and spherical coordinates.
15. Use the given information to find the position and velocity vectors of the particle. Look at problems 13.6.15-18.
16. Find the curvature for 2D function.
17. Find the equation of the plane
18. Convert the cylindrical equation into rectangular coordinates.

## Take Home Portion

1. Find the altitude in miles of a communications satellite that is in geosynchronous orbit about the moon. Look at example 1 in section 13.7 and problems 13.7.7-8. (5 points)
2. Find the speed in miles per hour of a satellite that is in geosynchronous orbit about the moon. Look at problems 13.7.7-8. (5 points)
3. Work problem 13.6.71. Show work (except for part b and then show what you're integrating). (5 points)

## Notes

• Several of the problems are directly from the text.
• There is a page at the end of the test with r, r', and r" as well as ||r||, ||r'||, and ||r"|| for questions 10-13. It is intended to help speed up the process. That page also contains the formulas for the curvature in 2 dimensions. It does not contain the formulas you will need for questions 10-13, just some of the values you'll need.
• There is a take home portion of the exam worth 15 points. It is due the day after the exam.
• You will need to do some research to answer questions 1 and 2 on the take home exam.

## Points per problem

 # Pts 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Total 6 6 6 6 6 6 10 6 4 5 5 5 5 8 8 6 6 6 110