Recent additions and changes will usually be above the first horizontal line. Course policies

Your final exam has been graded and I encourage you to pick it up in the Main Math Office. Your letter grade in the course in on its back page and will be on your "myinfo" site Wednesday, May 13.

Have a great summer!

April 27

Final Exam, Thursday, May 7, of exam week, at 4:00 pm in the usual classroom.

  • Here is a copy of last year's final exam.
  • No HW due Friday, May 1.
  • HW due Wednesday, April 29: Section 10.5, page 464: 1, 3 (an excellent problem), 6, and use level curves to sketch a graph (picture) illustrating a polynomial function R2 -> R1 which has a strict local minimum at the origin along every line through the origin but the origin is not a local minimum. 

April 21

HW due Friday, April 27:  [This is not long] Think of a linear scale for rating how good some team (individual) is (any sport you want) and figure out some way to covert ratings of two teams (individuals) into the probability one team beats the other. What sort of ratings would be typical for regular participants in your system?  100s?  0 to 1?  If rating A is larger than rating B, how can you convert that to a probability A beats B?

HW due Monday, April 27: [Based on material covered in class.]  Use the gradient and "the method of steepest ascent" to maximize the log likelihood we derived in the case when the the data is as follows. Begin at some initial point, with, say, a = 1/3 and b = 1/2. Change (a, b) using a small multiple of the gradient and then redo it. Repeat until the answer stabilizes.

[Do this with a calculator or computer, and exhibit enough programming so I know you know how it works. Do not use a powerful equation-solving program or function-maximizing program that does it by itself without showing you what it does.]

 
A
B
C
D
1
A*
B
C
 
2
A
B
 
D*
3
 
B*
C
D
4
A*
B
 
 
5
 
B*
 
D
6
A
B*
 
D

The six voters are listed in the left column. They vote for the "best" movie. The four movies they vote among are in columns. The ones they actually saw are listed in the row, and their favorite among the ones they saw is starred.

  • The model:  Assume the probability is px that a voter who has seen all four movies votes for movie X. If the voter has seen a subset of the movies, the probability is px divided by the sum of the probabilities of the movies the voter has seen, if X was seen, and 0 otherwise.
  • Because no one voted for C, we eliminate C from the computations (as if C's rating were 0). Then, because the probabilities sum to 1, the probability for D is a function of the probabilities for A and B.
  • The likelihood function to maximize in this example is a2b3(1-a-b)/((a+b)2(1-a)2). We decided to maximize its log instead, using the method of steepest ascent, in the region where a and b are probabilities such that a+b < 1.
  • Start with, say, a = 1/3 and b = 1/2.

April 9


HW due Friday, April 10:  9.5:  1, 2, 5, 6

HW due Monday, April 13: Skim Section 10.1 (We don't study "connected").  Do 10.1, page 436, 4, 7, 8
[Monday at 10:00 you are invited to my talk on RSA cryptography to my Methods of Proof class  in Wilson 1-142.]

HW due Wednesday, April 15: Read Section 10.2 and do page 445: 1b, 3b

HW due Friday, April 17:  10.2, page 446:  4, 8 (a reason, but not a formal proof required), 14

HW due Monday, April 20: Section 10.3, page 454:  1a,b,c [for part c), change the value to 0 {from 1} at (x, y) = (0, 0)], 3a (use the definition, as he requests. Omit 3b).

Theorem:

Let f: R2 -> R1. If f is differentiable at (x0, y0), then fx(x0, y0)and fy(x0, y0) exist.

This can be used to show a function is not differentiable at a particular point (x0, y0), but will not suffice if fx(x0, y0) and fy(x0, y0) exist.

Theorem:

If fx(x0, y0) and fy(x0, y0) exist, define ε [a function of (x, y)] by

*   f(x, y) = f(x0, y0) + fx(x0, y0)(x - x0) + fy(x0, y0)(y - y0) + ε ||(x-x0, y-y0)||.

Then, f is differentiable at (x0, y0) if ε -> 0 and f is not differentiable at (x0, y0) if ε does not go to 0.

  • Regardless of whether f is differentiable or even continuous, there is an ε such that equation * holds. If (x, y) is not (x0, y0), equation * can be solved for ε. The theorem above is very similar to the definition of "differentiable"--it just notes the role of the partial derivatives.

