Functions of a Single Variable

**Volume:** 30 lectures @ 90 minutes

**Syllabus:** Download Course Description (PDF)

**Textbook:** Michael Spivak, *Calculus*, 3rd Edition, Cambridge University Press,

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**Prerequisites:** None. The course is an introductory math course for freshman students.

**Background and Goals:** The sequence Honors Math Vv186-285-286 is an introduction to calculus at the honors level. It differs from the Applied Calculus sequence in that new concepts are often introduced in an abstract context, so that they can be applied in more general settings later. Most theorems are proven and new ideas are shown to evolve from previously established theory. Initially, there are fewer applications, as the emphasis is on first establishing a solid mathematical background before proceeding to the analysis of complex models.

The present course covers the calculus of functions of a single real variable.

**Key Words:** Elements of logic; set theory; properties of rational, real and complex numbers; sequences, convergence, completeness of metric spaces; functions, convergence and continuity; the derivative and applications; normed vector spaces; sequences of functions; series and power series; transcendental functions; the regulated, Darboux and Riemann integrals with applications; Taylor's theorem; Stirling's formula.

**Detailed Content:** This course consists of four distinct parts. We first fix the rules of language we will be using, giving a brief introduction to mathematical logic and statements in mathematics. We introduce the rational numbers and show various aspects of their insufficiency. We extend them to the set of real numbers and from there to the complex numbers. On the way, we will learn important basic concepts essential for all future mathematical studies.

The second part of the course introduces the concepts of function, convergence and continuity. Starting from the simplest non-finite maps, sequences of real numbers, we familiarize ourselves with the concept of convergence. This is then extended to functions of real variables and the notion of continuity is introduced. Many properties of continuous functions are studied. This leads naturally to the next part of the course.

We next study the linearization of functions and define the derivative of real-valued functions. It subsequently becomes necessary to discuss functions with values that are not real (e.g., complex) and to this end we introduce the notion of normed vector spaces. This allows us to discuss sequences of functions, and we learn about pointwise and uniform convergence. In order to extend the classes of known functions (which heretofore have been limited to rational functions) we discuss series in vector spaces and power series. This leads to the definition of the exponential function and from it the logarithmic and trigonometric functions.

In the final part of the lecture we introduce the notion of the regulated integral. We first define this integral for step functions, where its meaning and value is intuitively clear, then extend it to all functions that can be uniformly approximated by step functions. This procedure allows us to immediately obtain an integral for vector-space-valued functions, which is not possible using the Riemann integral. We will then introduce the Riemann integral as the (equivalent) Darboux integral building on our work with the regulated integral.

After a discussion of the practical issues associated with integration, we continue with a discussion of improper integrals, and applications, such as the definition of the Euler Gamma function and Taylor's theorem with several expressions for the remainder. The culmination will be a proof of Stirling's estimate of the factorial.

**Alternatives:** Vv156 (Applied Calculus II) is an applications-oriented course, which covers much of the same material.

**Subsequent Courses:** Vv285 (Honors Mathematics III) is the natural sequel.

**Overview:**

Lecture | Lecture Subject |
---|---|

1 | Elements of Logic |

2 | Set Theory |

3 | Natural, Rational and Real Numbers |

4 | Complex Numbers |

5 | Sequences and Convergence |

6 | Sequences and Convergence |

7 | Metric Spaces, Real Functions |

8 | Limits and Landau Symbols |

9 | Properties of Continuous Functions |

10 | Properties of Continuous Functions |

11 | First Midterm Exam |

12 | The Derivative |

13 | The Derivative |

14 | Properties of Differentiable Functions |

15 | Properties of Differentiable Functions |

16 | Vector Spaces |

17 | Sequences of Functions |

18 | Series |

19 | Series |

20 | Power Series |

21 | The Exponential and Logarithm Functions |

22 | The Trigonometric Functions |

23 | Second Midterm Exam |

24 | Step Functions and Regulated Functions |

25 | The Regulated and Darboux Integrals |

26 | The Riemann Integral |

27 | Integration in Practice |

28 | Integration in Practice |

29 | Applications of Integration |

30 | Final Exam |

The exact location of the exam classes may vary according to the term schedule. The exams in the list above are placed so that the exam topics are precisely those of the preceding lectures.

**Lecture Slides (2015 version; PDF):**

(There will always be minor modifications from one iteration of the course to the next; if you are presently taking the course, it is not advisable to print out all these slides at the beginning of the term.)

**Assignments (2015 version; PDF files):**

(There will always be minor modifications from one iteration of the course to the next; if you are presently taking the course, your assignments may differ from these.)