MATH 4130 Honors Introduction to Analysis I

Transcription

1 MATH 430 Honors Introduction to Analysis I Professor: Robert Strichartz Spring 206 Malott Hall 207 Tuesday-Thursday 0:0 :25 Office: 563 Malott, Office Hours: Mon 0:30 :30, Wed 0:30 :30 str@math.cornell.edu Office Phone: T.A.: Sasha Patotski Office: 20B Malott, Office Hours: Mon 5:00 6:00, Wed 5:00 6:00, 28 Malott ap744@cornell.edu Text: R.Strichartz, The Way of Analysis, revised edition (2000) Course description: Introduction to the rigorous theory underlying calculus, covering the real number system and functions of one variable. Based entirely on proofs. The student is expected to know how to read and, to some extent, construct proofs before taking this course. Topics typically include construction of the real number system, properties of the real number system, continuous functions, differential and integral calculus of functions of one variable, sequences and series of functions. The course will cover the first eight chapters of the text. Grading and Exams: Class participation (25%) Written homework - weekly assignments due in class on Tuesdays (25%) Midterm - in class March 7 (20%) Final exam, to be scheduled (30%) A Cornell student s submission of work for academic credit indicates that the work is the student s own. All outside assistance should be acknowledged, and the student s academic position truthfully reported at all times... Cornell University Code of Academic Integrity. How the course will work: For each class there will an assigned reading from the text and a set of discussion questions. You will do the reading and think about the questions before the class. Most of the class time will be devoted to discussing the questions. We will also discuss questions that you bring up, and may

2 go over some homework problems. It is expected that everyone will participate in the discussion. Approximately 25% of your grade will be based on your enthusiastic participation. The goal is to get each student to understand the material through a learning process that is messy, challenging, individualized, frightening, thrilling and ultimately transcendental. Homework: You are allowed to work together with other students on HW, provided you write up the solutions on your own. Please write the names of the students you worked with on the top of your HW paper. WARNING: numbers listed below are for the 2nd edition of the textbook. If you have older edition, please, check if you are doing the right problems. A copy of the 2nd edition of the textbook is available in the Malott library. Comments on the homework: HW 2 Due date Problems HW February c, g; 2f, h, j; 3d.2.3-2, 4, 5, , 4, 5, 6, 7, 8, 9 HW2 February ,4 5, 8, , 6 HW3 March , , 2, 5, 7, 9, , 4, 5, 8, 0 HW4 March , 2, 3, 4, 5, , 3, 7, 4, , 7, 9, 0,, 7 HW5 March , 3, 6, 8, , 8, 9,, 3 HW6 April , 3,, 2, 3, 4, , 4, 5, HW7 April , 6, 9, 0, , 3, 5, 6 HW8 April , 3, 5, 6, , 3, 7, 0, 4 HW9 April , 2, 3, 4, 7, 8 HW0 May , 3, 4, 5, 8, , 2, 3, 6, HW May , 4, 9,, , 6, 7, 9, 0 Problem 2.2.4(4): you can use infinite decimal expansions if you wish; Problem 2.2.4(9): there is a typo, parentheses need to be around x 2 + y 2 ; Problem 2.3.3(6): this problem is optional. 2

3 HW There is no class on May 2. However, HW 2 is due May 2. Exams: In class Midterm: March 7. Final: May 24, at 9 am in Malott 25. These will be closed book exams. 3

