Math 409
Advanced Calculus I
Sections 200 and 501
Spring 2011
Course Schedule (subject to change)
January
- Tuesday, January 18
- Sections 1.1–1.6. The real numbers form a complete, ordered field.
- Thursday, January 20
- Sections 1.7–1.10. Properties of the natural numbers and the rational numbers: the Archimedean property, induction, density of the rationals. Distance on the real line.
- Tuesday, January 25
- Sections 2.1–2.4. Sequences, convergence of sequences. Section 2.3 on countable sets is optional for students in Section 501.
- Thursday, January 27
- Sections 2.5–2.7. Divergent sequences; boundedness of convergent sequences; algebraic properties of limits of sequences.
February
- Tuesday, February 1
- Sections 2.8–2.9. Order properties of limits, Squeeze Theorem, convergence of bounded monotonic sequences.
- Thursday, February 3
- Sections 2.10–2.13.
Examples of limits; subsequences, Bolzano–Weierstrass theorem; Cauchy's criterion for convergence. Section 2.13 on upper and lower limits is optional for students in Section 501.
- Tuesday, February 8
- Sections 4.1–4.3. Open and closed sets, interior points, boundary points, accumulation points.
- Thursday, February 10
- Sections 4.4–4.5.1.
Properties of open and closed sets; Bolzano–Weierstrass property.
- Tuesday, February 15
- Sections 4.5.2–4.6. Notions of compactness: Cantor's theorem on nested sets, Cousin's covering lemma, Heine–Borel property. Countable sets. Sections 4.5.2–4.5.4 are optional for students in Section 501.
- Thursday, February 17
- Catch-up and review.
- Tuesday, February 22
- First examination, covering Chapters 1,
2, and 4.
Update: after the exam, solutions were posted.
- Thursday, February 24
- Section 5.1. Limits of functions.
March
- Tuesday, March 1
- Section 5.2. Properties of limits of functions.
- Thursday, March 3
- Sections 5.3–5.4. Continuity. Section 5.3 on limits superior and inferior is optional for students in Section 501, as is Section 5.4.4 about continuity on a set.
- Tuesday, March 8
- Sections 5.5–5.8. Properties of continuous functions, extreme-value property, intermediate-value property; uniform continuity.
- Thursday, March 10
- Section 5.9. Discontinuities; monotonic functions. Section 5.9.3 is optional for students in Section 501.
- March 14–18
- Spring Break
- Tuesday, March 22
- Sections 7.1–7.3.1. Definition of the derivative, algebraic rules. Section 7.2.3 is optional for students in Section 501.
- Thursday, March 24
- Sections 7.3.2–7.5.
Chain rule, inverse functions, powers, discontinuous derivatives, local extrema.
- Tuesday, March 29
- Sections 7.6–7.7.
Mean-value theorems, monotonicity.
- Thursday, March 31
- Sections 7.8–7.10.
Dini derivates (Section 7.8 is optional for students in Section 501), intermediate-value property of derivatives, convexity.
April
- Tuesday, April 5
- Sections 7.11–7.12. L'Hôpital's rule, Taylor polynomials.
- Thursday, April 7
- Catch-up and review.
- Tuesday, April 12
- Second examination, covering Chapters 5 and 7.
Update: after the exam, solutions were posted.
- Thursday, April 14
- Sections 8.1–8.2. The integral of a continuous function.
- Tuesday, April 19
- Sections 8.3–8.5. Properties of the integral, improper integrals.
- Thursday, April 21
- Section 8.6. Riemann's concept of the integral. Sections 8.6.2–8.6.4 are optional for students in Section 501.
- Tuesday, April 26
- Sections 8.7–8.9. Properties of the Riemann integral, improper integrals, fundamental theorem of calculus. Section 8.9 is optional for students in Section 501.
- Thursday, April 28
- Catch-up and review; last class day for this course.
May
- Tuesday, May 3
- This day is redefined as a Friday, so our class does not meet.
- Wednesday, May 11
- Comprehensive final examination,
8:00am–10:00am. Update: after the exam, solutions were posted.
