Chapter 8

Wednesday

• Topic: Section 8.1: 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. We conclude with an example that involves computing limits of certain sequences using a special algebraic technique.
• Class materials: Lecture notes, worksheet
• After class: Work on Homework 0.

Friday

• Topic: Section 8.2: Infinite 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, worksheet
• After class: Try today’s worksheet and work on Homework 1.

Chapter 8

Monday

• Topic: Section 8.2: Infinite Series, continued. We wrap up our discussion on geometric series and introduce the \(n\)th term test for divergence.
• Class materials: Lecture notes
• After class: Read Definition 8.2.3, Theorem 8.2.2, and Example 8.2.3 in Section 8.2.

Wednesday

• Topic: Section 8.3: Comparison Tests. We introduce \(p\)-series and begin formulating the comparison test, which allows us to make conclusions about convergence of series with complicated terms by comparing them with simpler versions whose behavior we understand.
• Class materials: Lecture notes, worksheet
• After class: Read the statements of the Comparison Test and Limit Comparison Test in Section 8.3.

Friday

• Topic: Section 8.3: Comparison Tests, continued. 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: Try the worksheet problems above and work on Homework 2.

Chapter 8

Monday

• Topic: Section 8.4: Ratio Test. We introduce a new test for convergence 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: Work on the worksheet above.

Wednesday

• Topic: Section 8.4: Ratio Test, 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: Finish today’s worksheet and read Section 8.5 up through Example 8.5.1.

Friday

• Topic: Section 8.5: Absolute Convergence and Alternating Series Tests. We introduce the notions of alternating series, absolute convergence, and conditional convergence. We also introduced our last convergence tests for the moment, the Absolute Convergence Test and the Alternating Series Test, giving a picture based proof of the latter.
• Class materials: Lecture notes, worksheet
• After class: Finish today’s worksheet and start on Homework 3

Chapters 5, 8

Monday

• Topic: Section 8.5: Absolute Convergence and Alternating Series Tests, continued. We get some more practice identifying absolute and conditional convergence and discuss the Riemann Rearrangement Theorem.
• Class materials: Lecture notes, worksheet
• After class: Read Examples 5.1.1, 5.1.2, and 5.2.2.

Wednesday

• Topic: Chapter 5: Basics of Integration review. 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: Read Theorem 5.4.3 and Example 5.4.7.

Friday

• Topic: Chapter 5: Basics of Integration review. We discuss finding areas between two curves using the definite integral.
• Class materials: Lecture notes, worksheet
• After class: Begin studying for Exam 1 by working on old homework and worksheet problems, and similar problems from the textbook.

Chapter 5

Monday

• Topic: Section 6.1: 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: Study for Exam 1.

Wednesday

• Topic: Exam review.
• Class materials: worksheet, solutions
• After class: Study for Exam 1.

Friday

• Topic: Exam 1.
• After class: Enjoy your break!

Chapter 6

Monday

• Topic: Fall break, no class.

Wednesday

• Topic: Section 6.1: 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: Read Section 6.2 up through Example 6.2.3.

Friday

• Topic: Section 6.2: 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: Read Section 6.5 up through Example 6.5.1.

Chapter 6

Monday

• Topic: Section 6.5: 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: Read Example 6.1.12 and Section 6.8 up through Example 6.8.1.

Wednesday

• Topic: Section 6.8: Improper integration. We wrap up our discussion of partial fractions, and how to handle terms that require the use of arctangent. We begin to discuss definite integrals over infinitely long integrals.
• Class materials: Lecture notes, worksheet
• After class: Read Definition 6.8.2 and Example 6.8.2.

Friday

• Topic: Section 6.8: 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: Read Examples 7.1.1, 7.1.2, and 7.1.3.

Chapter 7

Monday

• Topic: Section 7.1: Area Between Curves. We continue a past discussion on using integrals to find areas between curves. One focus will be on the idea that it’s often possible to set up these integrals with respect to either the \(x\) variable or the \(y\) variable.
• Class materials: Lecture notes, worksheet
• After class: Read Examples 7.2.2 and 7.2.3.

