MATH 250: Calculus III

April 17, 2025

Test # 2 Review Topics

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

1. Tangential and normal components of acceleration

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

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

4. Limits and continuity of functions of several variables

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

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

7. Gradient

8. Chain rule for paths

9. Directional derivatives

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

11. Chain rule for functions of several variables

12. Critical points; local extrema

13. Global extrema theorem

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

15. Integration in two variables; double Riemann sum

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

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

18. Changing the order of double integration

19. Average value of a function of two variables

20. Integration in three variables; triple Riemann sum

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

22. Triple integrals over general regions

23. Triple integrals over regions between surfaces

24. Various orders in triple integration

25. Computing volumes with double and triple integration

26. Double integrals in polar coordinates

27. Triple integrals in cylindrical and spherical coordinates

28. Average value of function in a 3D region

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

30. 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