This is a (very) brief summary of what was (or will be) covered in each class.
April 29
Lecture overview: Reviewed for Final Exam.
The Final Exam will be given in Bannow 340 on Monday, May 6, from 8:00 - 11:00 a.m. A Review Outline for the Final Exam is available as well as additional Practice Problems Involving Trigonometric Functions. See Announcements for more information.
I will hold office hours on Wednesday, May 1, 11:00 a.m. - 1:00 p.m. and on Friday, May 3, 11:00 a.m. - 1:00 p.m.
April 25
Lecture Overview: Showed more examples of deriving Taylor series from known series, including the series for cos(x); computed an example of approximating an integral using a Taylor series. Introduced L'Hopital's Rule and sketched the proof of the 0/0 case using Taylor polynomials; worked out several examples of limits having indeterminate forms that may or may not exist.
The lecture material is contained in Sections 12.5 and 12.7 of your text.
Quiz 4 was returned and briefly reviewed in class, and solutions were distributed.
Homework 9 is due on Monday, April 29. See Homework Assignments.
The Final Exam will be given in Bannow 340 on Monday, May 6, from 8:00 - 11:00 a.m. There will be a review session in class on Monday, April 29. Come prepared with questions! See Announcements for more information.
April 22
Lecture overview: No class due to Easter break.
Homework 9 is due on Monday, April 29. See Homework Assignments for more information.
April 18
Lecture Overview: Introduced Taylor series of a function based at 0; if the series converges on an open interval, then it converges to its function on that interval; the largest such interval is called the interval of convergence, and its radius is called the radius of convergence for the Taylor series; for x in the interval of convergence, we have a series reperesentation for the function. Computed the Taylor series and interval of convergence for ex, 1/(1-x), and sin(x) using the Ratio Test; using these Taylor series, we can easily compute the Taylor series of many related functions; worked out series for 1/(1+x), 1/(4-x), and ln(1+x); using substitution, we can often derive Taylor series much more quickly than taking deritavtives one at a time.
The lecture material is contained in Section 12.5 of your text.
Homework 9 was assigned and is due Monday, April 29. See Homework Assignments.
Quiz 4 was given during the first 20 minutes of class.
April 15
Lecture Overview: Answered questions from Homework 8. Worked out a third example of applying the ratio test to a series involving a factorial. Introduced Taylor polynomials as generalization of linear approximation; derived formula for nth degree approximation by matching coefficients of the polynomial with derivatives of the function; calculated Taylor polynomials up to degree 3 for f(x) = ex and graphed them; calculated Taylor polynomials up to degree 3 for f(x) = sin(x) and graphed them; noted the pattern of coefficients for sin(x) and used this to write the Taylor polynomials up to degree 7 and beyond, which involve only the odd powers of x.
The lecture material is contained in Section 12.3 of your text, as well as a handout containing some examples and the proof of the Ratio Test.
Homework 8 is due Thursday, April 18. See Homework Assignments.
Quiz 4 will be given in class on Thursday, April 18. See Announcements for more information.
April 11
Lecture Overview: Reviewed geometric sequences, and introduced the problem of adding the terms of an infinite sequence; the sum of these terms is called an infinite series; this is related to several paradoxes in Greek philosophy, including Xeno's paradox. Derived formula for the nth partial sum of a geometric series; proved the series converges when r < 1 and diverges when r >= 1 , and found the sum of the convergent series; calculated several exmaples. Introduced the ratio test, and proved the case when R < 1; worked through 2 examples.
The lecture material is contained in Sections 12.1 and 12.4 of your text, as well as a handout containing some examples and the proof of the Ratio Test.
Homework 8 was assigned and is due Thursday, April 18. See Homework Assignments.
Quiz 4 will be given in class on Thursday, April 18. See Announcements for more information.
April 8
Lecture Overview: Introduced infinite sequences; discussed examples of sequences defined in closed form and also recursively; defined geometric sequences; worked out an application to depreciation; defined convergence and divergence of a sequence, and discussed examples in both closed and recursive form.
The lecture material is contained in Section 12.1 of your text.
Exam 1 was returned and briefly reviewed in class, and solutions were distributed. Please see me to discuss your exam.
April 4
Exam 2 was given in class.
