We discussed the real numbers and the notion of function (see sections 1.1 and 1.2 in the textbook). We did an exercise on reading and writing proofs (see also comments on reading and writing proofs).
The homework is to read Chapter 1 in the textbook and to look at the following problems: on pages 5-6, numbers 4, 6(b), 10, and 12; and on page 14, numbers 8 and 9. Each group should be prepared to present one problem next class, as follows.
We discussed several of the homework problems and the notions of cardinality and countability. We looked at Cantor's diagonal argument showing that the real numbers cannot be put into one-to-one correspondence with the natural numbers (Theorem 1.3.6 on page 18 of the textbook).
The homework is to read section 2.1 in the textbook and to type solutions of the following problems: number 9 on page 5 (section 1.1), number 3(c) on page 19 (section 1.3), and number 6 on page 24 (section 1.4).
We discussed the formal definition of the limit of a sequence of real numbers, and we looked at some examples (including problem 2 on page 33) of applying the definition.
Sample solutions are available for the homework due today.
The homework for next class is to read section 2.2 in the textbook and to be prepared to present problem 8 on page 20 (section 1.3) and problems 5 and 8 on page 34 (section 2.1).
We discussed the homework problems, and also the three related methods for understanding convergence of a sequence: the formal definition from section 2.1, the limit theorems from section 2.2, and the notion of Cauchy sequence from section 2.4.
The homework is to read section 2.4 (we are skipping section 2.3) and to type solutions of the following problems: number 8 on page 20 (section 1.3), number 3(a) on page 33 (section 2.1), and number 8 on page 39 (section 2.2).
We discussed homework problem 8 on page 39 and the concepts of supremum and infimum from section 2.5 of the textbook. We solved problem 1 on page 54 and considered the question of whether or not the equations and hold for sets.
The homework is to read section 2.5 and to be prepared to present problems 5, 10, and 12 on pages 50-51 (section 2.4).
We discussed homework problems 5 and 10, the notion of limit point of a sequence, and the Bolzano-Weierstrass theorem.
The homework is to read section 2.6 and to be prepared to present the following problems: number 7 on page 34 (section 2.1), number 6 on page 50 (section 2.4), and number 1 on page 59 (section 2.6).
We worked on a quiz and discussed the homework problems.
The homework is to type solutions to the following problems: number 2 on page 54 (section 2.5); number 4 on page 59 (section 2.6); and a problem not from the book, namely to write carefully in terms of epsilon and the negation of " -> ".
We discussed the notion of continuity and its two equivalent formal definitions (see formulas (1) and (2) in section 3.1 of the textbook). We worked on problems 7, 8, and 9 on page 79 as examples of applying the formal epsilon-delta definition.
The homework has three parts: (A) read section 3.1; (B) in preparation for the upcoming exam, make a list of (i) the main definitions and (ii) the main theorems from Chapters 1 and 2; (C) prepare to present problems 3 and 12 on page 79 (section 3.1).
We discussed the homework and reviewed for the examination to be held next class on sections 1.1-1.4, 2.1, 2.2, 2.4, 2.5, 2.6, and 3.1 in the textbook.
Congratulations on your strong performance on the examination! We discussed some examples of continuous functions and three important properties of a continuous function on a closed, finite interval : such a function must (1) be bounded; (2) attain a maximum value and a minimum value; and (3) satisfy the intermediate value property (if the function takes both positive values and negative values, then the function must take the value . We worked on problems 3, 4, and 5 on page 86.
The homework is to read section 3.2 and to type solutions to the following problems: number 7 on page 86 (section 3.2), number 11 on page 79 (section 3.1), and number 9 on page 59 (section 2.6).
We further discussed properties of a continuous function on a closed, finite interval and the notions of uniform continuity and Lipschitz continuity. We looked at the examples of the absolute value function on the real line (a nondifferentiable but Lipschitz and hence uniformly continuous function) and the square root function on the interval (a uniformly continuous but not Lipschitz function). We warmed up to the Riemann integral by computing by hand the area under the parabola from to .
The homework is to read section 3.3 and to prepare to present the following problems: number 13 on page 95 (section 3.3), number 10 on page 86 (section 3.2), and number 13 on page 80 (section 3.1).
We continued the discussion of the Riemann integral and looked at some examples both of integrable and of non-integrable functions.
The homework is to read section 3.5 and to type solutions to the following problems: number 3 on page 94, number 2 on page 111, and this problem that is not in the book: Suppose that and is continuous on the open interval . Prove that if is uniformly continuous on the open interval and on the open interval , then is uniformly continuous on the interval .
We discussed some recent homework problems.
The homework is to read section 3.6 and to prepare to present the following problems: number 1 on page 117 (section 3.6), number 7 on page 112 (section 3.5), and number 9 on page 94 (section 3.3).
We discussed the notion of an improper integral in the context of some of the homework problems. We looked at the following comparison theorem: If and are continuous functions on the interval , if is less than or equal to for all positive , and if the improper integral from to of exists (that is, converges), then the improper integral from to of exists too.
The homework is to type solutions to the following problems: numbers 4 and 6 on page 118 (section 3.6) and number 8 on page 94 (section 3.3).
The key concept in this course is the notion of limit, and the three main classes of functions under consideration -- continuous functions, Riemann integrable functions, and differentiable functions -- are all defined in terms of limits. We worked on constructing examples of sequences of nice functions that converge to limit functions that lack these properties. This motivates our future study of different notions of convergence of functions.
There is no homework to do over spring break.
We discussed the definition and elementary properties of the derivative, including the chain rule.
The homework is to read sections 4.1 and 4.2 and to type solutions to the following problems: numbers 4 and 6 on pages 127-128 (section 4.1) and number 1 on page 133 (section 4.2).
We discussed some of the important theorems about differentiation, including Rolle's theorem, the mean-value theorem, the fundamental theorem of calculus, and Taylor's theorem.
The homework is to read section 4.3, to prepare a list of the named theorems in chapters 3 and 4, and to study the proofs of these theorems, one of which you will be required to reproduce on the examination next Thursday.
We reviewed for the examination.
Second examination. Partial solutions are available.
We discussed the notions of pointwise convergence and uniform convergence of sequences of functions.
The homework is to read section 5.1 and to type solutions to problems 1, 3, and 7 on pages 168-169 (section 5.1).
We looked at examples of pointwise and uniform convergence, and we discussed the supremum norm in the space of continuous functions on a closed, finite interval.
The homework is to read section 4.5, pages 147-150, and section 5.3, pages 175-181, and to prepare problem 3 on page 150 and problems 4 and 5 on page 181.
We continued exploring convergence of sequences of functions, and we discussed the homework problems.
The homework is to read section 5.2 and to type solutions to problems 1 and 3 on page 173 (section 5.2) and problem 4 on page 150 (section 4.5).
We discussed the notions of lim sup and lim inf.
The homework is to read section 6.1 and to prepare problems 1, 2, and 10 on pages 227-228 (section 6.1).
We discussed convergence of infinite series and the ratio and root tests.
The homework is to read section 6.2 and to type solutions to problems 3 and 5 on page 236 (section 6.2) and problem 6 on page 227 (section 6.1).
We continued discussing convergence of infinite series, including Cauchy's condensation test (not in the book), and we talked about uniform convergence of series of functions, including the Weierstrass M-test.
The optional homework is a crossword puzzle.
We discussed the convergence theorem for power series and also Taylor series.
The homework is for various groups to make flash cards of the boldface terms and named theorems in the sections that we covered.
This was our last regular class meeting. We reviewed for the final examination.