HW due Wednesday, April 22:  Section 10.4, page 460: 1a,b,c, 2, 4

HW due Friday, April 24: Section 10.5, page 464: 1, 3 (an excellent problem), 6, and use level curves to sketch a graph (picture) illustrating a polynomial function R2 -> R1 which has a strict local minimum at the origin along every line through the origin but the origin is not a local minimum.

March 30

Monday, April 6: Exam on Chapter 8.

HW due Wednesday, April 8: 9.3: 16, 17  [Comments below on 9.1-9.7 have been expanded. Look at them again.]  List some of what you learned in differential calculus (Now M 171) that is not "how to do calculations." (Work with a friend to expand your list.) That is, if I gave you a calculation device that could do all the derivatives, take all the limits, etc., what did you learn that that device cannot replace?  [Note:  This list will help us identify what there is to learn about multivariate calculus.]

Chapter 9 is "Vector Calculus" which we will cover rapidly so we can move on to Chapter 10, "Functions of Two Variables,"  where we discuss differentiation in multiple dimensions. Nevertheless, Chapter 9 has some interesting and important results.

  • 9.1.1 The Pythagorean Theorem (distance) in 3-D
  • 9.2.4  perpendicular, parallel, the "direction" of a vector
  • 9.3.1-3  dot product, properties, angle between vectors
  • 9.3.8-12  cross product (notation typo in old printings: 9.3.9b should, obviously, have "x" [cross] for "+") Last line above "Exercises 9.3" has a formula for area (I don't know why this was not labeled as a theorem).
  • 9.4 and 9.5 parametric equations (9.4, especially of lines, 9.5) which are used in differentiation (especially 9.5.3), Example 9.4.2 generalizes to 3-D.
  • 9.5, amazingly, has an asterisk on it ("9.5*")which suggests it is optional. Not so. Theorem 9.5.3 is critical, since it gives the equation of a plane in 3-D, and the definition of "differentiable" refers to approximation by planes.
  • 9.6  mostly very parallel to previous material. It treats a function in 1-D with 3 images (such as a parametric representations of a 3-D path over time) as 3 functions (one for each dimension), each with one image. Theorem 9.6.13 "The Fundamental Theorem of Calculus" is a perfect parallel to the 1-D version we already studied.
  • 9.7  arc length, Suppose a particle is moving in 2 or 3-D. We describe the path parametrically (with time as the parameter). The distance it moves is arc length.
  • Many of the results for functions of two variables are close parallels to the corresponding results for functions of one variable, but some are not close parallels. We will pass rapidly over the ones that are close parallels, and dwell on the ones that are not.
  • We do not need much of 10.1. Anyway, the results through 10.1.8 are close parallels to what we have studied. 10.1.10-10.1.13 we don't need.
  • In 10.2, we use 10.2.1, 10.2.6, 10.2.7, 10.2.8 10.2.9, and 10.2.13, which are all close parallels to things we know.  10.2.10 is a topological way of dealing with continuity, and 10.2.14 is also a topological thought, both of which we will not use.
  • Examples 10.2.2 and 10.2.4 are complicated, but illustrate the types of functions that appear in this subject. In place of 10.2.4, I will do  xy/(x2+y2 ) which is a simpler, and therefore better, example. He finds another reason to use this function in 10.3.2c, which is a major example.
  • 10.3, "Partial Derivatives," is where the action begins. Some things are complicated and long, but they are important. Be clear that 10.3 is not yet about differentiation (10.4 is "Differentiation."). 10.3 is aboutpartial differentiation, which is quite a bit simpler than differentiation. I will clarify 10.4.1 through 10.4.6 when we get there.

HW due Friday, April 10:  9.5:  1, 2, 5, 6

HW due Monday, April 13: Skim Section 10.1 (We don't study "connected").  Do 10.1, page 436, 4, 7, 8

March 30

Last year's exam on Series

  • It was from a different text, but the flavor is not far off. However, this time we will emphasize series of functions because we already had an exam on series of numbers.  See the quiz we had Monday, March 30, for questions about series of functions.

March 16

Much of the point of "Introduction to Analysis" is to teach you to be analytical. It is to teach you to notice details that may matter. Lawyers have to do this. Computer programmers have to do this. Spreedsheet analyists have to do this. It is very important in medical billing and insurance. I hope your improving ability to notice critical details will improve the rest of your life!

I hope you don't think this course is just math. I hope you don't think we are just studying theorems, the details of which are particular to them and unimportant in the big picture. We are also studying how to notice what matters (in any context) by becoming careful and analytical.

