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Appendix B Hints for Exercises
1 Vectors and Geometry in Two and Three Dimensions
1.1 Points
Exercises
1.2 Vectors
1.2.9 Exercises
1.2.9.2.
1.2.9.3.
1.2.9.4.
1.2.9.5.
1.2.9.15.
1.2.9.16.
1.2.9.17.
1.2.9.18.
1.2.9.19. (✳).
1.2.9.20.
1.2.9.30. (✳).
1.3 Equations of Lines in 2d
Exercises
1.3.1.
1.3.2.
1.3.3.
1.3.4.
1.3.5.
1.3.6.
1.3.7.
1.3.9.
1.4 Equations of Planes in 3d
Exercises
1.4.1.
1.4.2.
1.4.4. (✳).
1.4.5.
1.4.6.
1.5 Equations of Lines in 3d
Exercises
1.5.3.
1.5.4.
1.5.7. (✳).
1.5.8.
1.5.9.
1.5.12. (✳).
1.6 Curves and their Tangent Vectors
1.6.2 Exercises
1.6.2.1.
1.6.2.2.
1.6.2.3.
1.6.2.4.
1.6.2.5.
1.6.2.6.
1.6.2.7.
1.6.2.8.
1.6.2.9. (✳).
1.6.2.10. (✳).
1.6.2.11.
1.6.2.12. (✳).
1.6.2.13.
1.6.2.14.
1.6.2.15. (✳).
1.6.2.16. (✳).
1.6.2.17. (✳).
1.6.2.19. (✳).
1.6.2.20.
1.6.2.23. (✳).
1.6.2.24. (✳).
1.6.2.25. (✳).
1.6.2.26.
1.7 Sketching Surfaces in 3d
1.7.2 Exercises
2 Partial Derivatives
2.1 Limits
2.1.2 Exercises
2.1.2.1.
2.1.2.2.
2.1.2.3.
2.1.2.4.
2.1.2.5.
2.1.2.6.
2.1.2.7. (✳).
2.1.2.8. (✳).
2.1.2.9. (✳).
2.1.2.10.
2.1.2.11. (✳).
2.1.2.12.
2.2 Partial Derivatives
2.2.2 Exercises
2.2.2.1.
2.2.2.2.
2.2.2.3. (✳).
2.3 Higher Order Derivatives
2.3.3 Exercises
2.4 The Chain Rule
2.4.5 Exercises
2.4.5.1.
2.4.5.2.
2.4.5.3. (✳).
2.4.5.4.
2.4.5.5.
2.4.5.6.
2.4.5.7.
2.4.5.9. (✳).
2.4.5.10. (✳).
2.4.5.11. (✳).
2.4.5.12. (✳).
2.4.5.13. (✳).
2.4.5.14. (✳).
2.4.5.15. (✳).
2.4.5.17. (✳).
2.4.5.19. (✳).
2.4.5.21. (✳).
2.4.5.23. (✳).
2.4.5.24. (✳).
2.4.5.25. (✳).
2.5 Tangent Planes and Normal Lines
2.5.3 Exercises
2.5.3.1.
2.5.3.2.
2.5.3.3.
2.5.3.4.
2.5.3.11. (✳).
2.5.3.12. (✳).
2.5.3.13. (✳).
2.5.3.14.
2.5.3.15.
2.5.3.17. (✳).
2.5.3.18. (✳).
2.5.3.19.
2.6 Linear Approximations and Error
2.6.3 Exercises
2.6.3.5.
2.6.3.6. (✳).
2.6.3.8.
2.6.3.9.
2.6.3.11.
2.6.3.15. (✳).
2.7 Directional Derivatives and the Gradient
2.7.2 Exercises
2.7.2.4.
2.7.2.5.
2.7.2.6. (✳).
2.7.2.9.
2.8 A First Look at Partial Differential Equations
2.8.3 Exercises
2.8.3.4. (✳).
2.8.3.5.
2.8.3.6.
2.8.3.7.
2.8.3.8.
2.8.3.10.
2.8.3.12.
2.8.3.13.
2.8.3.14.
2.9 Maximum and Minimum Values
2.9.4 Exercises
2.9.4.4. (✳).
2.9.4.13.
2.9.4.30.
2.9.4.31.
2.10 Lagrange Multipliers
2.10.2 Exercises
3 Multiple Integrals
3.1 Double Integrals
3.1.7 Exercises
3.1.7.9. (✳).
3.1.7.10. (✳).
3.1.7.14. (✳).
3.1.7.15. (✳).
3.1.7.16. (✳).
3.1.7.18. (✳).
3.1.7.19. (✳).
3.1.7.22. (✳).
3.1.7.24. (✳).
3.1.7.26. (✳).
3.2 Double Integrals in Polar Coordinates
3.2.5 Exercises
3.2.5.2.
3.2.5.3.
3.2.5.4.
3.3 Applications of Double Integrals
3.3.4 Exercises
3.4 Surface Area
Exercises
3.5 Triple Integrals
Exercises
3.6 Triple Integrals in Cylindrical Coordinates
3.6.4 Exercises
3.7 Triple Integrals in Spherical Coordinates
3.7.5 Exercises
3.7.5.19. (✳).
3.7.5.26. (✳).