Math 107H Fall 2008 Course Log and Cumulative Homework List

Transcription

1 Date: 8/25 Sections: 5.4 Math 107H Fall 2008 Course Log and Cumulative Homework List Log: Course policies. Review of Intermediate Value Theorem. The Mean Value Theorem for the Definite Integral and the Fundamental Theorem of Calculus, part I. Date: 8/26 Sections: 5.4 Log: Using the Fundamental Theorem to differentiate functions of the form G(x) = u(x) a f(t) dt. Antiderivatives. The Fundamental Theorem of Calculus, part II. Assignment: Section 5.4, problems 7-31 odd, odd, 74, 75. Sections: 5.5 Log: The indefinite integral. The method of substitution. Assignment: Section 5.5, problems odd, 55, 57, 67. Read examples 7 and 8. Date: 8/28 Sections: 5.5, 7.1 Log: Definite integrals and substitution. Integration by parts (IBP). Repeated integration by parts. Introducing a factor of 1 in the integrand and then integrating by parts. Integrating by parts more than once and then solving algebraically for the integral. Preliminary substitution followed by integration by parts. Assignment: Section 7.1, problems 1-17 odd, 29, 31a, 33, 39, 41. Date: 8/29 Sections: 7.2 Log: Trigonometric integrals. Using trigonometric identities in integration. Integrands containing products and powers of trigonometric functions. Assignment: Section 7.2, problems 5-35 odd. Read example 6. Date: 9/2 Sections: 7.3 Log: Trigonometric substitutions. Integrands containing terms of the form a 2 + x 2, a 2 x 2 and x 2 a 2. Assignment: Section 7.3, problems 5-23 odd, 31, 33. Date: 9/3 Sections: 7.4 Log: The partial fraction decomposition of a proper rational function. Linear factors. Date: 9/4 Sections: 7.4 Log: Partial fractions. Irreducible quadratic factors. Assignment: Section 7.4, problems 1-21 odd. Read example 3. Date: 9/5

2 Sections: 7.5, 7.6 Log: Integral tables. Reduction formulas, special functions. Numerical integration. The Riemann sum approximation to the definite integral. The trapezoid rule. Assignment: Read examples 1-4 in 7.5. Date: 9/8 Sections: 7.6 Log: The derivation of the trapezoid rule. The Lagrange interpolating polynomial of degree 2. The derivation of Simpson s rule. Date: 9/9 Sections: 7.6 Log: Applications of Simpson s rule and the trapezoid rule. Digression on parrots. Error bounds for Simpson s rule and the trapezoid rule. Assignment: Section 7.6, problems 9, 17, 25, 27, 28, 31. Notes: Exam 1 will be given on Monday, 9/15. The exam will cover material from sections 5.4, 5.5, and 7.6. Date: 9/10 Sections: 7.6, 7.7 Log: Review of homework problems. Improper integrals. The failure of the Riemann sum definition of the definite integral when (I) the interval is infinite and (II) the integrand is unbounded. Improper integrals of type I. Date: 9/11 Sections: 7.7 Log: Improper integrals. Improper integrals over [a, ), (, b] and (, ). The comparison test. Date: 9/12 Sections: 7.7 Log: Improper integrals. The comparison test. Determining convergence and divergence by inspection. Type II improper integrals: Unbounded integrands. Date: 9/15 Log: Exam 1. Date: 9/16 Sections: 6.1 Log: Volumes by slicing. Derivation of the volume integral by Riemann sums and by infinitesimals. Assignment: In section 7.7, read examples 7 and 8, and do problems 1-15 odd and odd. In 6.1, do problems 7, 11, 17, 19, 37, and 38. Date: 9/17 Log: Review.

