MATH 250: Calculus III

October 24, 2015

Review Topics for Test #2

1. Calculus of vector valued functions; differentiation rules; product rules; derivative as the tangent vector

2. Arc length; speed

3. Arc length parametrization

4. Unit tangent vector; curvature; unit normal vector

5. Motion in three-dimensional space; velocity and acceleration; finding velocity and path from acceleration; solving projectile problems

6. Tangential and normal components of acceleration

7. Functions of two variables; traces; level curves and contour maps

8. Functions of three variables; level surfaces and contour maps

9. Limits and continuity of functions of several variables

10. Partial derivatives; higher-order partial derivatives; Clairaut’s theorem

11. Linearizations; linear approximations; finding tangent planes to surfaces z=f(x,y) without using gradient

12. Gradient

13. Chain rule for paths

14. Directional derivatives

15. Significance of gradient: direction and value of maximum rate of increase; normal vector to level surface; finding tangent planes to level surfaces with the use of gradient

16. Chain rule for functions of several variables

17. Critical points; local extrema

18. Global extrema theorem

19. Finding local and global extrema for functions of two variables

Recommended (optional, not graded) problems from the review sections of the textbook:

Chapter review on calculus of vector-valued functions: 3, 7, 10, 17, 19, 23, 24, 25, 29, 31, 33.

Chapter review on differentiation in several variables: 3, 7, 11, 12, 21, 23, 25, 26, 31, 35, 37, 39, 41, 45, 49, 53, 57.