Record of daily activities and homework, Math 409-502, Advanced Calculus I, Fall 2004

Monday, August 30

We discussed the fundamental completeness property of the real numbers stating that a bounded monotonic sequence has a limit, and we looked at some examples of sequences of real numbers.
Homework for Wednesday:

1. Read Chapter 1 in the textbook (pages 1-12).
2. Group number n should be prepared to present in class the Exercises for section 1.(n+1) on pages 12-13.

Wednesday, September 1

We looked at solutions to some of the Exercises for Chapter 1.
Homework for Friday:

1. Read Chapter 2.
2. Do Exercise 2.3/1 on page 31 (to hand in).

Friday, September 3

We discussed various notions from Chapter 2: working with inequalities and absolute value, estimating size, finding upper and lower bounds, and proving boundedness or unboundedness of sequences.
Homework for Monday:

1. Read section 3.1, pages 35-38.
2. Do Exercise 2.6/3 on page 32 and Exercise 3.1/1(b) on page 46 (to hand in on Monday).
   In doing these exercises, you may find it useful to consult Question 2.6/3 on page 30 and its answer on page 34, and Question 3.1/1 on page 38 and its answer on page 49.

Monday, September 6

We looked at examples of proving existence of limits (including infinite limits).
Homework for Wednesday:

1. Read sections 3.2-3.5, pages 38-43.
2. Do Exercises 3.1/2 and 3.3/1(a) on pages 46-47.

Wednesday, September 8

We worked on Exercises 3.4/1-5 on page 47 concerning limits related to the limit of aⁿ as n tends to infinity.
Homework for Wednesday:

1. Read sections 3.6-3.7 (pages 44-45) and 4.1 (page 51).
2. Do Problem 2-1 on page 32.

Friday, September 10

We discussed limits of some special sequences, such as naⁿ and a^1/n.
Homework for Monday:

1. Read the rest of Chapter 4 (pages 52-57).
2. Do Problem 4-1 on page 59: namely, prove using the definition of limit that n^1/n→1 as n→∞.

Monday, September 13

We discussed various methods for proving the existence of limits, and we worked an example showing that the limit of integrals is not necessarily equal to the integral of the limit. The slides from the lecture are available.
Homework for Wednesday:

1. Do Exercise 3.6/1 on page 47 and Exercise 4.4/1 on page 58.
2. Read section 5.1, pages 61-64.

Wednesday, September 15

We discussed the homework problems and the squeeze theorem for limits.
Homework for Friday:

1. Finish the homework from last time.
2. Read sections 5.2 and 5.3, pages 64-68.
3. Do Exercise 5.1/1 on page 73.

Friday, September 17

We reviewed limit theorems for sequences and discussed the notion of subsequence. The slides from the lecture are available.
Homework for Monday:

1. Read sections 5.4 and 5.5, pages 68-73.
2. Do Problem 5-7, page 75.

Monday, September 20

We discussed the nested interval theorem, the Bolzano-Weierstrass theorem, and the notion of cluster point. The slides from the lecture are available.
Homework for Wednesday:

1. Read sections 6.1-6.3, pages 78-83.
2. Do Exercises 6.2/1 and 6.3/1 on pages 89-90.

Wednesday, September 22

We discussed Cauchy's criterion for convergence and the notions of supremum, infimum, limsup, and liminf.
Homework for Friday:

1. Read sections 6.4-6.5, pages 83-89.
2. Do Exercises 6.4/3 and 6.5/1 on page 90.

Friday, September 24

We discussed a comparison test for convergence of infinite series and looked at examples.
Homework for Monday:

1. Read sections 7.1 and 7.2, pages 94-100.
2. Do Exercise 7.2/2 on page 109 and Problem 5-5 on page 75.

Monday, September 27

We discussed convergence tests for infinite series, including the root test. The slides from the lecture are available.
Homework for Wednesday:

1. Read sections 7.3 and 7.4, pages 100-104.
2. In preparation for the examination, make a list of the main definitions and theorems in the course so far.

Wednesday, September 29

We reviewed for the examination to be given on Friday.
Homework for Friday: Study for the exam.

