MATH 250: Calculus III


December 5, 2025


Final Exam Review Topics


  1. Parametric equations of curves in the plane

  2. Slope of a tangent line to a curve given in parametric form

  3. Arc length of a curve given in parametric form

  4. Speed along a parameterized path

  5. Polar coordinates; conversions between polar and rectangular coordinates; graphing curves and regions in polar coordinates

  6. Arc lengths and areas in polar coordinates

  7. Conic sections; converting to standard forms

  8. Vectors in two dimensions; magnitude of a vector; vector algebra; basic properties

  9. Unit vectors, standard basis vectors

  10. Vectors in three dimensions; distance formula

  11. Parametric equations of a straight line in three dimensions; intersections of straight lines

  12. Dot product; basic properties (with proofs); angle between vectors; projection of a vector along another vector

  13. Determinants

  14. Cross product; basic properties (with proofs)

  15. Areas with the use of cross product

  16. Planes in three dimensions; planes through three given points; intersection of a plane and a line; intersections of planes; traces of a plane

  17. Cylindrical coordinates; conversions between cylindrical and rectangular coordinates; level surfaces; identifying surfaces and regions in cylindrical coordinates

  18. Spherical coordinates; conversions between spherical and rectangular coordinates; level surfaces; identifying surfaces and regions in spherical coordinates

  19. Basic quadric surfaces; traces of quadric surfaces

  20. Vector-valued functions: parametric curves in three dimensions; projections of curves onto coordinate planes

  21. Calculus of vector-valued functions

  22. Tangent lines to parameterized curves

  23. Arc length; speed

  24. Arc length parameterization

  25. Unit tangent vector; curvature; unit normal vector

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

  27. Tangential and normal components of acceleration

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

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

  30. Limits and continuity of functions of several variables

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

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

  33. Gradient

  34. Chain rule for paths

  35. Directional derivatives

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

  37. Chain rule for functions of several variables

  38. Critical points; local extrema

  39. Global extrema theorem

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

  41. Integration in two variables; double Riemann sum

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

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

  44. Changing the order of double integration

  45. Average value of a function of two variables

  46. Integration in three variables; triple Riemann sum

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

  48. Triple integrals over general regions

  49. Triple integrals over regions between surfaces

  50. Various orders in triple integration

  51. Computing volumes with double and triple integration

  52. Double integrals in polar coordinates

  53. Triple integrals in cylindrical and spherical coordinates

  54. Average value of function in a 3D region

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

  56. Change of variables; Jacobian

  57. Vector fields in two and three dimensions

  58. Line integrals of scalar fields; derivation and computation

  59. Line integrals of vector fields; derivation and computation

  60. Line integral as work

  61. Gradient vector fields; field as the gradient of a potential function

  62. Finding potential functions in two and three dimensions

  63. Fundamental theorem for gradient vector fields; path independence

  64. Surface parameterization

  65. Normal vectors of parametric surfaces; tangent planes

  66. Surface integrals of scalar functions

  67. Surface area of parametric surfaces

  68. Surface integrals of vector fields; flux of a vector field

  69. Divergence and curl of a vector field

  70. Scalar curl (curl in 2D)

  71. Green’s Theorem: illustrating the theorem

  72. Computing area via Green’s Theorem

  73. Stokes’ Theorem: illustrating the theorem

  74. Divergence Theorem: illustrating the theorem

  75. Computing one side of GT or ST or DT instead of the other

  76. The Big Picture of Calculus Theorems


Optional review problems:

12.Review: 11, 15, 21, 35, 47

13 Review: 17, 25, 37, 41, 49, 52, 65

14 Review: 23, 25, 29, 33

15 Review: 7, 11, 25, 33, 38, 53

16 Review: 15, 19, 27, 31, 35, 37, 43

17 Review: 9, 21, 23, 31, 35, 37, 47, 53, 57

18 Review: 5, 11, 18, 22, 23


Note: Calculators will not be needed and are not allowed on the exam. Also: No bathroom breaks!