MATH 250: Calculus III


October 3, 2025


Insanely Long List of Review Topics for Test #1


  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



Optional review problems:

Chapter 12 Review: 7, 11, 15, 33, 43, 52

Chapter 13 Review: 15, 17, 26, 29, 39, 49, 52, 55, 63

Chapter 14 Review: 3, 19, 23, 25, 29, 34

Chapter 15 Review: 5, 7



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