Math 2153 - Calculus II

MWF 9:30 am, room 355 PS

Instructor: Anatolii Grinshpan
Current office hours: 3:10-4 MW, 5:10-6 F, or by appointment, 406 MS

Program Outline

Jan 9   Course overview. Sequences: definition and examples.
Jan 11 Convergence of sequences. Operations with limits.
Jan 13 Examples. Sequences arising from functions. Squeeze theorem.
HW 1 due Jan 23: 12, 16, 20, 24, 34, 36, 38, 56, 65 (pages 747-48)
Jan 18 Monotone sequence theorem. Passage to the limit.
Jan 20 Examples. Exponential and logarithmic functions.   The number e
Jan 23 Exponential and logarithmic functions continued.
HW 2 due Jan 30: 8, 9, 13, 14, 18, 25, 28, 30, 38, 50, 64 (pp. 431-3), 3-8, 10, 13-18, 28, 30, 38, 44, 60, 69 (pp. 439-40), 2-12 (p. 449)
Jan 25 Differentiation and integration of exponentials and logarithms.
Jan 27 Examples of integration. Logarithmic differentiation. e as a limit.
Jan 30 Inverse trigonometric functions (arcsin, arccos, arctan).
HW 3 due Feb 6: 1-10, 12, 14, 22, 26, 36, 47, 62, 64, 66, 70 (pp. 483-5) 1-4, 6, 8, 10, 18, 26, 30, 40, 44, 50, 78 (pp. 501-2)
Feb 1  l'Hôpital's rule for limits.   l'Hôpital's textbook on differential calculus (German translation)
Feb 3  Indeterminate forms (0/0, ¥/¥, ¥-¥, 0´¥, 00, ¥0, 1¥): examples.
Feb 6  Integration by parts.
HW 4 due Feb 13: 3-35 (p. 516), 2, 4, 6, 12, 14 (pp. 524-5), 2, 4, 6, 10 (p.530)
Feb 8  Trigonometric integrals.
Feb 10 Trigonometric integrals (continued). Trigonometric substitutions.
Feb 13 Examples. Partial fraction decomposition.
HW 5 due Feb 20: 12, 14, 16, 18, 20, 22, 24 (p.531), 8, 10, 12, 14, 16, 18, 20, 22, 40, 42, 55, 56 (p. 540)
Feb 15 Examples of integration.
Feb 17 Review. A couple of integrals.
Feb 20 No class.
Feb 22 Exam I. Answers.
Feb 24 Approximate integration.
Feb 27 Improper integrals.
HW 6 due Mar 6: 12, 20 (p. 563), 2, 6, 8, 10, 12, 18, 26, 50, 59, 71 (pp. 574-5)
Mar 1  Introduction to series. Series with positive terms.
Mar 3  Convergence of series. Geometric series. Harmonic series.
Mar 6  Telescoping series. Divergence test. Integral test and p-series.
HW 7 due Mar 20: 10, 14, 18, 19, 20, 22 (p. 756), 2, 12, 15, 18, 19, 20 (p. 765), 4, 6, 8, 10, 12, 14, 18 (p. 770)
Mar 8  Integral test remainder estimate. Comparison test.
Mar 10 Practice session.
Mar 20 Limit comparison test.
HW 8 due Mar 27: 1, 2, 19, 22, 24, 26, 31 (pp. 770-1), 4, 5, 10, 14, 24, 35 (pp. 775-6).
Mar 22 Alternating series test. Remainder estimate.
Mar 24 Absolute convergence. Root and Ratio tests.
Mar 27 Power series about the origin. Interval and radius of convergence.
HW 9 due Apr 3: 1-16, 31, 33 (pp. 781-2), 3-14, 29, 30 (p. 789).
Mar 29 Power series about a given point. Operations with power series.
Mar 31 Representation of functions as power series.
Apr 3  Functions as power series. Taylor polynomials.
HW 10 due Apr 10: 1-8, 15-18, 24 (p. 795), 1-8, 12, 14, 47, 48 (pp. 806-7)
Apr 5  Taylor series. An applet.
Apr 7  Review.
Apr 10 Exam II. Answers.
Apr 12 The arc length. Johann Bernoulli. Arc length of xp.
Apr 14 Parametric description of curves. Brachistochrone.
Apr 17 Calculus of parametric curves.
HW 11 due Apr 24: 2, 10-12, 24, 28 (pp. 692-3), 1-4, 7-12, 15-17, 22, 26, 54 (pp. 713-4), 5-8, 47 (p. 719)
Apr 19 Polar coordinates.
Apr 21 Polar description of curves. Arc length and area in polar coordinates. Cardioid.
Apr 24 Ellipse, parabola, hyperbola.
Apr 26 Practice session. Review questions I.
Apr 28 Practice session. Review questions II.
Review questions III.
The review questions are selected from the homework. Do not neglect the rest of the homework.
Office hours: Apr. 28, 5:10-6 pm, Apr. 29, 1-2+ pm
May 1 Final Examination 8:00-9:50 am. Answers.