MATH 250: Calculus III
December 10, 2016
Final Exam Review Topics
1. Parametric equations of curves in the plane; paths versus curves
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 parametrized path
5. Polar coordinates; conversions between polar and rectangular coordinates; graphing curves and regions in polar coordinates
6. Arc lengths in polar coordinates
7. Areas in polar coordinates; area between two curves in polar coordinates
8. Conic sections: standard forms
9. Vectors in two dimensions; magnitude of a vector; vector algebra; basic properties; triangle inequality
10. Unit vectors, standard basis vectors
11. Vectors in three dimensions; distance formula
12. Parametric equations of a straight line in three dimensions; intersections of straight lines
13. Dot product; basic properties (with proofs); angle between vectors; projection of a vector along another vector
14. Determinants
15. Cross product; basic properties (with proofs)
16. Areas and volumes with the use of cross product
17. Planes in three dimensions; planes through three given points; intersection of a plane and a line; angle between planes; traces of a plane
18. Basic quadric surfaces; traces of quadric surfaces
19. Cylindrical coordinates; conversions between cylindrical and rectangular coordinates; level surfaces; identifying surfaces and regions in cylindrical coordinates
20. Spherical coordinates; conversions between spherical and rectangular coordinates; level surfaces; identifying surfaces and regions in spherical coordinates
21. Vector-valued functions: parametric curves in three dimensions; projections of curves onto coordinate planes
22. Calculus of vector-valued functions
23. Tangent lines to parametrized curves
24. Arc length; speed
25. Arc length parametrization
26. Unit tangent vector; curvature; unit normal vector
27. Motion in three-dimensional space; velocity and acceleration; finding velocity and path from acceleration; solving projectile problems
28. Tangential and normal components of acceleration
29. Functions of two variables; traces; level curves and contour maps
30. Functions of three variables; level surfaces and contour maps
31. Limits and continuity of functions of several variables
32. Partial derivatives; higher-order partial derivatives; Clairaut’s theorem
33. Linearizations; linear approximations; finding tangent planes to surfaces z=f(x,y) without using gradient
34. Gradient
35. Chain rule for paths
36. Directional derivatives
37. Significance of gradient: direction and value of maximum rate of increase; normal vector to level surface; finding tangent planes to level surfaces
38. Chain rule for functions of several variables
39. Critical points; local extrema
40. Global extrema theorem
41. Finding local and global extrema for functions of two variables, including applications
42. Iterated integrals; Fubini’s theorem
43. Double integrals over general regions; regions of Type I and Type II
44. Interchanging the order of double integration
45. Average value of a function of two variables
46. Derivation and definition of a triple integral
47. Iterated integrals and Fubini’s theorem for triple integrals
48. Triple integrals over general regions
49. Triple integrals over regions between surfaces
50. Changing the order of 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. Using double and triple integrals to find mass (charge, etc.) of an object
55. Change of variables; Jacobian
56. Vector fields in two and three dimensions
57. Line integrals of scalar fields; derivation and computation
58. Line integrals of vector fields; derivation and computation
59. Line integral as work
60. Gradient vector fields; field as the gradient of a potential function
61. Finding potential functions in two and three dimensions
62. Fundamental theorem for gradient vector fields
63. Path independence
64. Rotation (curl) of a vector field
65. Surface parameterization
66. Normal vectors of parametric surfaces; tangent planes
67. Surface area of parametric surfaces
68. Surface integrals of scalar functions
69. Surface integrals of vector fields
70. Flux of a vector field
71. Scalar curl (curl in 2D)
72. Green’s Theorem
73. Computing area via Green’s Theorem
74. Stokes’ Theorem: illustrating the theorem; computing one side instead of the other
75. Divergence of a vector field
76. Divergence Theorem: illustrating the theorem; computing one side instead of the other
77. The gradient operator unifying notation
78. The Big Picture of Calculus Theorems
Office hours 12/12 - 12/16:
Mon, 12/12: 1:20 - 2:20 and 4 - 5:30
Tue, 12/13: 2 - 4
Wed, 12/14: 1 - 2
Thu, 12/15: 2 - 4:30
Fri, 12/16: 12 - 2
Assorted optional review problems (many of which were assigned before):
Chapter 11.Review: 11, 15, 21, 33, 37, 43
Chapter 12.Review: 15, 27, 29, 41, 49, 60
Chapter 13.Review: 17, 23, 25, 29
Chapter 14.Review: 7, 12, 23, 25, 28, 31, 35, 41, 45, 53, 57
Chapter 15.Review: 13, 21, 31, 37, 41, 57, 61a
Chapter 16.Review: 13, 23, 25, 31, 37, 41, 49ab
Dr. Hoffoss' Review Problems:
http://home.sandiego.edu/~dhoffoss/teaching/math250/9-surface-int-prac.pdf
NOTE:
Calculators will not be needed and will not be allowed on the exam.