The statements and proofs of the theorems in Chapter 8 on series of functions are good examples; they help train you to think analytically.

HW due Friday, March 20: Section 8.4, page 357:  1a, b, e, h, i, j, 2b, 5,

We primarily use the Weierstrass M Test (T8.4.12). Do not worry about T8.4.13-14 now; they are too subtle. T8.4.15 is major. Compare it to Theorem 8.3.3.

HW due Monday, March 23: Section 8.4: 11 (important), 12a,b,c

HW due Wednesday, March 25: 8.4: 8, 12d,e,f (important. Do not do part (g)), 15a-d.

Comments on 8.5 and 8.6.

  • T8.5.5 is awkwardly stated. The key is the distance between the x-value for which it converges and the "a" in "x - a". That is, If it converges for x0 and |x - x0| = d, then it converges for all x such that |x - x0| < d.
  • C8.5.6 follows by contrapositive.
  • Definition 8.5.7 is similar to a theorem which says "There exists R such that ....".
  • T8.5.8b  should say (and does in recent printings) "≤ r", not "< r".
  • T8.5.9 is the old Limit Ratio Test redone in terms of the coefficients (ak used be a term; now it is a coefficient on a term).
  • T8.5.11 follows from T8.3.1.
  • T8.5.12 may be omitted now (but keep it in mind if you ever need to work with an endpoint).
  • T8.5.13 is a useful Corollary.
  • T8.5.15 is major. It is largely why we have put so much effort into series.
  • T8.5.15a:  the integral of the sum is the sum of the integrals.  [This is really about definite integrals.]
  • T8.5.15b:  the derivative of the sum is the sum of the derivatives.
  • T8.5.16 is used to prove 8.5.15 and belongs first.
  • Remark 8.5.18 follows T8.5.16 and is relevant to T8.5.15.
  • T8.5.20 I have actually used this, but it is too subtle for us here.
  • T8.6.1. My printing has a significant error at the end of page 370. It recalls Taylor's Theorem (5.4.8) and says the "remainder" is the tail of the series. No. This works for T8.6.1, but not in general. The "remainder" was the difference between the function being expanded and the polynomial approximation of it. If (and only if) that difference converges to zero the remainder equivalent to the "tail". We had one example, (page 211, Example 5.4.3, repeated as Example 8.6.2, page 371) where the power series did not converge to the function and the "remainder" of T5.4.8 is not the "tail" from page 370. T8.6.3.  Restated: When is the Taylor Series of a function equal to the function itself? When the remainder has limit zero.

HW due Friday, March 27: (8.5) 1, 2a,b,c [Do part (a) last; it is the hardest. Omit (d) for now.] 3a,b,c, 5 [typo in some printings; it is supposed to say "≤ r", not "< r"]

  • Think of T8.5.16 before T8.5.15, not after.

HW due Monday, March 30: (8.5) 6a (only part (a)), 8b, 11, 14b,c

HW due Wednesday, April 1:  (8.5) 13, 19, (8.6) 1a,b,c, 2a,b

Friday, April 3: University day, no classes

Monday, April 6: Exam on Chapter 8.

March 4

HW due Friday, March 6:  8.1, 2b through j.  [8.1 is loaded with critical examples], 3, 4

  • March 9-13, Spring Break, no classes.

HW due Monday, March 16:  Section 8.2, page 346: 1, 4, 5, 8a

HW due Wednesday, March 18: Section 8.3, page 350:  2 (Do not use T8.3.3), 3, 5 (for part c, you may consider the decreasing case instead if you prefer. I find it more natural to consider decreasing to the zero function.)  [Look at 8. This is important and discussed at length in measure theory, in Math 547.]

Feb. 18

HW due Friday, Feb. 20, revised: Homework handout you recieved in class  ( I added #9 to the pdf on-line, but you don't need to do it.). For Friday, do some of it, including at least one that isn't just an example with the harmonic or alternating harmonic, and fix something fixable.

All previously posted HWs have been postponed one class, with new dates below.

HW due Wednesday, Feb. 18:  [Conjectures numbers are from class, but do them in the reverse order.]

  • Conjecture 3:  If  ∑bn converges and  0 ≤ |an|≤ |bn| for all n, then  ∑ an converges. Conjecture 3 is false and was shown false in class.
  • Resolve Conjecture 2:  If 0 ≤ bn for all n and ∑(-1)nbn converges and 0 ≤ |an|≤ |bn| for all n, then ∑(-1)nan  converges.
  • Also, Conjecture 1:  If two series are strictly alternating and limit (an/bn) = L, 0 < L < ∞, then if one converges, so does the other.
  • All the previously posted assignments below have been postponed one day. I changed the dates below.

