This file will be frequently changed and added to as the semester goes along. Consult it frequently.

You may move to a particular lecture by clicking on its number: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

About the problems: In each assignment, you will find up to 3 lists of problems, labeled (O), (A) and (B).

• You will be asked to write up and turn in a number of the problems, chosen from lists (A) and (B), by a certain date. As you will see, you will have a choice as to how many problems and which problems you do. The (O) problems cannot be turned in for credit.
• The (A) problems are worth 10 points each. Each of the (B) problems will be worth at least 10.
• You should do all of the (O) and (A) problems on your own, or at least make sure you understand how to do them--even though you don't write all of them up to turn in. You are responsible for learning how to do such problems.

The following requests are mostly meant to help us process your papers efficiently and avoid losing papers.

• When you turn in a problem set, please use 8.5-by-11 sheets of paper and fold the sheets together lengthwise, with your name and the date written on the right-hand side of the back of the last sheet.
• Also, please write your name at the upper right on every sheet.
• Please turn in your (A) problems and your (B) problems in separate sets--because they may be turned over to different readers for grading.
• To protect yourself from identity theft and other mischief, do not write your Social Security Number on papers or tests. It's not needed.

In grading a problem on homework or a test, I use a numerical scale from 0 to 10, thus: 8.5 to 10, A; 6.5 to 8.4, B; 4.5 to 6.4, C; 2.5 to 4.4, D. Interpret all scores according to that conversion table, not as percentages. There's a statement of criteria for the grades, but the number/letter conversion table at the bottom of that statement is incorrect.
 
 

Read Chapter 1: Functions and Models, and also read Appendices A-E. This material is review. Even if you have not studied it all before, you will probably be able to read and understand everything with a bit of effort. During the course, as we go along and actually need various things that are in this reading assignment, I may devote some lecture time to some of them.

I plan to begin by talking about the number systems, starting with the Peano Axioms and the Principle of Induction. Your objective is to understand proofs by induction, and learn how to write such proofs. I will do some examples in class.
 

In the next few days, I plan to present more examples of proof by induction, ranging from really simple to somewhat interesting. You will come across problems in which the task is first to discover, then to check, a law or a formula, and in which the key is an understanding of the induction axiom.
 

During the rest of the week, I'll cover several topics concerning the real number system. These are topics not covered in the book.

I want you to know the definition of an algebraic number and to understand these facts: (1) Every rational number is algebraic. (2) Not every algebraic number is rational. (3) Not every real number is algebraic. One way to prove (3) is to show that the set of all algebraic numbers is countable, but the set of all real numbers is not. I may show you that argument a bit later. Another way to do it is to find one specific real number and prove it's not algebraic; that requires Liouville's Theorem, which depends on some calculus, and which we may study later. By the way, neither pi nor e is algebraic.

Problem: Given a polynomial with integer coefficients, find all of its rational zeros. There is a Theorem that provides a solution and a technique for finding it. Your objective is to understand the Theorem, know its proof, and be able to apply it. I will do some examples in class.
 

Today I'll re-state the Theorem mentioned on Wednesday, prove it, and work some more examples using it.
 

I'll talk about the Completeness (or Least Upper Bound) Axiom, which is part of the definition of the real number system, and which will be used repeatedly in the course. And I'll begin discussing sequences and limits of sequences. Your objectives: Know the definitions and understand examples.

Assignment due today: Write up and turn in at least 3 but no more than 6 problems. The problems you turn in must be chosen from lists (A) and (B). Always write your solutions so that they will be understandable to another member of the class. You may wish to consider volunteering to present one or more of your solutions at the blackboard.

The problems are from H1 (Handout 1), which I'll give you on the first day, and Exe1 (Exercise List 1).

Here are the three lists: (O) H1: 1, 2, 4, 5, 6, 9, 10. (A) H1: 3, 7, 8; and Exe1: E1, E2, E3. (B) Exe1: E4, E5, E6, E7.
 

I'll prove a couple of results about limits of sequences. These will be only slightly tricky exercises in making logical arguments, proving the obvious, but they'll prepare the way for subtler things. I'll also introduce some problems of determining whether certain important sequences converge, and identifying their limits.

Read Section 11.1 (in Stewart's book) with care. Omit Example 4 unless you're familiar with l'Hospital's Rule. There are mentions of Chapter 2 material, but you don't need to know it to read this Section.
 

I'll talk about tangent lines, derivatives, and limit statements. Read Section 2.1. You should do Exercise 3 in this Section carefully, using a calculator.
 

Limit statements. Theorems about limits. Identifying limits and values of derivatives.

Assignment, due today: Write up and turn in at least 4 but no more than 8 problems. Here are the lists. (O) 2.1: 3. 11.1: odds 1-19. (A) 11.1: 16, 20, 22, 24. Exercise List 1: E8, E9, E10, E11. (B) 11.1: 63 (double credit). E5, E6, E7. (Note that the last three were on the previous assignment; if you haven't done them, they're still available.)
 

