This is a (very) brief summary of what was (or will be) covered in each class.
December 17
Lecture overview: Showed the construction of the middle thirds Cantor set: although its "length" is zero, it can be put into a one-to-one correspondence with the entire unit interval; defined the Cantor function and proved that it is increasing and continuous and maps the Cantor set onto the interval [0,1]; its derivative is 0 almost everywhere, but it does not satisfy the Fundamental Theorem of Calculus; it is known as a "devil's staircase." Constructed an example of a function that is continuous everywhere, but differentiable nowhere on R. Constructed an example of a "space filling curve," i.e. a function from [0,1] to [0,1]x[0,1] which is continuous and onto.
The lecture material is contained in Section 9.4 and Appendix E of your text.
Quiz 5 was returned and solutions were distributed.
Problem Set 6 was collected.
December 10
Lecture Overview: Defined infinite series and convergence in terms of the sequence of partial sums; derived formula for the sum of a geometric series; proved the Integral Test and used it to determine convergence and divergence of p-series; proved the Ratio Test; stated the Root Test and the Alternating Series Test. Defined series of functions and related notions of absolute and uniform convergence; proved the Weierstrass M-Test for uniform convergence. Defined power series and radius and interval of convergence. Proved that a power series is uniformly convergent on any closed and bounded interval strictly inside its interval of convergence; this allows us to integrate and differentiate a power series term by term. Defined Taylor series; derived the Taylor series for sin x and proved that it converges to sin x for all x in R; Derived a series for sin(x2) and used it to obtain an infinite series expression for the integral of sin(x2) on the interval [0,1]; since this is an alternating series, we can easily estimate the error incurred by truncating the series at any finite term.
The lecture material is contained in Sections 9.1-9.4 of your text.
Problem Set 5 was returned and solutions to selected problems were distributed.
Quiz 5 was given during the first 20 minutes of class.
Problem Set 6 is due on Wednesday, December 17. See Problem Sets for more information.
December 3
Lecture overview: Proved the second half of Lebesgue's theorem: a bounded function that is continuous almost everywhere is Riemann integrable. Proved both versions of the Fundamental Theorem of Calculus; proved the substitution rule for integrals. Defined pointwise convergence of functions on a subset of R and looked at several examples, including an example in which the limiting function is not continuous; this motivates us to define uniform convergence of a sequence of functions; contrasted this definition with that of pointwise convergence and looked at several examples. Proved that the uniform limit of continuous functions is continuous; however, the converse is false: if the limit of continuous functions is continuous, this does not imply that the convergence is uniform. Proved that if a sequence of functions converges pointwise and the sequence of derivatives converges uniformly on a closed, bounded interval, then the limit of the derivatives is the derivative of the limit. Showed an example of how the limit of integrals can fail to be the integral of the limit. Proved that if a sequence of integrable functions converges uniformly to a function f on a closed, bounded interval, then f is integrable and the limit of the integrals is the integral of the limit.
The lecture material is contained in Sections 7.3, 8.1 and 8.2 of your text.
Quiz 4 was returned and solutions were distributed.
Problem Set 5 was collected and Problem Set 6 was assigned. Problem Set 6 is due Wednesday, December 17. See Problem Sets for more information.
Quiz 5 will be given during the first 15 minutes of class on Wednesday, December 10. See Announcements for more information.
Please review your notes and look over Sections 9.1 and 9.2 for next week.
November 26
No class - Happy Thanksgiving!
Problem Set 5 is due Monday, December 3. See Problem Sets for more information.
Please review your notes and look over Section 3 of Chapter 7 for next week.