4 February 2 Read.,.2,.3,.4,.5 Daily readings and questions:. Does every mathematical statement have to have quantifiers (perhaps only implicitly)? 2. Use quantifiers explicitly to formulate the statement there exist infinitely many primes. Do the same for the negation of the statement. 3. Does the order within a sequence of quantifiers of the same type matter? 4. Does an infinite subset of an uncountable set have to be uncountable? 5. How many different ways can you count the set of integers? 6. In the proof of a theorem do you have to use all the hypotheses? 7. What does the Axiom of Archimedes say for the rational numbers? Can you prove it? 8. How many representations p q are there for a fixed rational number? Is there a best choice? 9. Do you need the Axiom of Choice to make a finite number of choices? 0. To prove a or b, do you have to show explicitly which of the statements is true? February 4 Read 2.. What problem was Cauchy trying to solve with his criterion? 2. How do we know Q R? Here Q denotes the rational numbers. 3. Given a Cauchy sequence of rationals, is there a way to tell whether or not it represents a rational number? 4. In the proof of Lemma 2.., we made the tricky choice of errors so that the ultimate 2n error was + =. After the proof it is pointed out that we do not have to be tricky. 2n 2n n If the initial errors were chosen to be, then the ultimate error is 2 and this is just as n n good. Why is it just as good? Which proof do you prefer? 5. What principle allows us to consider errors of the form, n m gain by this? instead of ε, δ? What do we 6. Let x n = log n. Does lim n x n exist? Does {log n} satisfy the Cauchy criterion? What is the approximate size of log(n + ) log n? Does it go to 0? 4

5 7. What do we get if we consider the space of equivalence classes of Cauchy sequence of integers? February 9 Read 2.2. If we were to replace the formula on the bottom of p.39, with would it make any difference? x j y j x k y k = x j (y j y k ) + y k (x j x k ) 2. Are the proofs in this section merely follow your nose, or do they use surprising ideas? Which of them was the most difficult for you to understand? 3. Division by zero is forbidden in both Q and R. How does Lemma allow us to define division by a nonzero real without getting hung up by division by zero in the Cauchy sequences? 4. When we define an operation on real numbers, do you always have to check that the definition is independent of the choice of Cauchy sequence representing the number? In practice, is this ever difficult? 5. If x is represented by the Cauchy sequence {x n } and y by {y n }, what does the condition x j < y j for all j (or all sufficiently large j) tell you about x and y? 6. True or false: in a Cauchy sequence, the first million terms dont matter. 7. In what sense do the results of this section justify the definition of the real numbers in Section 2.? February Read 2.3. Numbers of the form a + b 2, where a, b are rational, form a field that contains the rationals and is contained in the reals. Could it be complete? 2. Why is it important to know that the reals are complete? 3. What is the intuitive idea behind the proof of completeness? Why can t you modify the proof to show that the rationals are complete? 4. In the proof of Theorem 2.3.2, why is it OK if some of the quotients x k y k are undefined? 5

6 5. How does the divide and conquer method in the proof Theorem produce a Cauchy sequence of rationals converging to x? Why can t you just take the Cauchy sequence x, x 2,... defining x and replace it by x, x 2,...? 6. Would you rather see an easier proof that leads to an inefficient algorithm, or a more difficult proof leading to an efficient algorithm? 7. Do you think you can prove the existence of cube roots? More generally, n-th roots? 8. A dyadic rational number is a number of the form k 2 m. What do you get if you consider all Cauchy sequences of dyadic rational numbers? February 8 Read 2.4. Is the same as ? 2. Given an infinite decimal expansion of x, can you decide whether or not x is rational? 3. How many infinite decimals represent 0? 4. What do you like better, Dedekind cuts or equivalence of classes of Cauchy sequences of rationals? Why? 5. What appeals to you more, the real number system, the nonstandard real number system, or the constructive real number system? Why? 6. How would you go from an infinite decimal expansion of x to the Dedekind cut L x? Equivalently, given a Dedekind cut, could you use it to produce an infinite decimal expansion? February 23 Read 3.. Why doesn t + + ( ) make sense? 2. Why is inf E = sup( E)? 3. The least upper bound of E is equal to sup E. Is this a theorem or a definition? 4. Are strict inequalities preserved in limits? 5. Why is the uniqueness part of Theorem 3.. obvious? 6. Give an example of a sequence that has a limit but is neither monotone increasing nor monotone decreasing. 6

