MATH 250: Calculus III

October 7, 2024

Test # 1 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

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

Optional review problems:

Chapter 11 Review: 7, 11, 15, 21, 36, 43, 52

Chapter 12 Review: 9, 15, 27, 29, 39, 48, 49, 50, 51, 52, 60, 63, 64, 65, 68

Chapter 13 Review: 4, 23, 24, 29, 30