April 1
Lecture Overview: Calculated several examples of improper integrals, including powers and exponentials; for power functions f(x) = x-p, p>0, classified as convergent all integrals having p>1 and as divergent all integrals having p<=1. Answered questions from Homework 7, and did problems to review for Exam 2.
The lecture material is covered in Section 8.4 of your text.
Homework 7 is due on Thursday, April 4. See Homework Assignments.
Exam 2 will be given in class on Thursday, April 4. See Announcements for more information.
March 28
Lecture Overview: Defined average value of a function on an interval and gave a brief derivation connecting the formula to our usual notion of average. Defined solid of revolution and derived formula for calculating volume of a solid in terms of the height function; calculated several examples, including those involving square roots and exponentials. Defined improper integral for finding the area under a horizontal asymptote; defined convergent and divergent integrals and computed two examples.
The lecture material is covered in Sections 8.2 and 8.4 of your text.
Homework 7 was assigned and is due on Thursday, April 4. See Homework Assignments.
Exam 2 will be given in class on Thursday, April 4. See Announcements for more information.
March 25
Lecture Overview: Answered questions from Homework 6. Continued discussing integration by parts; derived the antiderivative of ln(x) and presented two examples which we solved by integrating by parts twice. Students worked on some problems from a Worksheet on Integration and Approximation for about 25 minutes, then we reviewed the solutions.
The lecture material is covered in Sections 7.6, 8.1 and 13.3 of your text.
Homework 6 is due on Thursday, March 28. See Homework Assignments.
Exam 2 will be given on Thursday, April 4. See Announcements for more information.
March 14
Completed example comparing approximations using the Trapezoidal Rule and Simpson's Rule. Derived formula for integration by parts from the product rule. Computed several examples using exponentials, trigonometric functions and logarithms.
The lecture material is covered in Sections 7.6, 8.1 and 13.3 of your text.
Quiz 3 was returned and solutions were distributed.
Homework 6 is due on Thursday, March 28. See Homework Assignments.
March 11
Lecture Overview: Began discussing numerical integration; reviewed approximating areas using rectangles with heights evaluated at left and right endpoints and midpoints of each interval. Introduced and derived Trapezoidal Rule and computed an example; the estimate using the Trapezoidal Rule is always the same as that obtained by averaging the estimates using left and right endpoints. Introduced Simpson's Rule, which approximates the integral of a function using quadratic, rather than linear, approximations; derived pattern of coefficients; began computing an example.
The lecture material is covered in Section 7.6 of your text.
Quiz 3 was given during the first 20 minutes of class.
Homework 6 is assigned and is due Thursday, March 28. See Homework Assignments.
March 7
Lecture overview: Answered questions on Homework 5. Introduced idea of area between curves and computed several examples, including examples in which the curves cross in the interval we are asked to consider.
The lecture material is contained in Section 7.5 of your text.
Homework 5 is due Monday, March 11. See Homework Assignments.
Quiz 3 will be given in class on Monday, March 11. See Announcements for more information.
March 4
Class cancelled due to snow.
Homework 5 is postponed until Monday, March 11. See Homework Assignments.
Quiz 3 will be given in class on Monday, March 11. See Announcements for more information.
February 28
Lecture overview: Reviewed approximating area using rectangles and the definition of the definite integral as the limit of Riemann sums; stated and proved the Fundamental Theorem of Calculus; calculated several examples of definite integrals, including one using substitution.
The lecture material is contained in Sections 7.3 and 7.4 of your text.
Homework 5 was assigned and is due on Thursday, March 7. See Homework Assignments.
Quiz 3 will be given in class on Thursday, March 7. See Announcements for more information.
February 25
Lecture overview: Continued practicing substitution as a method of integration. Discussed an application to computing the root mean square voltage in a standard alternating circuit.
The lecture material is contained in Sections 7.2 and 13.3 of your text.
Exam 1 and Quiz 2 were returned in class and solutions were distributed. We went over two problems from the exam in class.
February 21
Exam 1 was administered in class.
Homework 4 is due on Thursday, February 28. See Homework Assignments.
February 19
Lecture overview: Reviewed antiderivatives and the indefinite integral; did various examples using the inverse power rule; introduction substitution as a method of integration; computed several examples of subsitution with square roots, exponential and trigonometric functions; substitution is a way to invert the chain rule for derivatives.