7 7. Why is it true that the sentence immediately following Definition 3.. is equivalent to the definition? 8. What are the limit points of the sequence {,, 2,, 2, 3,, 2, 3, 4,... }? 9. In what sense is a lim sup a limit of sup s? 0. Write y k = sup x j. How is inf y k related to lim y k? You might first ask: what kind of j>k sequence is {y k }?. For a sequence {x j }, is it true that every value of t satisfying lim inf x j t lim sup x j must be a limit point of {x j }? February 25 Read 3.2. What is the relationship between (0, ) (, 2) and (0, 2)? 2. Is the intersection of a finite number of open intervals equal to an open interval? 3. Why don t we just define a closed set to be the complement of an open set? 4. Give an example of a set that is neither open nor closed. 5. Can you describe the complement of the Cantor set? 6. Is it true that the union of all open sets contained in A is equal to the interior of A? How does this relate to the statement that the intersection of all closed sets containing A is equal to the closure of A? 7. Is it always true that A Q is dense in A? (Remember that Q denotes the rationals.) 8. Is the empty set closed? 9. Why is a finite set always closed? 0. If you remove a finite set of points from an open set is the result still open? What about adding a finite set of points to a closed set?. Denote the complement of a set B by B. What is (interior(a ))? What is (closure(a ))? March Read 3.3 Note that this is a very challenging section. There are essentially five proofs: (a), (b), (c) relatively easy (a) compact closed and bounded; 7

8 (b) HB closed and bounded; (c) Theorem (nested sequence property); and (d),(e) relatively hard (d) closed and bounded compact; (e) compact HB [Here HB means Heine Borel property: every open covering has a finite subcovering.] Please try to learn some of the proofs well enough to present them at the blackboard. I will ask for volunteers to do this.. Is the empty set compact? 2. In Definition 3.3., why is it required that the limit points belong to A? What would happen if we drop this requirement? 3. The first part of the proof of Theorem reduces to the case of countable open covers. Does this argument uses the compactness of A? 4. Before the proof of Theorem it is shown that HB closed and bounded. It is also possible to show that HB compact. Can you fill the details in the following argument: Suppose A satisfies HB, and suppose x, x 2,... is a sequence of points in A with no limit point in A. We will show that this is impossible. For each a A, there exists n (depending on a) such that the interval (a n, a + n ) contains only a finite number of x, x 2,.... Call I a the interval (a n, a + n ). Then the sets I a (as a varies in A) form an open cover of A. By HB there is a finite subcover, say I a,..., I am, so A I a I am. How many of the x, x 2,... can belong to A? March 3 Read 4.. What can you say about the intersection of the graph of a function with a vertical line? a horizontal line? Does it matter whether or not the function is continuous? 2. Is an implicit description of a function unique? (i.e. can there be more than one?) 3. Is the composition of continuous functions necessarily continuous? 4. What is uniform about uniform continuity? 5. Is f(x) = x 2 with domain [, ] uniformly continuous? What if the domain is R? 8

9 6. Informally, continuity means that you can control the error in the output ( f(x) ) by controlling the error in the input ( x ). Can you use this informal description to distinguish between continuity at a point, continuity on the whole domain, and uniform continuity? 7. Explain why, for continuous functions, it is the image of a convergent sequence that converges, while it is the inverse image of an open set that is open. 8. Give an informal description of the Lipschitz condition in terms of stretching. 9. Why do we want to exclude the value f(x 0 ) in the definition of lim x x0 f(x)? March 8 Read 4.2. How do you express min(f, g) in terms of max and minus signs? 2. There is an easier proof of Theorem Consider separately the cases f(x 0 ) = g(x 0 ) and f(x 0 ) > g(x 0 ). Try to give the argument for each case. In the second case, ask yourself if f(x) > g(x) for x near x Why do you think the function shown in Fig is Lipschitz? What Lipschitz constant would work? 4. Does the proof of the Intermediate Value Theorem give an effective way to find x? 5. Do the proofs of Theorems 4.2.3, and use the closed and bounded equivalent condition for compactness? 6. If f is continuous, must the inverse image of a compact set under f be compact? 7. Combine Theorem and the Intermediate Value Theorem to show that a continuous function maps a closed bounded interval to a closed bounded interval. 8. For a monotone function, does f(x 0 ) have to lie between the two -sided limits? 9. How does the last line of the proof of Cor yield the conclusion? March 0 Read 5.. In Definition 5.. on the top line of p.45 it says x x 0. Nevertheless f(x 0 ) must be defined in order for f (x 0 ) to be defined. Is there a contradiction here? 2. Of the two formulas on the top of p.45, which one do you like better? Why? 3. Could you define one-sided derivatives, analogous to one-sided limits? 9