Friday, October 1

First examination

Monday, October 4

The graded examinations were returned, and we discussed the asymptotic comparison test and the integral test for convergence of infinite series.
Homework for Wednesday:

1. Read sections 7.5-7.7, pages 104-109.
2. Do Exercises 7.4-7.5/1a,b,c on page 110 and 7.7/1 on page 111.

Wednesday, October 6

We reviewed convergence tests for infinite series, discussed Cauchy's condensation test (which is not in the textbook), and began a discussion of convergence of power series. The slides from the lecture are available.
Homework for Friday:

1. Read section 8.1, pages 114-117.
2. Do Exercise 7.6/1a,c on page 111.
3. Do Exercise 8.1/1g on page 123.

Friday, October 8

We worked on some exercises about convergence of power series (Exercise 8.1/1 on page 123).
Homework for Monday:

1. Read sections 8.2, 8.3, and 8.4 (pages 117-122).
2. Do Exercise 8.3/1 on page 123.
3. Do Problem 7-2 on page 111 (assume that a_n and b_n are non-zero for all n).

Monday, October 11

We discussed endpoint convergence of power series and the multiplication theorem for absolutely convergent series (including a counterexample for conditionally convergent series that is not in the book). The slides from the lecture are available.
Homework for Wednesday:

1. Read Chapter 9, pages 125-134.
2. Do Exercises 9.2/3 and 9.3/1, pages 134-135.

Wednesday, October 13

We discussed some of the homework problems and considered group structures on sets of functions. The slides from the lecture are available.
Homework for Friday:

1. Read sections 10.1 and 10.2, pages 137-142.
2. Do Exercises 10.1/2 and 10.2/1 on page 148.

Friday, October 15

We discussed approximation of functions and the contrast between local properties of functions and global properties of functions.
Homework for Monday:

1. Read sections 10.3 and 10.4, pages 143-147.
2. Do Exercise 10.3/2 on page 149.
3. Do Problem 10-2 on page 150.

Monday, October 18

We discussed the notions of limits of functions and continuity of functions. The slides from the lecture are available.
Homework for Wednesday:

1. Read sections 11.1 and 11.2, pages 151-158.
2. Do Exercises 11.1/5 and 11.2/1 on page 167.

Wednesday, October 20

We continued the discussion of limits and continuity, and we worked in groups on some problems about limits. The slides from the lecture are available.
Homework for Friday:

1. Read sections 11.3, 11.4, and 11.5, pages 158-167.
2. Do exercises 11.4/1 and 11.5/1 on page 168.

Friday, October 22

We discussed the theorem that a continuous function on a compact interval has a range that is again a compact interval. The slides from the lecture are available.
Homework for Monday:

1. Read sections 12.1 and 12.2, pages 172-177.
2. Do Exercises 12.1/1 and 12.1/5 on page 180.

Monday, October 25

We discussed continuity, the intermediate-value property, monotonicity, the existence of inverse functions, and relations among these properties. The slides from the lecture are available.
Homework for Wednesday:

1. Read sections 12.3 and 12.4 (pages 178-180) and sections 13.1 and 13.2 (pages 185-187).
2. Do Exercise 12.4/2 on page 181 and Exercise 13.1/1a,b on page 192.

Wednesday, October 27

We discussed two properties of continuous functions on compact intervals: boundedness and existence of extreme values. The slides from the lecture are available.
Homework for Friday:

1. Read sections 13.3 and 13.4, pages 187-190.
2. In preparation for the examination, make a list of the main definitions, concepts, and theorems from sections 7.5 through 13.4.

Friday, October 29

We reviewed for the examination and discussed some homework problems. The slides from the lecture are available.
Homework for Monday: Study for the examination.

Monday, November 1

Second examination

Wednesday, November 3

The graded examinations were returned, and we discussed some of the problems. The slides from the lecture are available.
Homework for Friday: Use the ε-δ definition of continuity to prove that

1. the function 1/x² is continuous at the point 1;
2. the function 1/x is continuous at the point 1/10.

Friday, November 5

We discussed uniform continuity and the theorem that a continuous function on a compact interval is uniformly continuous. The slides from the lecture are available.
Homework for Monday:

1. Read section 13.5, pages 190-192.
2. The interval [0,∞) is not compact. Show nonetheless that the function √x is uniformly continuous on this unbounded interval.
3. The interval (0,1) is not compact. Determine (with proof) whether sin(1/x) is uniformly continuous on this open interval.