HW due Monday, Feb. 23:  Section 7.1, page 302, 20. Also, read Theorem 7.2.1 and its proof and determine the role of the hypothesis "continuous." Section 7.2, page 310, 5, 7, Section 7.3, page 315: 2, 7a, 9a,b,c

Due Wednesday, Feb. 25:  Section 7.4, page 322: 2a, 3a, 5 (3 series, one for each part), 7, 8.  Memorize the proof of the alternating series test (T7.4.2).  Section 7.1, #16, using tools from Sections 7.2-7.4.

Friday, Feb. 27:  Section 7.3, page 317, 11, 12.  Section 7.5 [New rules: If it is true, just say so. If it is false, give a counterexample]  4, 5, 6, 9, 12, 16, 18, 19. We have seen three version of arguments for 7.2 HW 7a. Pick one, clean it up, and hand in a revised, very nice and clear, proof.

Previous exam on Chapter 7

Monday, March 2

No HW due. Exam on Chapter 7.

Due Wednesday, March 4: Section 8.1, page 340:  1b,e,g,h  [These are more interesting than they first appear. We will be taking limits of integrals and limits of derivatives and limits of continuous functions and not always getting what you might expect. Examples like these are relevant]

Later (dates to be announced):

  • Section 8.2, page 346: (8.1) 2b,c,d,e,g,h,j, 3, 4, 5, 8a
  • Section 8.3, page 350:  2 (Do not use 8.3.4), 3, 5 (for part c, you may consider the decreasing case instead if you prefer. I find it more natural to consider decreasing to the zero function.)  [Look at 8. This is important and discussed at length in measure theory, in Math 547, Real Analysis.]

Chapter 8 comments

  • All results for sequences will be converted to results for series. The sequence results then refer to the sequence of partial sums.
  • Example 8.1.3 is major (and illustrated on page 335).
  • Example 8.1.6 is an important counterexample.
  • Definition 8.2.1 of uniform convergence is a key to the chapter.
  • Example 8.2.4 (page 342f), pictured on page 344 at the top, is a major example.
  • Theorem 8.2.8 says there is a Cauchy Criterion for uniform convergence. We will use it.
  • Theorem 8.3.1 is important:  A sequence of continuous functions that converges uniformly converges to a continuous limit.
    • There is a corollary for series.  (Theorem 8.4.3)
    • There is a way to prove series are uniformly convergent (Theorem 8.4.11 -- the Weierstrass M-test)
  • Theorem 8.3.3 is important:  The limit of the integral is the integral of the limit, if the convergence is uniform and the functions are continuous.
    • There is a corollary for series (Theorem 8.4.15, page 355)
  • Theorem 8.3.4 is a refinement that we will not use.
  • Theorem 8.3.5 is worthy of study. When is the derivative of the limit the limit of the derivatives?
    • There is a corollary for series.  (Theorems 8.4.17 and 18)
  • Theorem 8.3.6 makes 8.3.5 slightly more convenient to actually use.  It is, however, missing a hypothesis. We need the derivatives integrable.
    • There is a corollary for series.
  • Theorem 8.3.7 is a good refinement that we will not have occasion to use.
  • There is a way to prove series are uniformly convergent (Theorem 8.4.11 -- the Weierstrass M-test)
  • The M-test is a sledgehammer that works most of the time. If you need a more-subtle test, there are some, but we will not use them (Theorems 8.4.13 and 14).
  • The integral of a sum is the sum of the integrals, if the conditions are right (Theorem 8.4.15)
  • The derivative of a sum is the sum of the derivatives, if the conditions are right (Theorems 8.4.17 and 18).

 


Previous exam on Chapter 6

Monday, Feb. 9: Exam on Chapter 6. No HW due,

Due Wednesday, Feb. 11:  Section 7.2, page 309: Learn the key theorems (listed below) and do these exercises [show the test] 1a, c, h, j, n, 2

HW for Friday, Feb. 13:  Homework handout. Do the task for Friday. Feel free to stop after two hours. Please make some conjectures (easy ones) and resolve them. Working with others in the class is okay.

Monday, Feb. 16: Monday is a holiday, no classes.