November 19
Lecture overview: Defined Riemann integral using the notion of tagged partitions, which is somewhat more general than the usual definition presented in Calculus texts; calculated from the definition the integral of a discontinuous, but piecewise constant function. Proved that a Riemann integrable function must be bounded; stated several important properties of the integral: the integral is linear, it preserves order, and it is additive. Defined step function. Proved a Cauchy criterion for the existence of an integral. Proved Riemann's criterion for integrability on the way to stating Lebesgue's criterion for the Riemann integrability of a function: that a bounded function is Riemann integrable if and only if it is continuous almost everywhere. As a consequence of the Lebesgue criterion, we have immediately that all continuous functions on bounded domains are Riemann integrable as well as functions which have at most a countable number of discontinuities, such as monotone functions. However, functions that are discontinuous almost everywhere are not Riemann integrable; the Lebesgue integral is a more general integral which can integrate some functions with an uncountable number of discontinuities. Proved the first half of Lebesgue's theorem, that Riemann integrability implies the function is continuous almost everywhere.
The lecture material is contained in Sections 7.1, 7.2 and Appendix C of your text.
Problem Set 4 was returned. Solutions to selected problems were distributed.
Quiz 4 was given during the first 20 minutes of class.
Problem Set 5 is due Monday, December 3. See Problem Sets for more information.
Please review your notes and look over Section 3 of Chapter 7 for next week.
November 12
Lecture overview: Stated and proved Cauchy's Mean Value Theorem, which is a generalization of the Mean Value Theorem. Stated two forms of L'Hospital's Rules and proved them using the Cauchy Mean Value Theorem. Defined Taylor polynomials, and derived the form of the coefficients if they exist; proved Taylor's Theorem by computing the error term using the Mean Value Theorem; worked out an example. Derived Newton's method for finding the roots of a function and proved its convergence; used the method to approximate the cube root of 2 to 14 decimal places.
The lecture material is contained in Sections 6.3 and 6.4 of your text.
Problem Set 4 was collected and Problem Set 5 was assigned. See Problem Sets for more information.
Quiz 3 was returned and solutions were distributed.
Quiz 4 will be given during the first 15 minutes of class on Wednesday, November 19. See Announcements for more information.
Please look over the first two sections of Chapter 7 for next week.
November 5
Lecture overview: Defined the derivative of a function at a point; discussed interpretations as an instantaneous rate of change and the slope of the tangent line. Proved that differentiablility implies continuity and proved the chain rule; reviewed familiar properties of the derivative such as linearity and the product and quotient rules; showed an example of determining whether a derivative exists at a point when the derivative is not continuous at that point. Proved Fermat's Theorem: if f has a relative extremum at x=c then either f '(c) does not exist or f '(c)=0. Proved Rolle's Theorem. Proved the Mean Value Theorem and several of its applications to locating local minimum and maximum values of a function on an interval; linked increasing to having a positive derivative and decreasing to having a negative derivative. Proved Darboux's Theorem, which is a type of IVT for f ' without assuming that the derivative is continuous.
The lecture material is contained in Sections 6.1 and 6.2 of your text.
Quiz 3 was given during the first 15 minutes of class.
Problem Set 3 was returned and solutions to selected problems were distributed. Problem Set 4 is due on Wednesday, November 12. See Problem Sets for more information.
Please review your notes and look over Sections 3 and 4 of Chapter 6 for next week.
October 29
Lecture overview: Recalled the definition of uniform continuity and calculated two examples from the definition. Proved that a continuous function on a closed interval is always uniformly continuous; defined Lipschitz function and proved all Lipschitz functions are uniformly continuous; the converse is false: there are uniformly continuous functions which are not Lipschitz continuous. Proved that the image of a Cauchy sequence under a uniformly continuous function is also a Cauchy sequence; proved a continuous function is uniformly continuous on an open interval if and only if it can be extended to a continuous function on the closure of the interval. Defined monotonic function and proved that for a monotonic function, one-sided limits exist at each point in the domain; proved that a monotonic function has at most countably many discontinuities; discussed inverses of monotonic functions and proved that a continuous, monotonic function has a continuuous, monotonic inverse; used this to define nth roots: the inverses of power functions to an integer power; from there, we can define rational and real power functions.
The lecture material is contained in Sections 5.4 and 5.6 of your text.