10 4. Can a tangent line be vertical? Can a vertical line be tangent to the graph of a function? Are these the same or different questions? 5. Can you express the Lipschitz condition in terms of big Oh and little oh notation? 6. Does the formula on the bottom of p.50 make sense for x = 0? Why, or why not? 7. Could you define a tangent line as a line that stays on one side of a graph? 8. Is our definition of f (x 0 ) any different from the definition in a standard calculus book? 9. If you create a function h(x) by gluing together differentiable functions { f(x), a x b h(x) = g(x), b x c what conditions do you need for h to be differentiable at b? March 5 Read 5.2. Is the condition f is monotone increasing at every point in the interval (a, b) the same as f is monotone increasing on the interval (a, b)? What about if you replace monotone increasing by strictly increasing in both statements? 2. Does monotone or strictly increasing imply continuity? 3. Is there a proof theme in this section: do 0 first? 4. Is Fermat s method genuinely different from solving f (x) = 0? 5. Why is Theorem a global result, while Theorem 5.2. is a local result? 6. How does Theorem 5.2.2(a) give us a stronger result than anything in Theorem 5.2.? 7. Does the MVT require one-sided derivatives to exist at the endpoints? 8. If the hypotheses of MVT hold on an interval [a, b], does it follow that they hold on any subinterval [c, d] [a, b]? What would the conclusion of MVT on all subintervals say? 9. Is the point x 0 whose existence is asserted in the conclusion of the MVT necessarily unique? March 7 In class prelim. Closed book. Covers materials through section 5.2. March 22 Read 5.3 0

11 . In the proof of Theorem 5.3. for quotients, we use the fact that differentiability implies continuity. Do we need to use this fact to prove the result for sums? For products? 2. In the chain rule, why are f and g are evaluated at different points? 3. Give an intuitive argument for the chain rule based on the intuitive idea that the derivative of a function is a multiplicative factor for differences. 4. Is the composition of two Lipschitz functions necessarily Lipschitz? What can you say about the Lipschitz constants? 5. What is the relationship between the two identities f f = I D and f f = I R? 6. If you know that the Local Inverse Function Theorem was true, could you use it to easily obtain the Inverse Function Theorem? 7. What is the relationship between the chain rule for f (f(x)) = x and the Inverse Function theorem? 8. Why is switching the x-axis and y-axis equivalent to reflecting in the line y = x? 9. Can f exist if f (x) = 0 for some points x? March 24 Read 5.4. Is Theorem 5.4. parts (c) and (d) the same as the second derivative test in calculus? f(x+h) 2f(x)+f(x h) 2. What is lim? h 0 h 2 3. Why is o( x x 0 2 ) a stronger condition than o( x x 0 ) but a weaker condition than o( x x 0 3 )? 4. What does Taylor s theorem tell you if f is C 4, f (x 0 ) = f (x 0 ) = f (x 0 ) = 0 and f (4) (x 0 ) > 0? 5. There is a typo in the middle of p.87 in the formula for f(x). The middle term is incorrect. g(x) What should it be? { sin x x 6. Let f(x) =, if x 0, if x = 0. What is f (0)? 7. The Lagrange Remainder Formula is an example of the cliché assume more, get more. What more are you assuming? What more are you getting? 8. If f is C k+, what is the relationship between T k+ (x 0, x) and T k (x 0, x)? 9. If f is C k and x 0, x are distinct points, what is the relationship between T k (x 0, x) and T k (x, x).

