1. Chapter 12 3-D functions For an animation of this topic visit: http://www.math.umn.edu/~nykamp/m2374/readings/lineparam/

2. Vector valued functions

3. This section could be titled graphs of 1 dimensional objects in 3 dimensional space Consider the graph. What would this graph look like? Sketch the curve. r(t) = cos(t)i + sin(t) j + (t)k

5. For an animation of the 3-D graphs visit http://www.math.umn.edu/~nykamp/m2374/r eadings/paramsurf/

6. Example 3

7. Example 3 solution

8. Example 4

9. Example 4 solution

10. Example 5

12. Problem 60 Sketch the curve represented by the intersection of the surfaces. Then represent the curve by a vector valued function.

13. Problem 60 solution 4 = (2cost) 2 + y 2 ____________ y = + √4 – 4 cos 2 t = 2 sint x = 2 cost y = 2 sint z = 4 OR r(t) = 2 cost i + 2 sint j + 4 k

14. All of the vector formulas that we learned in Calc BC can be expanded into 3 Dimensions see page 846, 851,869,870 and page 859 to see how the 2 D formulas can be expanded

15. Find the error in this problem if there is one: