# Calculus Revisited: Single Variable Calculus

## by MIT OpenCourseWare

To listen to an audio podcast, mouse over the title and click Play. Open iTunes to download and subscribe to iTunes U collections.

#### Description

Calculus Revisited is a series of videos and related resources that covers the materials normally found in a freshman-level introductory calculus course. The series was first released in 1970 as a way for people to review the essentials of calculus. It is equally valuable for students who are learning calculus for the first time. About the Instructor Herb Gross has taught math as senior lecturer at MIT and was the founding math department chair at Bunker Hill Community College. He is the developer of the Mathematics As A Second Language website, providing arithmetic and algebra materials to elementary and middle school teachers. You can read more about Prof. Gross on his website. Acknowledgements Funding for this resource was provided by the Gabriella and Paul Rosenbaum Foundation.

Name | Description | Released | Price | ||
---|---|---|---|---|---|

1 | VideoPreface | Calculus Revisited is a series of videos and related resources that covers the materials normally found in a freshman-level introductory calculus course. | 5/6/11 | Free | View In iTunes |

2 | VideoUnit I: Lecture 1: Analytic Geometry | Cartesian coordinates; curves as sets of points; graphs of functions; equations of straight lines; simultaneous linear equations. | 5/6/11 | Free | View In iTunes |

3 | VideoUnit I: Lecture 2: Functions | Notations; concepts of onto and one-to-one; the arithmetic of functions of a real variable; intervals and deleted neighborhoods; absolute values; composition of functions | 5/6/11 | Free | View In iTunes |

4 | VideoUnit I: Lecture 3: Inverse Functions | The concept of an inverse function; graphical interpretation; single-valued and multivalued functions; branches of functions. | 5/6/11 | Free | View In iTunes |

5 | VideoUnit I: Lecture 4: Derivatives and Limits | Instantaneous speed as an outgrowth of average speed; definition of limit; instantaneous speed as a limit; the formal definition of limit and some consequences. | 5/6/11 | Free | View In iTunes |

6 | VideoUnit I: Lecture 5: A More Rigorous Approach to Limits | A continuation of the previous lecture; important limit properties are developed as theorems from the formal definiton of limit. | 5/6/11 | Free | View In iTunes |

7 | VideoUnit I: Lecture 6: Mathematical Induction | The meaning of mathematical induction; some examples of what mathematical induction is and isn't; applications to limit theorems. | 5/6/11 | Free | View In iTunes |

8 | VideoUnit II: Lecture 1: Derivatives of Some Simple Functions | Definition of derivative; the derivative of x to the n where n is an integer; derivatives of sums, differences, products, and quotients. | 5/6/11 | Free | View In iTunes |

9 | VideoUnit II: Lecture 2: Approximations and Infinitesimals | Approximating delta y by f(x) delta x; discussion of that difference between delta y and f'(x) delta x; introduction to the chain rule. | 5/6/11 | Free | View In iTunes |

10 | VideoUnit II: Lecture 3: Composite Functions and the Chain Rule | Composition of functions; a graphical interpretation; applications to parametric equations; using the chain rule to extend the concept of finding derivatives. | 5/6/11 | Free | View In iTunes |

11 | VideoUnit II: Lecture 4: Differentiation of Inverse Functions | The concept of an inverse function; differentiation of an inverse function; when is a function invertible? | 5/6/11 | Free | View In iTunes |

12 | VideoUnit II: Lecture 5: Implicit Differentiation | Finding the derivative when the functional relationship is implied; application to the case of x to the n where n is a rational number; the use of implicit differentiation in the study of related rates. | 6/16/11 | Free | View In iTunes |

13 | VideoUnit II: Lecture 6: Continuity | Physical interpretation of continuity; the definition of continuity in terms of limits; a geometric interpretation of continuity; analytic consequences. | 5/6/11 | Free | View In iTunes |

14 | VideoUnit II: Lecture 7: Curve Plotting | Basic pre-calculus review; even and odd functions and other symmetries; the role of the first and second derivatives in curve plotting; stationary points; inflections. | 5/6/11 | Free | View In iTunes |

15 | VideoUnit II: Lecture 8: Maxima and Minima | High and low points of a curve; techniques for finding them; applications to finding maxima and minima of functions; physical applications. | 5/6/11 | Free | View In iTunes |

16 | VideoUnit II: Lecture 9: Rolle's Theorem and its Consequences | Statement of Rolle's Theorem; a geometric interpretation; some cautions; the Mean Value Theorem; consequences of the Mean Value Theorem. | 5/6/11 | Free | View In iTunes |

17 | VideoUnit II: Lecture 10: Inverse Differentiation | The "opposite" of differentiation; trying to find f(x) knowing f'(x); some examples; some formulas; notation. | 5/6/11 | Free | View In iTunes |

18 | VideoUnit II: Lecture 11: The "Definite" Indefinite Integral | The meaning of the definite integral as g(b) - g(a) where g'(x) = f(x); some applications. | 5/6/11 | Free | View In iTunes |

19 | VideoUnit III: Lecture 1: Circular Functions | Trigonometric functions without angles; the logic of radian measure; definition of circular functions; derivatives of sin x and cos x. | 5/6/11 | Free | View In iTunes |

20 | VideoUnit III: Lecture 2: Inverse Circular Functions | Meaning of arc sin x in terms of the sine function; the derivative of arc sin x in terms of the derivative of sin x; some applications. | 5/6/11 | Free | View In iTunes |

