Chapters 5 and 6

Wednesday

• Topic: Introduction. We give an overview of the class and then get refreshed on some material from the end of Calculus 1. In particular, we discuss antiderivatives, definite integrals, the Fundamental Theorem of Calculus, and some basic algebraic properties of the definite integral.
• Class materials: Lecture notes, worksheet
• After class: Finish today’s worksheet and take a look at Homework 0.

Friday

• Topic: Refresher. We conclude our refresher of material from Chapters 5 and 6.
• Class materials: Lecture notes, worksheet
• After class: Finish today’s worksheet and start working on Homework 1.

Chapter 7

Monday

• Topic: Section 7.1: Substitution. We introduce a new technique for finding antiderivatives. It comes from the chain rule for derivatives and is called \(u\)-substitution.
• Class materials: Lecture notes, worksheet
• After class: Finish today’s worksheet and read Examples 9 and 11 in Section 7.1.

Wednesday

• Topic: Section 7.1: Substitution, continued. We study more examples using the method of substitution, now focusing on definite integrals and changing the limits of integration.
• Class materials: Lecture notes, worksheet
• After class: For tomorrow’s quiz, make sure to know your list of antiderivatives to remember.

Friday

• Topic: Section 7.2: Integration by parts. We learn a method for integration that is derived from the product rule for differentiation. The main difficulty in this method is making a choice for which part of the integrand should be \(u\) and we use the mnemonic LIATE as a general rule of thumb.
• Class materials: Lecture notes, worksheet
• After class: Finish today’s worksheet and start working on Homework 2.

Chapter 7

Monday

• Topic: Section 7.4: Partial fraction decomposition. We discuss integrating rational functions using an algebraic technique that is akin to the reverse of adding fractions through finding common denominators.
• Class materials: Lecture notes, worksheet
• After class: Finish today’s worksheet. Read Examples 6 and 9 in Section 7.4.

Wednesday

• Topic: Section 7.4: Trigonometric substitutions. We discuss using trigonometric identities to simplify certain integrals.
• Class materials: Lecture notes, worksheet
• After class: For tomorrow’s quiz, make sure you know your list of antiderivatives to remember as well as how to find the area between two curves.

Friday

• Topic: Section 7.6: Improper integration. We discuss definite integrals over infinitely long intervals.
• Class materials: Lecture notes, worksheet
• After class: Finish today’s worksheet and start working on Homework 3.

Chapter 8

Monday

• Topic: Section 7.6: Improper integration, continued. We conclude our discussion on improper integrals by introducing a new type: integrals where the integrand has a vertical asymptote.
• Class materials: Lecture notes, worksheet
• After class: Finish today’s worksheet and read Examples 2 and 3 in Section 8.1.

Wednesday

• Topic: Section 8.1: Areas and volumes. We continue a past discussion on using integrals to find areas between curves. We also introduce the idea of using integrals to compute volumes.
• Class materials: Lecture notes, worksheet
• After class: Read Examples 1 and 3 in Section 8.2. For Friday’s quiz, you should expect to compute some integrals, definite or indefinite, using substitution or integration by parts.

Friday

• Topic: Section 8.2: Applications to geometry. We discuss volumes of three-dimensional solids formed by revolving a region in the \(xy\)-plane around an axis. Our approach will be to find the volumes of such solids by slicing them into thin disks and summing the volumes of these disks.
• Class materials: Lecture notes
• After class: Start working on Homework 4.

Chapter 8

Monday

• Topic: Section 8.2: Applications to geometry, continued. We continue our discussion on solids of revolution.
• Class materials: Lecture notes, worksheet
• After class: Read the first page of Section 8.5 and Examples 3, 4, and 5.

Wednesday

• Topic: Section 8.5: Applications to physics. We discuss another application of integration through the physical notion of work. Our focus will be on problems involving lifting a mass over a distance.
• Class materials: Lecture notes, worksheet
• After class: For Friday’s quiz, make sure you know how to do problems like those in Homework 3.

Friday

• Topic: Section 8.5: Applications to physics, continued. We continue our discussion on computing work done in pumping fluids from containers of various shapes.
• Class materials: Lecture notes
• After class: Begin studying for next week’s exam. Think about what formulas you might like to have on the exam cover sheet. Go through old homework problems, including the suggested problems, old worksheets, and examples from class.

Review and exam

Monday

• Topic: Review. We discuss the integration techniques we have learned and do practice problems identifying an appropriate method and carrying out the calculation.
• Class materials: worksheet, solutions
• After class: Finish today’s worksheet and continue your review.

Wednesday

• Topic: Review. We continue our review for the exam, focusing on improper integrals, areas, and volumes.
• Class materials: Lecture notes, worksheet
• After class: Finish today’s worksheet and continue your review.

Friday

• Topic: Exam 1.
• After class: Work on Homework 5.

Chapter 9

Monday

• Topic: Section 9.1: Sequences. We discuss the definition of a sequence (an infinite list of numbers) and get some intuition for what a sequence is and what it means for it to converge or diverge through examples.
• Class materials: Lecture notes, Desmos plotter
• After class: Finish today’s worksheet. Read the first two pages of Section 9.2.

Wednesday

• Topic: Section 9.1: Sequences, continued. We discuss algebraic techniques for finding limits of convergent sequences.
• Class materials: Lecture notes, worksheet
• After class: Finish today’s worksheet. For Friday’s quiz, make sure you know how to do integrals using trigonometric substitutions and I want you to be able to list the first 5 terms of a given sequence like in the first example of Monday’s lecture notes.