Topics: Selected problems. Continuity. Composite functions. Range and domain. Properties of functions.

Read Section 2.2. To check your understanding, do these problems: (O) 2.2: odds 5-9. If you have a graphing calculator, do 31.

Read Section 2.3. To check your understanding, do these problems: (O) 2.3: odds 1-9 and 37-45 and 53.

Read Section 2.4. Do these problems: (O) 2.4: 1, 3, 5, 33, 39.
 

Selected problems: Finding limits, computing derivatives, geometric problems using derivatives, proofs by induction.

Read Section 2.5. Do these problems: (O) 2.5: odds 1-5 and 15-19.

Read Section 2.6. Do these problems: (O) 2.6: odds 3-31.
 

Selected topics and problems: Applying the Rational Zeroes Theorem (problem E11). "Finding a delta" (2.4, problem 3)." Look for problems like these on Test One.

Assignment due today: Do at least 4 but no more than 8 of these problems. (O) p. 83: 1, 3, 5, 7. p. 179: 1, 3, 5. (A) 2.3: 58. Page 83: 2, 4, 6, 12. 2.6: 12, 14, 16, 20, 22, 30. Always justify your answers analytically. (B) 2.6: 52. p. 179: 8, 10.
 

Topics: Geometric and pictorial interpretations and uses of the derivative. Proof that a differentiable function is continuous, but a continuous function may not be differentiable. Introduction to the Chain Rule.

Read Section 2.7. Do these problems: (O) 2.7: 1, 3, 5ab, 15. I'm not interested in having you evaluate derivatives in a lot of cases by using the definition, but please make sure you understand the process by doing problem 5ab. Also, be sure you understand the geometric interpretations called for in problems 3 and 15.

Read Section 2.8. Do these problems: (O) 2.8: 3, 4, 5, 6. All these problems call only for geometric interpretations of the derivative.
 

In class I'll do selected problems, and problems by request.

Read Section 2.9. Do these problems: (O) 2.9: 1, 3, 4-12, 33, 39, 43, 45. I'm not giving you a large number of problems that call for finding the derivative "by the definition," and emphasizing instead geometric and pictorial problems. Make especially sure you can deal with problems like 4 through 12.
 

In class I'll do selected problems, and problems by request.
 

Test 1 will cover everything through Friday the 13th. The test may represent the assigned reading, the material discussed in lectures, and the problems assigned from (O) and (A) lists.
 

Today I talked about problems 1 and 4 on page 179. In connection with problem 1, you may wish to review Section 1.6. I made an error in doing problem 4; see the assignment for 9/23.
 

Read Section 3.1. Do these problems: (O) 3.1: odds 3-27 (do enough of these, checking your answers in the back, to be sure that you are proficient doing such problems by hand); 43, 45, 47.
 

Read Section 3.2. Do these problems: (O) 3.2: odds 3-21 (do enough of these, checking your answers in the back, to be sure you're proficient); 23, 31, 33, 39.
 

Read Section 3.4. Do these problems: (O) 3.4: odds 1-15 and 35-43 (do enough of these to be sure you're proficient); 21, 45. In class (Prof. Perlis): The Intermediate Value Theorem (p. 129).
 

Assignment due today: Do at least 6, no more than 12 of these problems: (A) 3.1: 44, 46, 50, 52; 3.2: 32, 40, 44 (see no. 42); 3.4: 38, 40, 42. (B) p. 179: 4 (do the same problem again for the function x^2/4, and again for the function x^3--these will count as 3 separate problems), 9, 10; 3.4: 46; p. 273: 2, 4, 6 (is this a formula to be established by induction?), 8, 10, 16 In class: Proof of one version of the Intermediate Value Theorem.
 

A problem involving the chain rule and tangent and normal lines.
 

Read Section 3.5. Do these problems: (O) 3.5: odds 7-41 (do enough of these to be sure you're proficient!); 51, 53, 55. Bring your questions to ask in class. If time allows, I'll talk about implicit differentiation and the differentiation of inverse functions.
 

Read Sections 3.6 and 3.7. Do these problems: (O) 3.6: 5, 15, 21, 25, 29, 41; 3.7: 1, 2, 3, 4.

Read Section 3.10. Do these problems: (O) 3.10: 1, 3, 9, 17.

Problem 17 in Section 3.10, was done in detail. The derivative of the arctangent (tangent-inverse) was discussed. A proof was given that if a function has a nonzero derivative at a point interior to its domain, then neither a relative max nor a relative min can occur at that point. This result is called Fermat's Theorem, and is proved in the book on page 280. Assignment due today: Write up and turn in these three problems: (A) E12, E13, and E14 from Exercise List 1.