Quiz 3 will be given in class on Wednesday, November 5. See Announcements for more information.
Problem Set 3 was collected.
Problem Set 4 was handed out and is due on Wednesday, November 12. See Problem Sets for more information.
Please review your notes and look over Sections 6.1 and 6.2 for next week.
October 22
Lecture overview: In some cases, when a function fails to exist at a point, we can define its value to make it continuous at that point; these are called removable discontinuities. Proved that the composition of continuous functions is continuous; this gives us many new ways to define continuous functions, including using square roots and absolute values. Proved that a continuous function on a closed, bounded interval is bounded. Defined absolute maximum and minimum and proved the Extreme Value Theorem; the proof of this theorem relies on the Completeness Axiom. Proved the Location of Roots Theorem; the proof gives us a practical algorithm to compute a root of a function which converges exponentially fast to the root; proved the Intermediate Value Theorem; proved that the image of an interval under a continuous function is always an interval and that the image of a closed and bounded interval is a closed and bounded interval. Defined uniform continuity and worked out an example of uniformly and nonuniformly continuous functions.
The lecture material is contained in Sections 5.2, 5.3 and 5.4 of your text.
Quiz 2 was returned and solutions were distributed. See me if you have questions about the quiz.
Problem Set 3 is due on Wednesday, October 29. See Problem Sets for more information.
Please review your notes and look over Sections 5.4 and 5.6 for next week.
October 15
Lecture overview: Defined one-sided limits and looked at several examples of convergence and divergence. Defined infinite limits and limits at infinity: when they exist, these correspond to vertical and horizontal asymptotes of a function, respectively; worked out several examples in detail. Defined continuity of a function at a point and gave equivalent characterization in terms of limits and sequences when the point is a cluster point of the domain; proved that sums, differences, products and quotients of continuous functions are continuous using the standard limit laws and the connection to sequences. Gave several examples of functions, including one which is not continuous at any point of R (Dirichlet's function) and one that is continuous at every irrational number and discontinuous at every rational number (Thomae's function).
The lecture material is contained in Sections 4.3, 5.1 and 5.2 of your text.
Problem Set 2 was returned and solutions to most problems were distributed. Look over the solutions and see me with questions.
Problem Set 3 is due on Wednesday, October 29. See Problem Sets for more information.
Please review your notes and look over Sections 2 and 3 of Chapter 5 for next week.
October 8
Lecture overview: Defined a cluster point of a set and discussed some examples; proved equivalent characterization of cluster point using sequences; defined the limit of a function at a point and proved a function has at most one limit at a point; used epsilon-delta definition to prove the convergence of several limits; proved an equivalence relating the limit of functions to the limit of sequences; this enables us to use many results regarding the limits of sequences when working with the limits of functions; in particular, the algebraic limit laws all transfer verbatim to the limits of functions; computed several examples; also, proved that the Squeeze Theorem for functions follows from the Squeeze Theorem for sequences. Stated an equivalent characterization of divergence of a limit of a function and the existence of a divergent sequence of function values and discussed two examples.
The lecture material is contained in Sections 4.1 and 4.2 of your text.
Problem Set 2 was collected in class. Problem Set 3 was assigned and is due on Wednesday, October 29. See Problem Sets for more information.
Quiz 1 was returned and solutions were distributed. Please come see me if you would like to discuss the quiz.
Quiz 2 will be given on Wednesday, October 15. See Announcements for more information.
Please review your notes and look over Sections 4.3 and 5.1 for next week.
October 1
Lecture overview: Introduced subsequences and proved that every subsequence of a convergent sequence converges to the limit of the sequence; the converse statement also holds: if all subsequences of a sequence converge to a common limit, then the sequence is convergent; this leads to the divergence criteria: every divergent sequence is either unbounded or has two subsequences converging to different limits. Proved that every sequence has a monotone subsequence; proved the Bolzano-Weierstrass Theorem: Every bounded sequence has a convergent (monotonic) subsequence. Defined properly divergent sequence and limits of + or - infinity. Defined the Cauchy Criterion and proved that a sequence is convergent if and only if it is Cauchy; did an extended proof of the convergence of a recursive sequence of weighted averages by showing the sequence is Cauchy.
