- Sketch the graph of the vector valued function and show the direction of increasing t. Look at problems 15.1.23-34*.
- Find the limit of the vector valued function. Look at problems 15.2.1-8*.
- Find the derivative of the vector valued function. Look at problems 15.2.11-14*.
- 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*.
- Evaluate the integral. Look at problems 15.2.37-48*.
- Find the derivative of a dot product. Look at problem 15.2.57*.
- Find the derivative of a cross product. Look at problem 15.2.58*.
- Determine whether
**r**is a smooth function of the parameter t. Look at problems 15.3.3-6*. - Calculate a derivative using the chain rule. Look at problems 15.3.7-9*.
- Find the arc length of the curve. Look at problems 15.3.11-16*.
- 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*. - 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*. - Find the binormal vector
**B**=**T**x**N**. Look at problems 15.4.29-30*. - Find the curvature at the indicated point. Look at problems 15.5.1-14*.
- Sketch the curve, calculate the radius of curvature at the indicated point, and sketch the osculating circle. Look at problems 15.5.39-45*.
- 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*. - Use the given information to find the position and velocity vectors of the particle. Look at problems 15.6.23-30*.
- Find the displacement and the distance traveled over the indicated time interval. Look at problems 15.6.33-38*.
- Find the scalar tangential and normal components of acceleration at the indicated time t. Look at problems 15.6.39-48*.

**Take Home Portion**

- Geosynchronous orbit. You will need to do some research to answer this question.
- 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)
- 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)
- Answer technology question 15.6 (5 points)
- Answer technology question 15.8 (5 points)
- Answer technology question 15.10 (5 points)

- 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