MATH 250: Calculus III


November 7, 2025


Test # 2 Review Topics


  1. Limits and continuity of functions of several variables

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

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

  4. Gradient

  5. Chain rule for paths

  6. Directional derivatives

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

  8. Chain rule for functions of several variables

  9. Critical points; local extrema

  10. Global extrema theorem

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

  12. Integration in two variables; double Riemann sum

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

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

  15. Changing the order of double integration

  16. Average value of a function of two variables

  17. Integration in three variables; triple Riemann sum

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

  19. Triple integrals over general regions

  20. Triple integrals over regions between surfaces

  21. Various orders in triple integration

  22. Computing volumes with double and triple integration

  23. Double integrals in polar coordinates

  24. Triple integrals in cylindrical and spherical coordinates

  25. Average value of function in a 3D region

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

  27. Change of variables; Jacobian

  28. Vector fields in two or three dimensions


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


Optional review problems:

Chapter 15 Review: 7, 11, 23, 24, 29, 33, 41, 49, 52, 53

Chapter 16 Review: 15, 19, 23, 25, 35, 37, 43, 59a, 60