12 April 5 Read 6.. In defining the integral, do the subintervals have to have equal lengths? Do the lengths have to go to zero? 2. Why do we insist on considering all partitions with maximum length going to zero? Would it suffice to know the limit exists for one particular sequence of partitions? 3. How does allowing arbitrary evaluation points in subintervals pay off in the proof of the Integration of the Derivative Theorem? 4. Why is there a factor of (n+)! in the Lagrange Remainder Formula and a factor of n! in the Integral Remainder Formula? 5. Think of the integration by parts formula as giving some information about g in terms of expressions involving only g, for different choices of f. What kind of information is this? 6. What does the change of variable formula say for f? 7. In the change of variable formula, what do the Cauchy sums for each integral look like? Can you relate them? April 7 Read 6.2, 6.3. There is a factor of 3 in Lemma Where does it come from? Why doesn t it matter? 2. In Theorem 6.2. part (c), what is the relationship between the values of the inf = sup and the integral? 3. If you attempted to graph Dirichlet s function, what would the graph look like? 4. What is the analogy between Theorem 6.2. parts (a) and (b) and the Cauchy criterion? 5. The integral of a product exists, but there is no formula for it. How would you explain this to a calculus class? Is the same true for f and max(f, g) in Theorem 6.2.2? 6. There is a typo in line 5 page 229. What is it? 7. Do the numerical integration formulas in Section 6..4 work for the Riemann integral? 8. If you change the values of a function at a finite number of points, does that change the integral? What about changing values at a countable number of points? 9. For which values of a does the improper integral 0 x a +x 2 dx exist? 2

13 April 2 Read 7.. The complex number system is complete in two different senses. What are they, and how are they different? 2. Can you give a geometric proof of the triangle inequality? 3. What are all possible values of z + z 2 if z = r and z 2 = r 2? 4. Give a geometric interpretation of the transformation z iz. 5. How would you relate C and the two-dimensional real vector space R 2? 6. How can you tell the difference between the complex numbers i and i? 7. The Mean Value Theorem does not hold for complex valued functions. What about the integration of the derivative theorem? Is there a moral to this story? 8. Do you think that F integrable implies F integrable? 9. In what way is the inequality in Theorem 7.. related to the triangle inequality? April 4 Read 7.2. What is the sum of the geometric series r k if we start the sum at k = 0 rather than k =? k=0 2. For an absolutely convergent series, is there any relationship between x n and x n? 3. If you want to use the comparison test to prove convergence of a given series x n, is there any strategy to pick the values of y n? 4. On p.255, why is it sufficient to understand the behavior of the partial sums k = 2 m? 5. To study the convergence of n= directly, but in the end we did end up using it. Explain. n a n= n= n= k b n for n= we did not use comparison with a geometric series 6. Absolute convergence is equivalent to unconditional convergence. Which property do you think is more useful? Which property do you think is easier to verify? 3

14 7. Why is the estimate at the bottom of p. 257 true? 8. In Theorem 7.2.5, is part a. a special case of part b.? 9. The integration by parts formula has boundary terms. What are the analogous terms in the summation by parts formula? April 9 Read 7.3 Corrections for p.273 and 274: Conversely, assume f n (x n ) f(x) if x n x. First, we claim that f is continuous. That means that we have to show f(x n ) f(x), or equivalently, f(x n ) f(x) 0. But look at the double comparison f(x n ) f(x) = [f(x n ) f m (x n )] + [f m (x n ) f(x)] where m depends on n as follows. Given an error we have f(x k n) f m (x n ) for all sufficiently large m, because f 2k m f at x n. Choose one such m, call it m(n). We need to show that f m(n) f(x) for all sufficiently 2k large n. Define a new sequence of points y j as follows: { x n if j = m(n) y j = x if j m(n) for any n (we can easily arrange that all m(n) are distinct). Clearly y j x so f j (y j ) f(x). But {f m(n) (x n )} is a subsequence of {f j (y j )} (corresponding to j = m(n)). So f m(n) f(x) for 2k all sufficiently large m. Thus f is continuous. [A curious observation is that this part of the argument did not use the continuity of the functions f n.] To show f n converges uniformly to f we consider the sets A m,n = {x D such that f n (x) f(x) m } The condition for uniform convergence that we want to prove is that for all m there exists N such that A m,n = D. Ordinary pointwise convergence would say that for all x and all m there exist N such that x is in A m,n. But we are assuming more than pointwise convergence, and we claim the following stronger conclusion: for each x in D and every m, there exists N such that A m,n contains a neighborhood of x. We prove this claim by contradiction. If it were not true, then there would exist a fixed x and m and a sequence of points {x N } converging to x such that x N is not in A m,n. But x n not in A m,n means there exists k(n) N such that f k(n) (x N ) f(x N ) > /m. Since f is continuous we can make f(x N ) f(x) /2m for N large, so f k(n) (x N ) f(x) > /2m by the triangle inequality. By inserting some terms equal to x into the sequence {x n } we can obtain a new sequence {y k } converging to x such that f k (y k ) f(x) > /2m for infinitely many k. Specifically, we take y k = x N if k = k(n), choosing the smallest N if there is more than one choice and y k = x otherwise. But this contradicts the hypothesis lim k f k (y k ) = f(x), proving the claim. 4