In class: Solution to problem E12 - proving the Intermediate Value Theorem. Summary of several important theorems: Extreme Value Theorem, Rolle's Theorem, Mean Value Theorem.

This week, read Sections 4.1, 4.2, and 4.3. Learn the definitions and the statements of all theorems. Read the examples with care. Problems to do on your own during the week: (O) 4.1: odds 1-17, 49, 51. 4.2: odds 1-7, 11, 15, 25. 4.3: 1, 3, 5, 7, 31, 41. In class: I plan to work problems by request, and to discuss the Chapter 4 theorems.

Assignment due today: Write up at least 3 but no more than 6 of these problems: (A) 4.2: 12, 14, 16, 24. 4.3: 32, 58, 68. In class: Solutions to problems E13 and E14, presented in class by Gina McClure and Laura Stevens. Hurricane Lili caused LSU to cancel classes on October 3 and 4.

Test 2: This test will be closed-book, closed-notes. No graphing calculators allowed. Calculators OK. You are responsible for all the material from the very beginning of the course, but the test will cover primarily Chapter 3, Sections 4.1-4.3, and other material discussed in class since Test One. Study the (O) and (A) exercises. Problems on the test may require you to make sketches by hand of certain graphs; to evaluate limits; to apply the Chain Rule and to do a "related rates" word problem; to do problems involving implicit differentiation; to state definitions, statements of theorems, and give proofs; and to find derivatives, using the "rules" and/or the definition.

Read Section 4.5 and do these problems: (O) 4.5: 1, 3, 7, 11, 33, 43. In class: The "curve-sketching theorems" related to intervals of strict increase or decrease, and tests for the kind of an extremum that occurs at a point where the derivative is zero.

In class, I'll answer questions, and then if time allows I'll show you some examples of curve-sketching problems and/or optimization problems.

Optimization problems. Snell's Law. A brief mention of the brachistochrone problem.

Review Sections 4.3 and 4.5. On Friday's test you may expect one problem to call for the "curve-sketching" analysis of a function. To prepare, study Examples 1 through 7 in Section 4.3, and Examples 1, 3, and 5 in Section 4.5. In class: More optimization examples. Inscribing a cylinder of maximum volume inside a sphere. Assignment due today: Write up and turn in at least two, no more than four, of these problems: (A) 4.3: 43, 45, 47, 49.

Read Section 4.7, and study Examples 1, 2, and 3 thoroughly. In class: More examples of optimization problems. Maximizing the volume of a cylinder with fixed surface area.

Assignment due today: Do at least 3, no more than 6 of these problems: (A) 4.7: 2, 10, 18, 26. (B) 4.7: 40, 48, 50. You are expected to write up your solution thoroughly and clearly, using complete sentences and explaining your logic.

Test 3: This test will be closed-book, closed-notes. No graphing calculators allowed. Calculators OK. You are responsible for all the material from the very beginning of the course, but the test will cover primarily Chapter 3, Sections 4.1-4.3, 4.5, 4.7, and other material discussed in class since Test One. Study the (O) and (A) exercises. Problems on the test may require you to make sketches by hand of certain graphs; to evaluate limits; to apply the Chain Rule and to do a "related rates" word problem; to do problems involving implicit differentiation; to state definitions, statements of theorems, and give proofs; and to find derivatives, using the "rules" and/or the definition; to be able to state, understand, and apply the theorems that pertain to curve-sketching; to carry out a complete curve-sketching analysis of a function, as in Sections 4.3 and 4.5; and to do optimization problems as in Section 4.7.

Deriving the properties of the exponential function from its differential equation. Direction fields.

Read Section 4.10 (O) 4.10: Odds 1-13; odds 29-35; 45. In class: Antiderivatives. Finding the most general function, the derivative being given.

In class: An introduction to Newton's Method.

Assignment due today: Turn in at least 4, at most 8. (A) 4.7: 42, 54; 4.10: 68, 72, 76. (B) p. 364: 6, 12, 16, 24. In class: Definition of the Riemann integral. Evaluation of the integral of the function x-squared over the interval [0,b]. Example of a function that isn't Riemann integrable.

In class: More on the Riemann integral, its properties. The Fundamental Theorem of Calculus. Read Section 4.9. Do these problems, or as many as you need to do to understand what's going on: (O) 4.9: 1, 3, 5, 7, 27. Study Newton's method with a view to being prepared to answer a test question like: Write an explanation of Newton's Method. Also, understand when Newton's Method fails; look at problems 31 and 32 in this section.

Newton's Law examples. The Fundamental Theorem of Calculus. Read Sections 5.1 through 5.3.

Applications of the Fundamental Theorem of Calculus.

