### Calculus

**Differential Equations**

Prof. Arthur Mattuck, MIT

1 y'=f(x,y): Direction Fields, Integral Curves.

2 Euler's Numerical Method for y'=f(x,y) and its Generalizations.

3 First-order Linear ODE's;

4 Bernouilli and Homogeneous ODE's.

5 Qualitative Methods, Applications.

6 Complex Numbers and Complex Exponentials.

7 First-order Linear with Constant Coefficients

8 First-order Linear Applications to Temperature, Mixing, RC-circuit, Decay, and Growth Models.

9 Solving Second-order Linear ODE's with Constant Coefficients: The Three Cases.

10 First-order Linear: Undamped and Damped Oscillations.

11 Second-order Linear Homogeneous ODE's: Superposition, Uniqueness, Wronskians.

12 Continuation: General Theory for Inhomogeneous ODE's. Stability Criteria for the Constant-coefficient ODE's.

13 Operator and Solution Formulas Involving Ixponentials.

14 Resonance.

15 Introduction to Fourier Series;

16 Fourier Series: More General Periods; Even and Odd Functions; Periodic Extension.

17 Resonant Terms; Hearing Musical Sounds.

19 Laplace Transform; Basic Formulas.

20 Derivative Formulas; Using the Laplace Transform to Solve Linear ODE's.

21 Convolution Formula.

22 Using Laplace Transform to Solve ODE's with Discontinuous Inputs.

23 Dirac Delta Function, Weight and Transfer Functions.

24 Solution by Elimination, Geometric Interpretation of a System.

25 Homogeneous Linear Systems with Constant Coefficients.

26 Continuation: Repeated Real Eigenvalues, Complex Eigenvalues.

27 2x2 Homogeneous Linear System with Constant Coefficients.

28 Matrix Methods for Inhomogeneous Systems.

29 Matrix Exponentials.

30 Decoupling Linear Systems with Constant Coefficients.

31 Non-linear Autonomous Systems.

32 Limit Cycles: Existence and Non-existence Criteria.

33 Relation Between Non-linear Systems and First-order ODE's.

**Single Variable Calculus**

Prof. David Jerison, MIT

1 Derivatives, slope, velocity, rate of change

2 Limits, continuity - Trigonometric limits

3 Derivatives of products, quotients, sine, cosine

4 Chain rule - Higher derivatives

5 Implicit differentiation, inverses

6 Exponential and log - Logarithmic differentiation; hyperbolic functions

7 Hyperbolic functions (cont.) and exam 1 review

8 Exam 1

9 Linear and quadratic approximations

10 Curve sketching

11 Max-min problems

12 Related rates

13 Newton's method and other applications

15 Differentials, antiderivatives

16 Differential equations, separation of variables

17 Exam 2

18 Definite integrals

19 First fundamental theorem of calculus

20 Second fundamental theorem

21 Applications to logarithms and geometry

22 Volumes by disks and shells

23 Work, average value, probability

24 Numerical integration

25 Exam 3 review

26 Exam 3

27 Trigonometric integrals and substitution

28 Integration by inverse substitution; completing the square

29 Partial fractions

30 Integration by parts, reduction formulae

31 Parametric equations, arclength, surface area

32 Polar coordinates; area in polar coordinates

33 Exam 4 review

34 Exam 4

35 Indeterminate forms. L'Hospital's rule

36 Improper integrals

37 Infinite series and convergence tests

38 Taylor's series

39 Final reviewv

**Video Calculus**

Dr Selwyn Hollis , Armstrong Atlantic State University

1. Limits and Graphs.

2. Calculation of Limits

3. Trigonometric Limits.

4. Continuity.

5. The Derivative.

6. Calculation of Derivatives.

7. Derivatives of Trigonometric Functions.

8. Leibniz Notation and the Chain Rule.

9. Rates of Change and Related Rates.

10. Implicit Differentiation.

11. Rectilinear Motion.

12. Higher-Order Derivatives.

13. The Mean-Value Theorem and Related Results.

14. Critical Numbers and the First Derivative Test.

15. Concavity and the Second Derivative Test.

16. Limits at inf and Horizontal Asymptotes.

17. Curve Sketching.

18. Extreme Values on Intervals.

19. Applied Optimization Problems.

20 Newton's Method.

21. The Area Under a Curve.

22. The Integral.

23. The Fundamental Theorem of Calculus.

24. Antidifferentiation and Inde?nite Integrals.

25. Change of Variables (Substitution).

26. Areas Between Curves

27. Volumes I Solids with specified cross-sections.

28. Volumes II Solids of revolution.

29. Volumes III The cylindrical shell method.

30. The Centroid of a Planar Region.

31. The Natural Logarithm.

32. The Exponential Function.

33. The Inverse Trigonometric Functions.

34. Integration by Parts.

35. Integration of Powers and Products of Sine and Cosine.

36. Integration of Powers and Products of Secant and Tangent, Cosecant and Cotangent.

37. Trigonometric Substitutions.

38. Partial Fraction Expansions.

39. Numerical Integration.

40. Arc Length and Surface Area.

41. Polar Coordinates and Graphs.

42. Areas and Lengths Using Polar Coordinates.

43. Parametric Curves.

44. The Conic Sections.

45. Improper Integrals.

46. Indeterminate Forms.

47. Sequences I.

48. Sequences II.

49. Series.

50. The Integral Test.

51. Comparison Tests.

52. Alternating Series and Absolute Convergence.

53. Power Series.

54. Taylor and Maclaurin Series.

55. Taylor's Theorem.

**The Practice of Mathematics**

Professor Robert P. Langlands, Princeton

**Solving Cubic Equations**

Professor Benedict H. Gross and Professor William A. Stein, Harvard

1. Introduction

2. Pythogorean Triples

3. Quadratic and Cubic Equations

4. Rational Solutions

5. Modular Prime

6. Conclusion: Gross-Zagier Theorem

7. Audience Question and Answer