The lecture material is contained in Sections 3.4 and 3.5 of your text.
Problem Set 1 was returned and solutions to selected problems were distributed.
Problem Set 2 is due on Monday, October 8. See Problem Sets for more information.
Quiz 1 was given during the first 20 minutes of class.
Please review your notes and look over Sections 1 and 2 of Chapter 4 for our next meeting.
September 24
Lecture overview: Showed every convergent sequence is bounded; stated the usual limit properties: limits respect the algebraic properties of the reals; proved that the limit of the sum of convergent sequences is the sum of the limits. Used the fact that a convergent sequence is bounded to prove that the limit of the product of convergent sequences is the product of the limits. Proved that limits respect the order properties of the reals and proved the Squeeze Theorem. Proved second form of triangle inequality, which is very useful. Defined monotone sequence and proved the Monotone Convergence Theorem. Did several examples of convergence using the Monotone Convergence Theorem including the calculation of square roots using a recursive sequence and the limit which defines the number e.
The lecture material is contained in Sections 3.2 and 3.3 of your text.
Problem Set 1 was collected in class. Problem Set 2 was assigned and is due on Wednesday, October 8. See Problem Sets for more information.
Quiz 1 will be given during the first 15 minutes of class on Wednesday, October 1. See Announcements for more information.
Please review your notes and look over Sections 4 and 5 of Chapter 3 for our next meeting.
September 17
Lecture overview: Defined what it means for a set to be countable; proved that the integers and rational numbers are countable; proved the Nested Intervals Property and discussed its equivalence with the Completeness Axiom; the last problem on Problem Set 1 is to show that the Nested Intervals Property implies the Completeness Axiom; used the Nested Intervals Property to show that the real and irrational numbers are uncountable; showed that both rational and irrational numbers are dense in R; defined the set of algebraic numbers, which contains the rationals plus all possible finite combinations of roots of rational numbers - our first examples of irrational numbers; the fact that the set of algebraic numbers is countable means that the "majority" of the real numbers lie outside this set. Introduced sequences of real numbers and defined what it means for a sequence to converge or diverge; proved that limits are unique; practiced proving convergence to a limit from the definition in several examples; proved a useful lemma to prove convergence by bounding the difference |xn - x|.
The lecture material is contained in Sections 2.5 and 3.1 of your text. The material on countable sets is contained in Section 1.3.
Problem Set 1 is due on Wednesday, September 24. See Problem Sets for more information.
Please review your notes and look over Sections 2 and 3 of Chapter 3 for next week.
September 10
Lecture overview: Reviewed some basic properties of the absolute value and proved the Triangle Inequality; introduced the picture of the real number line and the notions of distance and neighborhood. Defined upper and lower bounds of a set and discussed several examples; defined supremum (least upper bound) and infimum (greatest lower bound) of a set and proved several properties. Stated the Completeness Property of the real numbers and proved that it implies the Archimedean Property: the Natural Numbers are unbounded in R. Proved several consequences of the Archimedan Property. Proved that the square root of 2 exists as a real number and that it is irrational.
The lecture material is contained in Sections 2.2-2.4 of your text.
Problem Set 1 is due on Wednesday, Setpember 24. See Problem Sets for more information.
Please review your notes and look over Sections 2.5 and 3.1 for next week.
September 3
Lecture overview:
The lecture material about properties of the real numbers is contained in Sections 2.1 and 2.2 of your text.
Handed out a list of the Algebraic Properties of R and an example of a field with only 4 elements.
Problem Set 1 was assigned and is due on Wednesday, September 24. See Problem Sets for more information.
Please review your notes and read Sections 2.2 and 2.3 of your text for our next meeting.