HW due Wednesday, Feb. 18:  Section 7.1, page 302, 20. Also, read Theorem 7.2.1 and its proof and determine the role of the hypothesis "continuous." Section 7.2, page 310, 5, 7, Section 7.3, page 315: 2, 7a, 9a,b,c

Due Friday, Feb. 20:  Section 7.4, page 322: 2a, 3a, 5 (3 series, one for each part), 7, 8.  Memorize the proof of the alternating series test (T7.4.2).  Section 7.1, #16, using tools from Sections 7.2-7.4.

Jan. 25

Due Wednesday, Jan. 28: 

  • Newly added:  Prove: sup|f(x)| - inf|f(x)| ≤  sup f(x) - inf f(x)
  • Section 6.4,  page 263:  5, 7L, 10d,e,
  • Theorem 6.4.6 is stated with c and d where many authors would use a and b (and vice versa). Restate it with the original integral from a to b and redo the entire proof to fit that (better) statement.


Friday, Jan. 30: Prove two claims:  Claim 1: If the integral of g2 is 0, then the integral of the absolute value of g is zero. Claim 2:  If f is Riemann integrable, then f 2 is Riemann integrable.

  • Note: We can reduce the second to proving Claim 3:  If f is Riemann integrable, then sup(f 2) - inf(f 2) is at most some constant times sup(f) - inf(f).
  • Limitation:  Spend at most two hours on this. Hand in something.

We had two versions of the Fundamental Theorem of Calculus: 6.4.2 and 6.4.4b. Memorize the proof of 6.4.2 and a proof (maybe a variant of the given one) of 6.4.2b. After all, it is the Fundamental Theorem of Calculus.

Monday, Feb. 2:  Hand in: Section 6.4,  page 263:  10f, 20 (a neat, non-trivial, problem confirming work of Archimedes who died in the third century BC). Note the existence of problems 6.4: 14 and 16 about the remainder in Taylor's Theorem. (I do not ask you to do them).


Due Wednesday, Feb. 4: Section 6.5:  page 270:  2, 3, 9a, b  [Note: Definitions 6.5.1 and 6.5.4 are the key parts of Section 6.5. We will not emphasize 6.5.16-18, but 6.5.19 is interesting.]

  • Section 6.7 (Instructions:  Say if it is true [no proof required], and if it is false, give a counterexample)  page 283: 2, 3, 4, 7, 8

Due Friday, Feb. 6:  Section 7.1, page 300: 1a, c (this is telescoping; find the sum), 3a,c, 4, 5, 9 [We will do #10 with the letters a and b interchanged, in class]

Previous exam on Chapter 6

Monday, Feb. 9: Exam on Chapter 6. No HW due,

Comments:

  • Only two types of series are easy to sum explicitly, geometric series (page 297, Theorem 7.1.14, which is very important) and telescoping series (which will be covered more thoroughly in class than he does. He barely mentions it at Example 7.1.7, page 296.)
  • Example 7.1.13 is very important. We can prove it diverges two distinct ways.
  • Theorems 7.1.8-9 are just series versions of previous sequence results.
  • Corollary 7.1.21 is important (worthy of being called a theorem), and its converse is false.
  • Theorem 7.2.1 is important and has a good picture. Example 7.2.3, p-series, is important.
  • 7.2.7, the Limit Comparison Test, is important.
  • We can get along without Theorem 7.3.1. 7.3.3 is basically a comparison with the geometric, and can be proven that way without 7.3.1.
  • 7.3.5b is obvious, since the terms don't go to zero.  7.3.5a is a Corollary to 7.3.3. Do you see why?
  • Theorem 7.3.13 is much the same as Corollary 7.3.5, with "absolute convergence implies convergence" stuck on at the end.
  • 7.3.8 and 9 parts b are obvious for the same reason as above; the terms don't go to zero.
  • Theorem 7.4.2 on alternating series is very important (and the proof could be expanded. Do you see why all those assertions are true?).
  • Remark 7.4.11c is really an important theorem (actually, several theorems), which is needed to explain 7.4.16, which is amazing. [This explains why we have to be very careful about the order of the sum of an infinite number of terms. This even bears on why we don't allow partitions for integrals to have an infinite number of points.]
  • Theorem 7.4.15, which says something about rearrangements, has a non-trivial proof.
  • Theorem 7.4.7 is Corollary 7.1.21, only stated employing a new term.

Jan. 14

This course is the second semester of a standard "Analysis" or "Advanced Calculus" course. The first semester covered material through differentiation and here is a copy of the final exam.