Assignment due today: Do at least 4, no more than 8. (A) p. 361: 49, 50, 51; 5.1: 18, 19; (B) p. 361: 52, 53, 54, 77(double credit). In class, Russell Russo and Tyler Moore will present solutions to p. 364:12, 4.10:68, and 362:77.

Read Section 5.4, returning to earlier sections of the chapter as necessary for review. Do these problems on your own, not to turn in: (O) 5.4: odds 1-9, odds 17-21, 29, 35, 41, 45.

Read Section 5.5 Try for proficiency at the process of finding antiderivatives. Problems to do on your own: (O) 5.5: odds 1-43.

Test 4 will cover everything we've dealt with since the last test; it will cover material from the last half of Chapter 4 and some from Chapter 5.

More on the definition of the natural logarithm as an integral, proofs of the logarithm's properties and its relation to the exponential.

Read Section 5.6. Do these problems to gain proficiency: (O) p. 428: odds 9-39, odds 45-49. Ch. 5 Review, p. 425-426: 1, 2, 3, 6, and the True-False Quiz.

In class: I've asked three of you to present and explain some problems from page 427 at the board. Tiffany Scavo: 10, 16, 22, 28, 34, 40. Amy Wormsley: 12, 18, 24, 30, 36, 46. Scott Sciacca: 14, 20, 26, 32, 38, 48.

Assignment due today: Do at least four, no more than eight. Write up your problems making your procedure clear, and be neat. Do it as if you were presenting the methods and solutions to the class. (A) 5.5: 74, 78, 80, 82; p. 427: 22, 24, 32, 40. In class: Integrals that give volumes of solids of revolution. The methods of "disks" and the method of "cylindrical shells"

The volume of a generalized cone. The area of a surface of revolution.

Read Sections 6.1 and 6.2. Problems to do on your own: (O) 6.2: 1, 3, 11, 31, 35, 41, 47. In class, I plan to do problem 44 on page 455.We'll get organized for some of you to present problems at the blackboard on Thursday.

Assignment due today: Do at least 2, no more than 6. Write up each solution thoroughly, as you would present it to the class, for maximum clarity and ease of understanding. (A) 6.2: 2, 6, 20, 24, 28. (B) 6.2: 50, 59, 62, 63.

Read Section 6.3. Do these problems on your own: (O) 6.3: 1, 5, 23, 29, 31, 41. 2, 6, 16. Students present solutions at the board.

Readpages 558-562 and study the examples there. Understand Pappus's Theorem and how it can save work, sometimes, in finding volumes of solids of revolution. In class: Students present solutions at the board.

Assignment due today: Do at least 2, no more than 6. Write up each solution thoroughly, as you would present it to the class, for maximum clarity and ease of understanding. (A) 6.3: 2, 6, 20, 24. (B) 6.2: 52, 54, 58, 66. 6.3: 42. Test 5 will cover applications of the integral.

Read Section 7.1 on Integration by Parts. Problems to do on your own: (O) 7.1: 1, 5, 7, 9, 11, 17, 19, 21, 31.

Read Section 6.4 on Work. Problems to do on your own: (O) 6.4: 1, 5, 7, 11, 13.

Assignment due today: At least 3, no more than 8. (A) 7.1: 6, 18, 20, 22, 24, 26. (B) 7.1: 40, 42, 44, 46.

Read Section 7.2 on Trigonometric Integrals. Do these problems on your own: (O) 7.2: 1, 3, 7, 9, 19, 21, 23.

Read Section 7.3 on Trigonometric Substitutions. Do these problems on your own: (O) 7.3: odds 1-9, 31. In class: Examples of improper integrals.

Assignment due today: At least 6, no more than 12. (A) 7.2: 2, 4, 6, 12, 32, 40, 44, 46; 7.3: 2, 4, 6, 8, 18, 30, 32. Read Section 8.1: Arc Length. (O) 8.1: 1, 7, 15. You'll notice that these problems are carefully designed so that you can carry out the evaluation by hand! Today in class: More on improper integrals.

Read Section 7.8: Improper Integrals. In class: Examples by request. If time allows: Arclength. Integration of Rational Functions by Partial Fractions. (O) 7.8: 1, 5, 9, 11.

More on improper integrals, more on partial fraction decomposition.

This test will cover the material from Chapters 6 and 7 identified above.

Read Section 4.4, L'Hopital's Rule. Problems to do on your own: (O) 4.4: odds 1-29.

Read Section 8.2, on the area of a surface of revolution. Consider this as a review of visualization, setting up integrals, evaluating integrals. Problems to do on your own: (O) odds 1-9; and 27.
 
 No class on Thursday.

Review, and advice about preparing for the final exam. Assignment due today: Do at least 2, no more than 10. (A) 7.2: 8, 22. 7.3: 10, 12. 7.5: 6, 44. 7.8: 40, 42. 8.2: 28, 30.