The lecture material is contained in Sections 13.3, 7.1 and 7.2 of your text.
Quiz 2 was given during the first 15 miutes of class.
Homework 4 was assigned and is due on Thursday, February 28. See Homework Assignments.
Exam 1 will be given in class on Thursday, February 21. See Announcements for more information.
February 14
Lecture overview: Calculated several derivatives of trigonometric functions combined with product, quotient and chain rules; noted the cyclic nature of higher derivatives of sine and cosine. Recalled antiderivatives and the notation of the indefinite integral. Found antiderivatives of sin(x) and cos(x) and used substitution to find the antiderivative of tan(x).
The lecture material is contained in Sections 13.2, 13.3, 7.1 and 7.2 of your text.
Homework 3 is due on Tuesday, February 19. See Homework Assignments.
Quiz 2 will be given during class on Tuesday, February 19. See Announcements for more information.
Exam 1 will be given in class on Thursday, February 21. See Announcements for more information.
February 11
No class today.
Homework 3 is due Tuesday, February 19. See Homework Assignments.
Quiz 2 will be given during class on Tuesday, February 19. See Announcements for more information.
Exam 1 will be given in class on Thursday, February 21. See Announcements for more information.
February 7
Lecture overview: Calculated several examples using the differential and linear approximation. Began reviewing trigonometric functions; recalled definitions of degree and radian measures and how to convert between the two; defined trigonometric functions and recalled several identities relating them; discussed graphing the functions and recalled the graphs of sine and cosine. Used the definition of the derivative to find the derivative of sin x; used this and the chain rule to find the derivative of cos x; used the quotient rule to find the derivative of tan x; calculated two examples of derivatives using the chain rule and one using the product rule in combination with the derivative of sin x.
The lecture material is contained in Sections 6.6, 13.1 and 13.2 of your text.
Homework 2 was due at the start of class today. Homework 3 is assigned and is due on Tuesday, February 19. See Homework Assignments.
Quiz 2 will be given during class on Tuesday, February 19. See Announcements for more information.
Exam 1 will be given in class on Thursday, February 21. See Announcements for more information.
The Math Center is open! Visit the Math Center Website to see the schedule and to sign up for an appointment.
February 4
Lecture overview: Continued prracticing related rates, which extends implicit differentiation to the case where two or more quantities depend on an independent variable; calculated several examples, including some applications and word problems. Introduced linear approximation, derived the formula and computed an example; defined differential and computed an example.
The lecture material is contained in Sections 6.5 and 6.6 of your text.
Quiz 1 was returned in class. Solutions were distributed and briefly discussed. Please see me if you have any questions.
Homework 2 is due Thursday, February 7. See Homework Assignments.
January 31
Lecture overview: Continued discussing implicit differentiation; worked through several examples with exponentials and logarithms. Introduced related rates; the idea here is similar to that used with implicit differentiation except that we consider both variables x and y to be implicit functions of a third variable, t.
The lecture material is contained in Sections 6.4 and 6.5.
Quiz 1 was given during the first 15 minutes of class.
Homework 2 was assigned and is due Thursday, February 7. See Homework Assignments.
January 28
Lecture overview: Reviewed solutions to a handout containing problems for reviewing derivative rules, which we started during the last class; students presented solutions to the first four problems and the next three we did together at the board. Introduced implicit differentiation as a method for finding dy/dx from an equation when we cannot solve for y; the main idea behind this technique is to view taking the derivative as an operation which can be performed on both sides of an equation; did several examples of polynomial functions and an exponential function; found the equation of a tangent line in one example.
The review material is contained in Chapter 4 of your text. The material on implicit differentiation is contained in Section 6.4.
The Orientation Exercises are due by the end of the day today. Homework 1 is due Thursday, January 31. See Homework Assignments.
Quiz 1 will be given in class on Thursday, January 31. See Announcements for more information.
January 24
Lecture overview:
The review material is contained in Chapter 4 of your text.
The Orientation Exercises (Assignment 0) are due on Monday, January 28. Homework Assignment 1 is assigned and is due on Thursday, January 31. See Homework Assignments for more information.