Due Friday, Jan. 16, 2015:  Correct your final exam and hand it in with the original, and come with questions about it. (If you were not enrolled last semester and therefore did not take that final exam, look it over and hand in problems 1, 2, 3, 4, and two of 6-13.)

Also, read 6.1. Hand in Section 6.1, page 245: 2, 4.

Monday, Jan. 19 is Martin Luther King Day, a holiday with no classes.

Due Wednesday, Jan. 21:  Read 6.2. Let f(x) = x on [0, 2]. Find a partition P such that U(f, P) - L(f, P) < ε.

Also, Section 6.2, page 249:  7, 8 [pictured on page 130], 10

Also, find a Calculus text and see how it motivates the definition of "integral."  On your homework name the text and its author and make a short comment on whether the motivation is solely finding the area under a curve or if there is some additional motivation.

Remark 6.2.8 is far from trivial. T6.2.7 says there is one. R6.2.8 changes that to "all".  R6.2.8 can be useful if you already know that f is integrable. Then it tells you that any sequence of partitions with norm converging to zero will work.  We usually pick a sequence with n subintervals of equal width. By 6.2.7 alone we would not know that the equal-width-subinterval sequence would be the one that works.

Page 258, Theorem 6.4.4b (and Remark 6.4.5b) is extremely important. It is also called "The Fundamental Theorem of Calculus" (there is more than one version of the FTC). I will sketch an illuminating picture and give a proof. Be sure you can sketch the picture and give the proof. After all, this is the Fundamental Theorem of Calculus.

In 6.3, Properties of the Riemann Integral, Theorem 6.3.4 is very difficult to prove and we will not do it. Furthermore, Corollary 6.3.5 depends upon that theorem and the proof of fg being Riemann integrable (part b) is not simple, unlike all the proofs (limit, continuity, derivative) with fg that we have previously done. It really does use part (a) in a clever fashion.

Then, Theorem 6.3.8 through 6.3.10 are of less use and we will skip them.

Typos in older printings that have been fixed in recent printings (ignore if your printing is number 5 or 6):  Page 247, in section 6.2, the last line of the proof of 6.2.1 in some older printings begins "Subtracting ....".  It should read, "Since L(P,f) and L(P,f)+epsilon are within epsilon of one another, so must be the upper and lower integrals, proving the desired result."

Page 257, section 6.4, Theorem 6.4.2:  If the third last line on your page begins "If  mk ...", the last three lines on the page are correct, but you could skip them (using Lemma 6.1.3) and resume on the next page.

Comments on Chapter 6:  Chapter 6 is on the definition of the integral. Definition 6.1.1 is important, as are Lemmas 6.1.3 and 6.1.5.  Note how often "sup" and "inf" appear on pages 243 and 244. When the sup of the lower sums and the inf of the upper sums agree, the function is Riemann integrable (Def. 6.1.6).  Theorem 6.2.1 is used to prove a function is integrable. The two major and simple cases are given in Theorems 6.2.2 and 6.2.4.  Be sure you understand those two proofs. (Finally, uniform continuity does some good!)  6.3 takes some time to prove properties of integrals. T6.3.1a takes more work than you might expect. T6.3.4 is important but not easy. Corollary 6.3.5b is interesting.

We thoroughly cover 6.1, 6.2, 6.3 through 6.3.9, 6.4 (including the Fundamental Theorem of Calculus, two ways), and 6.5.

Due Friday, Jan. 23:  Section 6.3: 2, 3b, 5, 6, (9 or 10, your choice)  [They are not assigned here, but 13, 14, and 15 are important named theorems studied in Math 505.]

Due Monday, Jan. 26: Section 6.4, page 262:  Don't do #1, but explain why is "a > 0" is necessary for #1. Do 3 (this is slightly different from the theorems in the text), 4, 6c, 10a

Due Wednesday, Jan. 28: Section 6.4,  page 263:  5, 7L, 10d,e,

Theorem 6.4.6 is stated with c and d where many authors would use a and b (and vice versa). Restate it with the original integral from a to b and redo the entire proof to fit that (better) statement.

The Beginning

The course:

  • Math 384, "Introduction to Analysis II".

Class Hours:

  • 11:00, MWF, Wilson Hall, 1-139.

