MATH 250: Calculus III
May 14, 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 (note that some of them have
been assigned before):
Chapter 12 Review: 15, 18,
27, 41, 49
Chapter 13 Review: 21, 23,
25, 29
Chapter 14 Review: 7, 12, 23,
25, 31, 35, 45, 52, 53
Chapter 15 Review: 19, 23,
31, 37, 41, 59
Chapter 16 Review: 13, 15,
25, 33, 47, 57
Chapter 17 Review: 5, 11, 17,
19, 22, 23
Note: Calculators will not be needed and are not allowed on the exam.