## Math 221 - Chapter 15 Study Guide

1. Sketch the graph of the vector valued function and show the direction of increasing t. Look at problems 15.1.23-34*.
2. Find the limit of the vector valued function. Look at problems 15.2.1-8*.
3. Find the derivative of the vector valued function. Look at problems 15.2.11-14*.
4. Find parametric equations of the line tangent to the graph of the vector valued function at the given point. Look at problems 15.2.23-26*.
5. Evaluate the integral. Look at problems 15.2.37-48*.
6. Find the derivative of a dot product. Look at problem 15.2.57*.
7. Find the derivative of a cross product. Look at problem 15.2.58*.
8. Determine whether r is a smooth function of the parameter t. Look at problems 15.3.3-6*.
9. Calculate a derivative using the chain rule. Look at problems 15.3.7-9*.
10. Find the arc length of the curve. Look at problems 15.3.11-16*.
11. Find parame-tric equations for the curve using arc length s as parameter. Use the point on the curve where t=0 as reference. Look at problems 15.3.19-27*.
12. Find the unit tangent vector T and the unit normal vector N for the given value of t. Look at problems 15.4.9-16*.
13. Find the binormal vector B=TxN. Look at problems 15.4.29-30*.
14. Find the curvature at the indicated point. Look at problems 15.5.1-14*.
15. Sketch the curve, calculate the radius of curvature at the indicated point, and sketch the osculating circle. Look at problems 15.5.39-45*.
16. Given the position vector 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. Look at problems 15.6.1-6*.
17. Use the given information to find the position and velocity vectors of the particle. Look at problems 15.6.23-30*.
18. Find the displacement and the distance traveled over the indicated time interval. Look at problems 15.6.33-38*.
19. Find the scalar tangential and normal components of acceleration at the indicated time t. Look at problems 15.6.39-48*.

Take Home Portion

1. Geosynchronous orbit. You will need to do some research to answer this question.
1. Find the altitude in miles of a communications satellite that is in geosynchronous orbit about the moon. Look at problem 15.7.7a. (5 points)
2. Find the speed in miles per hour of a satellite that is in geosynchronous orbit about the moon. Look at problem 15.7.7b. (5 points)
2. Answer technology question 15.6 (5 points)
3. Answer technology question 15.8 (5 points)
4. Answer technology question 15.10 (5 points)

### Notes:

• Each question on the in-class portion of the exam is worth 4 points for a total of 76 points.
• Each part of the take home exam is worth 5 points for a total of 25 points.
• You may have notecards with the following formulae on them. No extraneous material may be on the notecards.
• 15.3 - 8, 9, 10, 16, 17
• 15.4 - 1-4
• 15.5 - 1, 2, 3, 8, 9, also rho=1/kappa
• 15.6 - 9, 10, 11, 12, 13, 14, 16, 27