Instructor:

  • Prof. Warren Esty, Wilson Hall 2-238, 994-5354. westy at math dot montana dot edu

Office Hours:

  • MWF 8:40 - 9:45 and many other hours when I am in the office. (I teach at 10:00.)
  • I expect to be in the office most other non-teaching hours, too, e.g. Tuesdays and Thursdays 9:00-11:30,  Mondays 1:30-2:00, Wednesday 1:30 - 3:00 and many afternoons.
  • If you want to meet some other hour, it is probably good. Just ask in class or call (994-5354) or simply show up. I will be happy to see you.

The textbook:

A Friendly Introduction to Analysis, Single and Multivariable, second edition, by Witold Kosmala. New copies are hardcover in an unnumbered printing. Many paperback copies are in the 6th printing. If you bought a used copy of the text, it may be an older printing which needs many minor typographical changes. You can tell what printing you have on the copyright page just above the ISBN number where it says "10 9 8 7 6" or a similar string. The last number in the string is the printing number. Prof. Kosmala and I correspond and he has sent me a list of changes in the text, most of which have been corrected in the latest printing. Here are the changes for the 6th edition, the changes for the 4th and 5th edition, the changes for the 3rd edition, and the changes for the 1st and 2nd edition.

I strongly recommend you make the changes in your text now rather than be confused by the errors later.

Prerequisite

M 383 (Introduction to Analysis, I) and M 242 (Methods of Proof).

Attendance:  Attendance every day is expected. More than a couple unexcused absences is unacceptable. Of course, excuses for academic reasons, illness, participation in university sporting events, and significant life events will be accepted. Every day in class you will learn about common mistakes and how to avoid them. Students who miss a day are missing a significant lesson that cannot easily be recovered from the text alone.

If you miss a day, I will not be able to recreate the class experience for you. Find a friend who can help you catch up, read the text thoroughly, and then I will be glad to help you with specific questions.

Exams

Exam dates will be announced on this site. Three unit exams will be 100 points each and a comprehensive final will be 200 points.

The final exam is 4:00-5:50 pm, Thirsday, May 7, as in the MSU Handbook. Arrange your summer-break schedule so you can take the final at the scheduled time.

Exam times for all other classes

Grading

Your course letter grade will be based almost entirely on exams and quizzes. However, homework is mandatory and you will hear from me if it is not coming in regularly and displaying serious effort.

It is important that it be attempted on time. The work you hand in need not be all correct, but it must display serious effort. More than a few late homeworks is not acceptable. I will give you important and useful feedback on all the HW you do on time.

Homework is intended to help you learn and its impact on your grade is primarily that it serves as evidence of your attempt to learn, and, of course, it is necessary practice so you can do well on the exams. Getting a few problems wrong or incomplete will be noted, but it will not lower your letter grade if you display appropriate effort. Late homework will be accepted, but because the solutions will be discussed in class, late homework will receive less than full credit.

Homework:

Homework will be listed at the top of this page. Every homework assignment includes reading the sections thoroughly, learning the concept images, concept definitions, and theorems, and understanding the proofs.

If something on your homework is wrong, I will mark it wrong with a big X at the place where it goes wrong. Please make sure you understand why. Do not treat your homework as just part of your grade. Treat it as an occasion to learn. Anything you got wrong must be looked at again and studied much harder than anything you easily got right. Some things are easy. It is not much of an accomplishment if you can learn the easy stuff. Some things are harder. Put substantial effort into making sure you understand the harder stuff too.

Each day each significant homework problem will be written on the board by some student (self-selected) before the beginning of class. We will go over the homework in class, using the work on the board for discussion, so homework must be turned in before class (unless you are using it to write your answer on the board, in which case you hand it in when you finish copying it).

I encourage you to study and work with others currently in the class. If you have difficulties come see me.

Occasionally come a bit early and put a problem on the board (It does not have to be all right--we use them for discussion).

Etiquette

If you must miss an exam, you must inform me (Dr. Esty, 994-5354) well in advance. I prefer not to give makeup exams and I do so only for very good reasons approved well in advance.

Cell phones must be turned off during class. Headphones or other electronic devices may not be used during class. As is obvious, students must behave so that others are not distracted during class.

Students must adhere to the Student Conduct Code.

Calculators. Calculators play no role whatsoever in this class. This is not a computation class.

The Course

This "Analysis" course is the same as many junior-level courses titled "Advanced Calculus" elsewhere. In the first semester the emphasis of this course is only partly on mathematical results that are new to the students. The focus is primarily on learning the modern rigorous approach to mathematics in the context of calculus and deriving the results of calculus.

The calculus component includes sets, real numbers, sequences, limits, continuity, and differentiation.