15 . In Theorem 7.3., it is observed that the Cauchy criterion for the sequence of functions implies the Cauchy criterion for the sequence of values at any given point x. Is it true that the Cauchy criterion for {f n (x)} for every x implies the Cauchy criterion for the sequence of functions? 2. Suppose f n 0 uniformly. What can you say about the graphs of the functions f n (in terms of containment in rectangles)? 3. Can a uniform limit of discontinuous functions be continuous? b 4. In Theorem 7.3.3, could we conclude that lim f n (x) f(x) dx = 0? n a 5. Which part of the Fundamental Theorem of Calculus is used in the proof of Theorem 7.3.4? 6. Suppose the functions f n are assumed to be in C 2 [remember this means twice continuously differentiable], with f n f pointwise. What would we need to assume in order to conclude that f is C 2? 7. In the proof of Theorem in the top of p.273 there are two displayed inequalities. Why is the second one useful while the first one is not? Is the first inequality still true? 8. In the proof of the converse portion of Theorem 7.3.5, what are we assuming that is stronger than pointwise convergence? In particular, why does it imply pointwise convergence? April 2 Read 7.4., Can you interpret the geometric series as a special case of power series? 2. Does the radius of convergence of a power series depend on the first N terms? 3. Suppose you multiply a power series by a polynomial. What do you get? 4. Suppose you add two power series about a point x 0. Is the result a power series, and what can you say about the radius of convergence? 5. Compare the properties of power series and Taylor expansions. In what ways are they similar? In what ways are they different? 6. The formula for b k pn p.282 only depends on a n for n k. Why? 7. Is the formula on p.284 for the power series of x about x 0 meaningful for x 0 >? 8. What is the power series for +x 2 about x = 0? What is its radius of convergence? 9. Where is f(x) = x analytic? 5

16 April 26 Read 7.4.3, 7.4.4, Using the distance to the nearest complex singularity method, what should the radius of convergence of +x 2 about x 0 be? 2. Why is the formula on the top of p.288 true? Why is it < ( + x 2 0) n? 3. What should the power series of +x 2 about x 0 be, using the derivatives formula? 4. What should the radius of convergence of f(x) = p(x) about x = x 0 be, where p(x) is a polynomial? 5. Why is the power series for g(x) a special case of Theorem 7.4.5? 6. Can you use the chain rule to determine the power series of g f? 7. Is the polynomial of degree n that interpolates n data points unique? 8. q k and Q k are polynomials with only real roots. Is the same true for the Lagrange interpolation polynomial P? April 28 Read 7.5.2, 7.5.3, and the revised version of p.300 (see below). Note: the results on p.300 in the book are correct, but the definition of approximate identity given is too restrictive, and is not strong enough for the application in

