MATH 250: Calculus III

November 9, 2024

Test # 2 Review Topics

1. Motion in three-dimensional space; velocity and acceleration; finding velocity and path from acceleration

2. Tangential and normal components of acceleration

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

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

5. Limits and continuity of functions of several variables

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

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

8. Gradient

9. Chain rule for paths

10. Directional derivatives

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

12. Chain rule for functions of several variables

13. Critical points; local extrema

14. Global extrema theorem

15. Finding local and global extrema for functions of two variables, including applications

16. Integration in two variables; double Riemann sum

17. Iterated integrals; Fubini’s theorem for double integrals

18. Double integrals over general regions; vertically and horizontally simple regions

19. Changing the order of double integration

20. Average value of a function of two variables

21. Integration in three variables; triple Riemann sum

22. Iterated integrals; Fubini’s theorem for triple integrals

23. Triple integrals over general regions

24. Triple integrals over regions between surfaces

25. Various orders in triple integration

26. Computing volumes with double and triple integration

27. Double integrals in polar coordinates

28. Triple integrals in cylindrical and spherical coordinates

29. Average value of function in a 3D region

30. Using double and triple integrals to find mass (charge, etc.) of an object

31. Change of variables; Jacobian

Note: Calculators will not be needed and are not allowed on the test.

Optional review problems:

Chapter 13 Review: 25, 28

Chapter 14 Review: 5, 6, 7, 11, 21, 23, 25, 31, 35, 37, 41, 43, 45

Chapter 15 Review: 15, 17, 19, 22, 23, 31, 33, 35, 36, 37, 38