Calculus

LECTURE NOTES ON CALCULUS (DREXEL 2005-2006)

1. Derivatives, Integrals and the Fundamental Theorem of Calculus (A Brief Aerial View of the Terrain)

2. Review of Functions

3. Limits (Intuitive Approach)

4. Techniques for Computing Limits I

5. Techniques for Computing Limits II

6. Continuous Functions

7. Limits of Trigonometric Functions (The Squeeze Theorem)

8. Tangent Lines and Rates of Change

9. The Derivative

10. Differentiation Techniques

11. Derivatives of Trigonometric Functions

12. The Chain Rule

13. Related Rates Problems

14. Local Linear Approximations

15. Implicit Differentiation

16. Derivatives of Logarithmic, Exponential and Inverse Trigonometric Functions

17. L’Hôpital’s Rule

18. Increasing, Decreasing, Concavity

19. Extrema and Graphing

20. Absolute Maxima and Minima

21. Applied Maximum-Minimum Problems

22. Newton’s Method

23. Mean Value Theorem

24. Areas and Antiderivatives

25. Areas and Riemann Sums

26. Indefinite Integrals

27. Integration by Substitution

28. Definite Integrals

29. Fundamental Theorem of Calculus

30. Rectilinear Motion

31. Definite Integrals by Substitution

32. Mass and Areas Between Curves

33. Volumes I

34. Volumes II

35. Arc Length

36. Work and Energy

37. Hyperbolic Functions

38. Hyperbolic Form of the Lorentz Transformations and “Addition of Velocities”

39. Basic Integration Techniques

40. Integration by Parts

41. Trigonometric Integrals

42. Trigonometric Substitution

43. Partial Fractions

44. Improper Integrals

45. Introduction to Ordinary Differential Equations

46. Geometrical and Numerical Methods

47. Some Mathematical Models

48. Second Order Linear Differential Equations

49. Polynomial Approximation of Functions

50. Infinite Sequences

51. Monotone Sequences

52. Infinite Series

53. Convergence Tests

54. Series of Non-Negative Terms

55. Alternating Series

56. Power Series

57. Convergence of Taylor Series

58. Differentiation and Integration of Series

59. Series Solutions of Differential Equations

60. Polar Coordinates

61. Area in Polar Coordinates

62. Cartesian Coordinates in 3-Space

63. Surfaces in Space

64. Functions of Two or More Variables

65. Limits and Continuity for Multivariable Functions

66. Partial Derivatives

67. Chain Rule for Multivariable Functions

68. Introduction to Partial Differential Equations