Calculus I, Fall 2024

Course Information

• NTU COOL
• Lectures: Tuesday and Thursday, 13:20 ~ 15:10 at Freshman Classroom Building 204
• Office hours: Monday 14:20 ~ 15:20, at Astro-Math 458
• Grading scheme:
  • Midterm 30% (on Thursday, October 24)
  • Final 35% (on Thursday, December 19)
  • Quiz 10% (on September 26 and November 14)
  • Homework 20%
    You have two jokers: the lowest two grades will be discarded.
  • Webwork 5%
• Textbook: Tom Apostol, Calculus vol.1 - One-variable Calculus with an Introduction to Linear Algebra
• Problem session: Thursday 17:30 ~ 18:20
• Teaching Assistants:
  • 連焌凱: problem session at Freshman Classroom Building 204.
    Office hour: Wednesday 13:20 ~ 14:20 at Astro-Math 446.
  • 郭立生: problem session at Freshman Classroom Building 203.
    Office hour: Wednesday 15:30 ~ 16:30 at Astro-Math 405.
  • 黃篆: problem session at Freshman Classroom Building 202.
    Office hour: Monday 17:30 ~ 18:30 at Astro-Math 405.

Lecture summaries and references

• (week 1)
  • completeness of real numbers: upper bound, least upper bound/supremum. reference: I 3.8, I 3.9
  • partition, step function and its integral. reference: 1.9 ~ 1.13
  • integral of general functions. reference: 1.16, 1.17
  • monotone functions and their integrability. reference: 1.20 ~ 1.23
• (week 2)
  • properties of integrals. reference: 1.24 ~ 1.27
  • *(Riemann) measurable set. reference: 1.6
  • integration of cosine. reference: 2.1 ~ 2.6
  • polar coordinate. reference: 2.9, 2.10
  • indefinite integral. reference: 2.18
  • continuity. reference: 3.1 ~ 3.3
• (week 3)
  • basic limit theorems. reference: 3.4, 3.5
  • composition and continuity. reference: 3.7
  • intermediate value theorem and strictly monotone functions. reference: 3.9 ~ 3.14
• (week 4)
  • Heine-Borel theorem, uniform continuity, properties of continous functions on closed and bounded intervals. reference: 3.16 ~ 3.19
  • tangent line and derivative. reference: 4.7
  • basic properties of derivatives. reference: 4.3 ~ 4.5
  • chain rule. reference: 4.10
• (week 5)
  • chain rule and its applications. reference: 4.10, 4.11
  • extreme value. reference: 4.13
  • mean value for derivative and its applications. reference: 4.14 ~ 4.17
• (week 6)
  • fundamental theorem of calculus. reference: 5.1 ~ 5.4
  • integration by substitution. reference: 5.7
  • integration by parts. reference: 5.9
  • Wallis formula. reference: Courant & John Vol.I p.280 ~ 281
• (week 7)
  • natural logarithm. reference: 6.1 ~ 6.7
  • polynomial approximation to logarithm. reference: 6.10
  • exponential function. reference: 6.12 ~ 6.16
  • hyperbolic trig functions. reference: 6.18 ~ 6.21
  • partial fractions. reference: 6.23, 6.24
• (week 8)
  • Taylor polynomial and error term formula. reference: 7.1 ~ 7.7
  • little o notation and Peano's form. reference: 7.9
  • Midterm
• (week 9)
  • application of Taylor's theorem to limit. reference: 7.9, 7.10
  • L'Hopital's rule. reference: 7.12
  • extend the notations by including the infinity. reference: 7.14, 7.15
  • behavior of log x and exp x for large x. reference: 7.16
• (week 10)
  • sequence and convergence. reference: 10.2, 10.3
  • Cauchy sequence. reference: Courant & John Vol.I p.74~75 and p.94~97
  • series, telescoping series. reference: 10.5 ~ 10.8
  • comparison test. ratio test and root test. reference: 10.11, 10.12, 10.15
  • integral test. Euler's constant. reference: 10.13
  • alternating series. criterion of Leibniz and Dirichlet. reference: 10.17 ~ 10.19
• (week 11 and 12)
  • first order ODE. reference: 8.1 ~ 8.4
  • second order ODE. reference: 8.8 ~ 8.16
• (week 13)
  • Abel's test. reference: 10.9
  • improper integral. reference: 10.23
  • uniform convergence: continuity, integral. reference: 11.1 ~ 11.5
  • Weierstrass M-test. reference: 11.5
  • power series, its integration and derivative. reference: 11.6, 11.8
• (week 14)
  • Taylor series generated by a function. reference: 11.9
  • two convergence criteria of Taylor series. reference: 11.10, 11.12
  • application of power series to ODE. reference: 11.14
  • periodic functions. reference: Courant & John Vol.I 8.1
  • Fourier coefficient, basic estimate on the coefficiet. reference: Courant & John Vol.I 8.4
  • the improper integral of sin(x)/x. reference: Courant & John Vol.I 8.4

Homework

• Homework 01 (due September 12): [1.15] #1 (d) (e), #4 (a) (b), #11, #12, #16.
• Homework 02 (due September 19): [1.26] #23, #27; [2.8] #31; [2.11] #8, #13, #14.
• Homework 03 (due September 26): [3.6] #5, #18, #22, #27; [3.8] #17, #21.
• Homework 04 (due October 3): [3.6] #31; [3.20] #2, #4, #7, #8.
• Homework 05 (due October 8): [4.6] #36; [4.9] #10; [4.12] #32.
• Homework 06 (due October 17): [4.15] #2, #5, #8(a), #10; [6.9] #19, #26, #28.
• Homework 07 (due October 31): [5.5] #19, [5.10] #16, [5.11] #21, #35, [6.17] #42.
• Homework 08 (due November 7): [6.17] #30, [6.25] #16, #31, [6.26] #23, [7.11] #24.
• Homework 09 (due November 14): [7.4] #10, [7.8] #2, [7.11] #32, [7.17] #26, [10.4] #2, #5.
• Homework 10 (due November 21): [10.4] #34, #35 (b), [10.9] #9, #23, [10.14] #8, [10.16] #12.
• Homework 11 (due November 28): [8.5] #1, #9, [10.16] #15, [10.20] #13, #47, #52 (b).
• Homework 12 (due December 5): [8.14] #5, #10, [10.20] #32, #51, [10.22] #5, [10.24] #11.
• Homework 13 (due December 12): [11.13] #5, #6, #12, #19, #23, [11.16] #2.