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.