17 The following is the correct version of page 300. All the corrections are colored in red. because I have been vague as to what is meant by the condition that g n gets more concentrated near zero as n. In fact, depending upon the context, there are several ways of making this precise. Here is one: suppose g m 0 satisfies g m (y)dy = and /n as m for all n. Then f g m f(x) = /n + /n + /n g m(y)dy 0 (f(x y) f(x))g m (y)dy /n + /n f(x y) f(x) g m (y)dy f(x y) f(x) g m (y)dy. We are assuming f is continuous and vanishes outside a bounded interval. This implies that f is bounded and uniformly continuous. Thus f(x) M for all x, and given any error /k there exists /n such that f(x y) f(x) < /k for all y /n. Substituting these estimates into the integrals (using f(x y) f(x) f(x y) + f(x) 2M in the y > /n integral) we obtain f g m (x) f(x) 2M g m (y)dy + y /n k 2M g m (y)dy + y /n k /n /n since /n g /n m(y)dy g m(y)dy =. But we are assuming g y /n m(y)dy 0 as m (this is our concentrating hypothesis), so we can make 2M g y /n m(y)dy /k by taking m large enough and, hence, f g(x) f(x) 2/k if m is large enough. We can summarize this result as follows: Definition 7.5. A sequence of continuous functions on the line {g m } satisfying. g m (x) 0, 2. g m(x)dx =, 3. lim g m x /n m(x)dx = 0 for all n g m (y)dy 7

18 . In the first displayed equation on p.299, how is the fair averaging property used? 2. In the corrected p. 300 there are two parameters named n and m. Which one goes to infinity first? 3. If f is increasing and g is one of the g m from an approximate identity, is f g increasing? 4. Can you give an intuitive explanation why f g is a polynomial if g is a polynomial? 5. What is the intuitive idea behind the proof of the approximate identity lemma? 6. Is the proof of the Weierstrass Approximation Theorem constructive? 7. In the displayed equation at the bottom of p. 304, what does the constant c depend on? 8. If f is C and you want to approximate both f and f, can you use the same polynomials as in the Weierstrass Approximation Theorem? 9. If f is C k and g is C m, how smooth is f g? 0. If f is defined and continuous on [a, b] [c, d] for b < c, can a single sequence of polynomials approximate f on both intervals? May 3 Read 7.6. How would you formulate a notion of equicontinuity for a sequence of functions f n at a single point x 0? 2. In the concept of uniform equicontinuity, what does the uniform refer to? What does the equi- refer to? 3. For a sequence {x n } of real numbers, what condition implies the existence of a convergent subsequence? 4. In the infinite matrix on p.32, how are the rows related to each other? What about the columns? 5. Where in the proof of the Arzela Ascoli Theorem is the uniform equicontinuity hypothesis used? 6. We say that a sequence of functions {f n } is equi-lipschitz if there exists a constant M such that f n (x) f n (y) M x y for all x, y and n. How does this concept relate to uniform equiconinuity? How does it relate to the uniform boundedness of the derivatives of {f n }? 8

19 May 5 Read 8.. There are at least two formulas for e. what are they? 2. Can you deduce e x > 0 for x < 0 from the power series? 3. What are all possible solutions to the differential equation f = f? 4. We have a power series for e x about x = 0 and log( + x) about x = 0. What do you get if you substitute both power series into log( + (e x ))? 5. If we were to take formula in Theorem 8..5 as the definition of log x, what would that tell us about d (log x)? About the derivative of the inverse function? dx 6. Why should you expect N k= k log N? 7. Compare regularization with the Weierstrass approximation theorem. 8. If f and g are C functions with the same Taylor polynomials about x 0 for all orders, does this imply f = g in a neighborhood of x 0? 9. In what sense is the infinite series defining f at the beginning of the proof of the Borel Theorem 8..7 really a finite sum? May 0 Read 8.2. Would the formula π = 2 arcsin be a reasonable way to compute π using numerical integration? 2. The function arcsin x is not differentiable at x =. Can you relate this to properties of sin x via the inverse function theorem? 3. Can you deduce C(x) 2 + S(x) 2 = from the power series on p.343? 4. What are all solutions to the differential equation f = f? 5. Why should we believe that exp(z + z 2 ) = exp(z ) exp(z 2 ) for complex numbers z, z 2? 6. What is 0 dt? +t 2 7. Why is tan θ periodic of period π while sin θ and cos θ are periodic of period 2π? 8. If you were to graph the partial sums of the power series for C(x) and S(x), would you be led to conjecture that C(x + 2π) = C(x) and S(x + 2π) = S(x)? 9. What differential equation does f(x) = tan x satisfy? 0. What does the graph of arctan x look like? 9