Monday, November 8

We discussed the definition of the derivative and the mean-value theorem. The slides from the lecture are available.
Homework for Wednesday:

1. Read Chapter 14 (pages 196-204) and section 15.1 (pages 210-211).
2. Do Exercise 14.1/3 on page 205.
3. Do Exercise 15.1/4 on page 218.

Wednesday, November 10

We discussed Cauchy's mean-value theorem and l'Hôpital's rule. The slides from the lecture are available.
Homework for Friday:

1. Read sections 15.2-15.4, pages 212-217.
2. Do Exercise 14.3/2 on page 206.
3. Do Exercise 15.4/2 on page 219.

Friday, November 12

We discussed Taylor's formula. The slides from the lecture are available.
Homework for Monday:

1. Read sections 17.1-17.3, pages 231-236.
2. Suppose you were to plot the functions y=cos(x) and y=1-x²/2 on the same graph with the x and y axes scaled in inches (1 inch = 1 radian) using a line thickness of 1 point (where 1 inch = 72 points).
   
   Over what interval of the x-axis would the two curves be indistinguishable? Why?

Monday, November 15

We discussed Taylor's theorem, Taylor series, and analytic functions. The slides from the lecture are available.
Homework for Wednesday:

• Read section 17.4, pages 236-238.
• The third examination is scheduled for Wednesday, December 1.
  One of the problems on the exam will be to prove a version of l'Hôpital's rule selected from the following eight possibilities: {0/0 or ∞/∞} and {x→a or x→∞} and {lim f'(x)/g'(x)=L or lim f'(x)/g'(x)=∞}.
  Work on proofs of two cases (to discuss in class).
  The method of dividing the cases that I suggested in class does not work. Here is a new scheme:
  1. Cases (x→a, 0/0, f'(x)/g'(x)→L) and (x→∞, ∞/∞, f'(x)/g'(x)→∞)
  2. Cases (x→a, 0/0, f'(x)/g'(x)→∞) and (x→∞, ∞/∞, f'(x)/g'(x)→L)
  3. Cases (x→a, ∞/∞, f'(x)/g'(x)→L) and (x→∞, 0/0, f'(x)/g'(x)→∞)
  4. Cases (x→a, ∞/∞, f'(x)/g'(x)→∞) and (x→∞, 0/0, f'(x)/g'(x)→L)

Wednesday, November 17

We discussed the notion of (Riemann) integrability. The slides from the lecture are available.
Homework for Friday:

1. Read sections 18.1 and 18.2, pages 241-244.
2. Consider the integrable function f(x)=x on the interval [1,2]. How small must the mesh of a partition be in order to guarantee that the upper sum and the lower sum differ by less than 1/10?
3. Do exercise 18.2/3 on page 248.

Friday, November 19

We proved that monotonic functions are integrable and that continuous functions are integrable. The slides from the lecture are available.
Homework for Monday:

1. Read sections 18.3 and 18.4, pages 244-248.
2. Prove that if a function f is defined and bounded on [a,b] and is continuous except at one point, then f is integrable.
3. Do exercise 18.4/2 on page 249.

Monday, November 22

We discussed Riemann sums and the fundamental theorem of calculus. The slides from the lecture are available.
Homework for Wednesday:

1. Read sections 19.1-19.3, pages 251-256.
2. Work on proving various versions of l'Hôpital's rule.
   A sketch of one case is at the end of today's slides.

Wednesday, November 24

We worked on proofs of l'Hôpital's rule.

Monday, November 29

We reviewed for the examination to be given on Wednesday, and we discussed the second part of the fundamental theorem of calculus.
Homework for Wednesday: study for the examination.

Wednesday, December 1

Third examination

Friday, December 3

We discussed uniform convergence and three theorems about uniform convergence. The slides from the lecture are available.
Homework for Monday: Read sections 22.1-22.5, pages 305-318.

Monday, December 6

We looked at some additional examples related to uniform convergence. The slides from the lecture are available.
Reminder: the final examination is scheduled for Tuesday, December 14, 8:00-10:00 am.

Tuesday, December 14

Final examination