3 Date: 9/18 Sections: 6.2 Log: Volumes by the method of cylindrical shells. Assignment: In 6.2, do problems 1, 3, 7, 13, 21, 25, 35. Date: 9/19 Sections: 6.3 Log: Parametric curves in the plane. Derivation and application of the arclength integral for curves given parametrically and in the form y = f(x). Assignment: In 6.3, do problems 1, 2, 3, 8, 9, 11, 17. Date: 9/22 Sections: 6.5 Log: First-order, separable ordinary differential equatons (ODEs). Nonseparable, separable and autonomous ODEs. The initial value problem. Solving separable ODEs. Date: 9/23 Sections: 6.5 Log: The general solution to a first-order ODE. The initial value problem. Radioactive decay. Carbon dating. Date: 9/24 Sections: 6.5 Log: The Malthus and logistic population models. Newton s law of cooling. Assignment: Section 6.5, problems 3, 5, odd, 24, 31, 38, 39, 41. Date: 9/25 Sections: 6.6 Log: Work. Derivation applications of the work integral. Assignment: Section 6.6, problems 3, 5, 9, 10, 23, 25, 29, 35. Read examples 3 and 5. Date: 9/26 Sections: 8.1 Log: Sequences. Convergent and divergent sequences. The Sandwich Theorem. Using L Hôpital s rule to determine the limit of a sequence. Upper bounds and nondecreasing sequences. Assignment: In section 8.1, read Theorem 5 and do problems odd, 87, 97, 101, 103. Date: 9/29 Sections: 8.2 Log: Infinite series. Partial sums. Convergent and divergent series. Telescoping series. The geometric series. Assignment: Section 8.2, problems odd, odd. Notes: Exam 2, over material from sections , 6.5, 6.6, 7.7, 8.2 and 8.3, will be given on Monday, 10/6. Date: 9/30 Sections: 8.2, 8.3 Log: The nth term divergence test. The integral test.

4 Date: 10/1 Sections: 8.3 Log: The integral test. Using partial sums and integrals to estimate a series sum. Assignment: Section 8.3, problems odd, 39, 41. Date: 10/2 Sections: 8.4 Log: The comparison and limit comparison tests. Assignment: Section 8.4, problems 5-31 odd. Date: 10/3 Log: Review. Date: 10/6 Log: Exam 2. Date: 10/7 Sections: 8.5 Log: Review of exam problems. The ratio test. Date: 10/8 Sections: 8.5 Log: The ratio and root tests. Factorials. Absolute and conditional convergence. Assignment: Section 8.5, problems 1-19 odd, 27, 39. Date: 10/9 Sections: 8.6 Log: Alternating series. The alternating series test. Approximation of alternating series by partial sums. Assignment: Section 8.5, problems 1-19 odd, 27, 39. Date: 10/10 Sections: 8.6 Log: Absolute and conditional convergence. Absolute convergence impies convergence. Rearranging terms in absolute and conditionally convergent series. Assignment: Section 8.6, problems 1-21 odd, 47. Notes: The project, distributed today, is due in class on Friday, 12/5. Date: 10/13 Log: Review of alternating series. Introduction to power series. Convergence of power series. The radius of convergence.

5 Date: 10/14 Log: Convergence of power series. The radius of convergence and interval of convergence. Using the ratio test to determine the radius of convergence. Examples of power series with radii of convergence R = 0, R = and R finite and positive. Term-by-term integration and differentiation of power series. Assignment: Section 8.7, problems 5-25 odd. Date: 10/15 Log: Using the ratio and root tests to determine the radius of convergence of a power series. Term-by-term integration and differentiation of power series. Behavior of power series at the endpoints of the interval of convergence. Power series for 1/(1 x), 1/(1 + x), 1/(1 + x 2 ), ln (1 + x) and arctan x. Date: 10/16 Log: Power series. Radius and interval of convergence of various series. Multiplcation of power series. Assignment: Section 8.7, problems odd, 41, 43. Date: 10/17 Sections: 8.8 Log: Taylor series, Taylor coefficients. Derivation of the Taylor series. Calculation of Taylor series for e x, sin x and cos x about a = 0, and for x 1 about a = 1. Taylor polynomials. Assignment: Section 8.8, problems 1-21 odd, 33, 37. Date: 10/22 Sections: 8.8, 8.9 Log: Approximation by Taylor polynomials. The convergence of Taylor series. Date: 10/23 Sections: 8.9 Log: Taylor polynomials and the remainder. Taylor s formula. Proof of Taylor s formula. Notes: Exam 3, covering material from sections , will be given on Monday and Tuesday of next week. Date: 10/24 Sections: 8.9 Log: Review. Taylor polynomials and the remainder. Completetion of the proof of Taylor s formula. Notes: Exam 3, covering material from sections , will be given on Monday and Tuesday of next week. Date: 10/27 Log: Exam 3, part 1.

6 odd, odd, 74, odd, 55, 57, , 11, 17, 19, 37, , 3, 7, 13, 21, 25, , 2, 3, 8, 9, 11, , 5, odd, 24, 31, 38, 39, , 5, 9, 10, 23, 25, 29, odd, 29, 31a, 33, 39, odd odd, 31, odd , 17, 25, 27, 28, odd, odd odd, 87, 97, 101, odd, odd odd, 39, odd odd, 27, odd, odd, 41, odd, 33, 37. Cumulative Homework List