Friday

• Topic: Section 9.2: Geometric Series. We introduce the notion of infinite series (a sum of an infinite list of numbers) and focus our attention on a particular type of infinite series, geometric series, where the ratio of successive terms is constant.
• Class materials: Lecture notes
• After class: Enjoy your break!

Spring Break

Monday

• Topic: Spring break, no class.

Wednesday

• Topic: Spring break, no class.

Friday

• Topic: Spring break, no class.

Chapter 9

Monday

• Topic: Section 9.2: Geometric Series, continued. We discuss what it means for a series to converge and continue our discussion of geometric series.
• Class materials: Lecture notes, worksheet
• After class: Finish today’s worksheet. Read up through Example 3 in Section 9.3.

Wednesday

• Topic: Section 9.3: Convergence of Series. We introduce the \(n\)th term test for divergence and the integral test.
• Class materials: Lecture notes, worksheet
• After class: For Friday’s quiz, be prepared to do a problem involving work and a problem involving volumes of revolution.

Friday

• Topic: Section 9.3: Convergence of Series, continued. We work on the worksheet from last time.
• Class materials: See last time’s notes and worksheet.
• After class: Work on Homework 7.

Chapter 9

Monday

• Topic: Section 9.4: Tests for Convergence. We give the statements of the Comparison Test and Limit Comparison Test and work on examples where we try to prove a given series converges or diverges using these two theorems and prior knowledge about \(p\)-series and geometric series. We put an emphasis on how to structure our solutions in order to communicate clearly.
• Class materials: Lecture notes, worksheet
• After class: Finish today’s worksheet and read about the ratio test and Example 6 in Section 9.4.

Wednesday

• Topic: Section 9.4: Tests for Convergence, continued. We introduce a new test for convergence, called the Ratio Test, that says when the terms of a series have an asymptotic common ratio, the series converges under criteria similar to that for a geometric series.
• Class materials: Lecture notes, worksheet
• After class: Finish today’s worksheet and read about the Absolute Convergence Test and Alternating Series test and Examples 5 and 8 in Section 9.4. Study Homework 6 for Friday’s quiz.

Friday

• Topic: Section 9.4: Tests for Convergence, continued. We learn how to use the ratio test and work on examples that get us to practice identifying which test is appropriate to use.
• Class materials: Lecture notes, worksheet
• After class: We will work on today’s worksheet in class Monday. Now that we’ve covered all the series tests, start making yourself a list of the tests and examples of series that work and don’t work with each test. Work on Homework 8.

Chapter 9

Monday

• Topic: Section 9.4: Tests for Convergence, continued. We get some more practice identifying absolute and conditional convergence and discuss the Riemann Rearrangement Theorem.
• Class materials: Lecture notes
• After class: Read the first 3 pages of Section 9.5.

Wednesday

• Topic: Section 9.4: Tests for Convergence, wrap up. We discuss general advice for identifying when each test we’ve learned should be use. We spend most of our time working on Exercises 64-92 in Section 9.4.
• Class materials: Lecture notes
• After class: Prepare for Friday’s quiz by studying Homework 7 material and focus on the integral test and \(n\)th term test.

Friday

• Topic: Section 9.5: Power Series. We introduce examples of series whose terms consist of powers of a variable \(x\), as in polynomials. We focus on determining the domain of values of \(x\) for which series converge; a notion called the interval of convergence.
• Class materials: Lecture notes
• After class: Begin studying for next week’s exam which will cover Homework 5-8 material. Make sure to practice using the series tests; I recommend working through Exercises 64-92 in Section 9.4.

Chapter 9 and Exam 2

Monday

• Topic: Section 9.5: Power Series. We continue our discussion of power series.
• Class materials: Lecture notes, worksheet
• After class: Continue studying for the exam. If you have some time work on Homework 9.

Wednesday

• Topic: Review. We spend the day working on review problems in preparation for Exam 2 on Friday.
• Class materials: Lecture notes, worksheet
• After class: Get some rest!

Friday

• Topic: Exam 2.
• After class: Work on Homework 9.

Chapter 10

Monday

• Topic: Section 10.1: Taylor polynomials. We introduce a generalization of linear (or tangent line) approximations of functions. We discuss how to find polynomials of arbitrary degree whose derivatives at a point match those of a given function.
• Class materials: Lecture notes
• After class: Work on finding Maclaurin polynomials for the cosine function.

Wednesday

• Topic: Section 10.1: Taylor polynomials, continued. We continue our discussion from last time with more examples.
• Class materials: Lecture notes, worksheet
• After class: Finish today’s worksheet and work on Homework 9.

Friday

• Topic: Section 10.2: Taylor Series. We discuss the concept of a Taylor series, thinking of them as a power series with coefficients prescribed by the same coefficients that define Taylor polynomials. We focus on how to find Taylor series for some well known functions.
• Class materials: Lecture notes, worksheet
• After class: Finish today’s worksheet and work on Homework 10.

Chapter 10

Monday

• Topic: Section 10.3: Taylor series, continued. We discuss deriving new Taylor series using known ones, as well as how to use Taylor series to approximate integrals and compute limits that would otherwise be difficult to work with.
• Class materials: Lecture notes, worksheet
• After class:

Wednesday

• Topic: Section 10.4: The Error in Taylor Polynomial Approximations. We discuss bounds on the error between a function’s value and its Taylor polynomial approximation’s value.
• Class materials: Lecture notes, worksheet
• After class:

Friday

• Topic: Section 10.4: The Error in Taylor Polynomial Approximations, continued. We practice using Taylor’s theorem.
• Class materials: Lecture notes, worksheet
• After class:

Review

Monday

• Topic: Review.
• Class materials: Lecture notes, worksheet
• After class: Study for finals! Enjoy your summer! Keep in touch!