21 | VideoUnit IV: Lecture 1: The Definite Integral | Axiomatic approach to area; area approximations by upper and lower bounds; the method of exhaustion; using limits to find areas of nonrectilinear regions; piecewise continuity; trapezoidal approximations. | 5/6/11 | Free | View In iTunes |

22 | VideoUnit IV: Lecture 2: Marriage of Differential and Integral Calculus | First Fundamental Theorem of Integral Calculus; some applications; Second Fundamental Theorem of Integral Calculus; some applications; significance of the two theorems. | 5/6/11 | Free | View In iTunes |

23 | VideoUnit IV: Lecture 3: Three-Dimensional Area | Extending the axioms of area to volume; some applications; the method of cylindrical shells. | 5/6/11 | Free | View In iTunes |

24 | VideoUnit IV: Lecture 4: One-Dimensional Area | The main difference between arc-length and either area or volume; the limit definition of arc-length; approximating errors and their magnitude when we use infinite sums. | 5/6/11 | Free | View In iTunes |

25 | VideoUnit V: Lecture 1: Logarithms without Exponents | The concept of the natural logarithm; the notion of the rate of change being proportional to the amount opresent; the general concept of a logarithmic function; ln x in terms of differential and integral calculus... | 5/6/11 | Free | View In iTunes |

26 | VideoUnit V: Lecture 2: Inverse Logarithms | The invertibility of the logarithmic function; e to the x as the inverse of ln x; a dicussion of exponential functions; some applications. | 5/7/11 | Free | View In iTunes |

27 | VideoUnit V: Lecture 3: What a Difference a Sign Makes | Hyperbolic functions; comparisons with circular functions; relationship between hyperbolic functions and exponential functions; applications of calculus to hyperbolic functions. | 5/7/11 | Free | View In iTunes |

28 | VideoUnit V: Lecture 4: Inverse Hyperbolic Functions | The theory of inverse functions applied to the hyperbolic functions; some formulas for differentiation and integration; some applications. | 5/7/11 | Free | View In iTunes |

29 | VideoUnit VI: Lecture 1: Some Basic Recipes | A review and extension of previous results for finding f(x) knowing f'(x); paricular emphasis on the case where f'(x) involves the sum and/or difference of two squares; completing the square. | 5/7/11 | Free | View In iTunes |

30 | VideoUnit VI: Lecture 2: Partial Functions | The concept of partial fractions; finding f(x) when f'(x) is the quotient of two polynomials; some notes about identities; application of partial fractions to the case where f is of the form f(sinx, cos x). | 5/7/11 | Free | View In iTunes |

31 | VideoUnit VI: Lecture 3: Integration by Parts | Using the identity d(uv) = udv + vdu to find the integral of udv knowing the integral of vdu; using the technique to evaluate certain integrals; reduction formulas; some applications. | 5/7/11 | Free | View In iTunes |

32 | VideoUnit VI: Lecture 4: Improper Integrals | The problem of trying to study the integral of f(x)dx when f(x) is not continuous on the interval [a,b]; what happens if the limits of integration are not finite; importance of improper integrals. | 5/7/11 | Free | View In iTunes |

33 | VideoUnit VII: Lecture 1: Many Versus Infinite | Discussion of how infinity differs from "very large"; some sublte and not-so-subtle consequences of the difference; the case against intuition; motivating infinite series in terms of finding area as a limit. | 5/7/11 | Free | View In iTunes |

34 | VideoUnit VII: Lecture 2: Positive Series | The special case wherein each term in the series is non-negative; the concept of convergence; the comparision test; the ratio test; the integral test. | 5/7/11 | Free | View In iTunes |

35 | VideoUnit VII: Lecture 3: Absolute Convergence | Non-absolute convergence; conditional and absolute convergence; a series converging when each of its negative terms is replaced by the absolute value of that term; geometric interpretation. | 5/7/11 | Free | View In iTunes |

36 | VideoUnit VII: Lecture 4: Polynomial Approximations | Using an nth degree polynomial to approximate a function f(x); how to choose the coefficients; power series; Taylor's Remainder Theorem; expressing functions in terms of power series. | 5/7/11 | Free | View In iTunes |

37 | VideoUnit VII: Lecture 5: Uniform Convergence | Pointwise convergence versus uniform convergence; some important consequences of uniform convergence; applications of uniform convergence to the study of power series. | 5/7/11 | Free | View In iTunes |

38 | VideoUnit VII: Lecture 6: Uniform Convergence of Power Series | Weirstrass M-test; using power series to evaluate definite integrals when we do not know the anti-derivative of the integrand. | 5/7/11 | Free | View In iTunes |

38 Items |

#### Customer Reviews

##### One of the best math teachers in the world!

This course is so well done and presented by one of the best math teachers in the world. The professor's ability to simplify and explain math terms and notations is incredible. Even though in black and white and on an old chalkboard you just find yourself learning calc every step of the way.

MIT please post more videos of Professor Herbert Gross if you have them in your archive.

##### Simple Explained

This podcast is an excellent source of know Calculus. The fact of it being older makes connections with modern day language fascinating.

##### Amazing resource!

Kudos for this professor, great teaching style and understandable examples.

- Free
- Category: Calculus
- Language: English
- http://ocw.mit.edu; Creative Commons Attribution-NonCommercial-ShareAlike 3.0; http://ocw.mit.edu/terms