Wednesday

• Topic: Section 7.2: Disk Method. 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, worksheet
• After class: Read Examples 7.2.2 and 7.2.3.

Friday

• Topic: Section 7.2: Washer Method. We expand our discussion of the disk method to include solids of revolution which have a hole in them. Instead of slicing the solids into thin disks, the solids will be sliced into washers, ie. disks with a hole in the middle.
• Class materials: Lecture notes, worksheet
• After class: Read Section 7.3 up through Example 7.3.1.

Chapter 7

Monday

• Topic: Section 7.3: Shell Method. We introduce a final method for finding volumes of revolution. This method involves slicing our region so that cylindrical shells are formed when a slice is revolved around the axis of revolution.
• Class materials: Lecture notes, worksheet
• After class: Read Section 7.5 up through Example 7.5.2.

Wednesday

• Topic: Section 7.5: Work. We discuss another application of integration through the physical notion of work. Our initial focus will be on problems involving lifting a mass over a distance and compressing springs.
• Class materials: Lecture notes, worksheet
• After class: Read Examples 7.5.5 and 7.5.6.

Friday

• Topic: Section 7.5: Work, continued. We do some more examples of computing work that involving pumping fluids from containers of various shapes.
• Class materials: Lecture notes, worksheet
• After class: Read Section 8.3 up through Example 8.3.1.

Chapter 8

Monday

• Topic: Section 8.3: Integral Test. We discuss how improper integrals are used to determine whether certain series converge.
• Class materials: Lecture notes, worksheet
• After class: Study old homework problems, worksheets, and lecture examples!

Wednesday

• Topic: Review. We spend the day working on review problems for Exam 2.
• Class materials: worksheet, solutions
• After class: Continue your review and study by working on new problems from the textbook.

Friday

• Topic: Exam 2.
• After class: Review the ratio test and read Example 8.6.2.

Chapter 8

Monday

• Topic: Section 8.6: 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, worksheet
• After class: Read the first two pages of Section 8.7.

Wednesday

• Topic: Section 8.7: 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: Finish Homework 8.

Friday

• Topic: Section 8.7: Taylor polynomials, continued. We work on a worksheet of material from last time and begin to discuss the possible error in using Taylor polynomials to approximate a function.
• Class materials: Lecture notes, worksheet
• After class: Read Examples 8.7.2 and 8.7.3.

Chapter 8

Monday

• Topic: Early break, no class.
• After class: Enjoy your break!

Wednesday

• Topic: November break, no class.

Friday

• Topic: November break, no class.

Chapter 8

Monday

• Topic: Section 8.7: Taylor polynomials, continued. We practice using Taylor’s theorem.
• Class materials: Lecture notes, worksheet
• After class: Read the first two pages of Section 8.8.

Wednesday

• Topic: Section 8.8: Taylor series. We discuss power series that come about by considering Taylor polynomials with infinitely many terms.
• Class materials: Lecture notes, worksheet
• After class: Read Theorem 8.8.2 and Examples 8.8.6 and 8.8.7.

Friday

• Topic: Section 8.8: 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: Work on Homework 10.

Applications

Monday

• Topic: Section 8.8: Taylor series, continued. We get some more practice working with Taylor series.
• Class materials: Lecture notes, worksheet
• After class: Study for Wednesday’s quiz.

Wednesday

• Topic: Application: Normal distribution. We discuss how to compute probabilities that involve data that is normally distributed (falls under a bell curve).
• Class materials: Lecture notes, worksheet
• After class: Finish Homework 10

Friday

• Topic: Application: Geometric distribution. We discuss the geometric distribution a model for understanding a random experiment that consists of repeated trials until a desired outcome occurs.
• Class materials: Lecture notes, worksheet
• After class: Study for Exam 3 by reviewing homework and quizzes since Exam 2.

Wrap up

Monday

• Topic: Review. We spend the day working on the review sheet linked below.
• Class materials: worksheet, solutions
• After class: Study for Exam 3. Enjoy your winter break! Keep in touch!