The proof component includes proof techniques, concept definitions (as opposed to vague concept images), precisely stated theorems, conditional statements, hypotheses and conclusions, logic for mathematics, truth and falsehood, conjectures, counterexamples when statements are false, and rigorous proofs.

List of the calculus topics.

The course continues in the second semester as Math 384.

Topics

The course will proceed straight through the text, with some omissions. [If the section titles are general and vague, some of the subjects that the section titles might not suggest are given in brackets.]

Chapter 6:  Riemann Integration
6.1:  Riemann Integral
6.2:  Integrable Functions
6.3:  Properties of Integration
6.4:  The Fundamental Theorem of Calculus

Chapter 7:  Infinite Series.
7.1:  Convergence
7.2:  Tests for Convergence
7.3:  Ratio and Root Tests
7.4:  Absolute and Conditional Convergence

Chapter 8:  Sequenes and Series of Functions
8.1:  Pointwise Convergence
8.2:  Uniform Convergence
8.3:  Properties of Uniform Convergence
8.4:  Pointwise and Uniform Convergence of Series
8.5:  Power Series
8.6:  Taylor Series

Chapter 9:  Vector Calculus
9.1-9.3, 9.5-9.6:  Cartesian Coordiates in R3, Vectors in R3, Dot and Cross Products, Lines and Planes, Vector-Valued Functions

Chapter 10;  Functions of Two Variables
10.1-2:  Basic Topologuy, Limits, and Continuity
10.3:  Partial Derivatives
10.4:  Differentiation
10.5:  Directional Derivatives
10.6:  Chain Rule

Projects with multiple variables

Advice

  1. Memorize all definitions we have used multiple times (in proper left-to-right order)
  2. Memorize the precise hypotheses and statements of all theorems we have used multiple times.
  3. Learn counterexamples to false statements that resemble results we have used multiple times, but fail to be true because some critical hypothesis is missing.
  4. Learn how proofs are written by
    1. studying, in order to be able to reproduce, the types of proofs we have done multiple times. 
    2. studying, in order to be able to reproduce, the simpler proofs of major results (I do not expect you to be able to prove the most complicated theorems we studied)
    3. Noting how they "follow the logic"
  5. Learn the basic logic we have used multiple times.

Comments on proofs

Proofs are sequences of prior results from which the statement to be proven is a logical consequence. The statement to be proven usually has some letters in it, and those are given and may be used in the proof. All others letters, such as bounds, ε, and δ, must be quantified or defined within the proof.  For example, usually ε is not mentioned in the theorem itself. You must "Let ε > 0" (for a representative case proof) before you can use ε in any other sentence. If you know {f(x)} is bounded, you cannot mention M for the first time by writing  "|f(x)| ≤ M, for all x."  You must, instead, say, "There exists M such that |f(x)| ≤ M for all x" and give a reason why.

When you read theorems, pay attention to the hypotheses. Note in the proof where and why they are used. Chances are a similar result is false if any hypothesis is missing. You should be able to think of counterexamples to similar statements with a hypothesis missing. Your attitude to mathematical statements must change. You should become skeptical.

Some things you know well and are certain about. Good. Anytime something new and similar appears, you must develop the attitude that it might be false. Perhaps you can prove the new variant from your list of prior results. Then you know it is true. Or, perhaps it is false and you can prove it false with a counterexample.

You need to learn how proofs work. Read the proofs in the book. See if you can anticipate where they begin and where they end. Every day your homework includes reading and studying the proofs of the theorems in the text. I will try to list which ones are more important and which ones less important so you can use your time efficiently. Nevertheless, it will be time-consuming. Previous students agree that Introdcution to Analysis required a lot of hours of work per week to do well.

When you study a theorem, you must learn to recognize when it applies and when it does not apply.  If we change the hypotheses slightly, it is possible that the conclusion will no longer hold. Examples of similar "conjectures" are extremely valuable in helping you learn the precise meaning of a theorem.

Homework notes:  If you are to prove an "iff" statement, there will likely be two proofs, one in each direction. When you do part, clearly label which direction you are doing. [I use "=>" and  "<=" to label the two directions].

When you are resolving a conjecture, first clearly state if it is true or false. Then prove your assertion.

On your homework, please put the section number and/or page number with each problem. Your homework often has problems from several different sections, and sometimes the same problem number in two different sections. Please identify each problem clearly. You will find this handy when you look over your old homework later. You can find the problems which you answered!