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:

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:

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:

Wednesday

• Topic: Section 7.4: Trigonometric substitutions. We discuss using trigonometric identities to simplify certain integrals.
• Class materials: Lecture notes, worksheet
• After class:

Friday

• Topic: Section 7.6: Improper integration. We discuss definite integrals over infinitely long intervals.
• Class materials: Lecture notes, worksheet
• After class:

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:

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:

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, worksheet
• After class:

Chapter 8

Monday

• Topic: Section 8.2: Applications to geometry, continued. We conclude our discussion on volumes and discuss arc lengths.
• Class materials: Lecture notes, worksheet
• After class:

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:

Friday

• Topic: Section 8.5: Applications to physics, 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:

TBA

Monday

• Topic: TBA
• Class materials: Lecture notes, worksheet
• After class:

Wednesday

• Topic: Review.
• Class materials: Lecture notes, worksheet
• After class:

Friday

• Topic: Exam 1.
• After class:

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

Wednesday

• Topic: Section 9.1: Sequences, continued. We continue our discussion on sequences.
• Class materials: Lecture notes, worksheet
• After class:

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, worksheet
• After class:

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 9.3: Convergence of Series. We discuss what it means for a series to converge and introduce the \(n\)th term test for divergence and the integral test.
• Class materials: Lecture notes, worksheet
• After class:

Wednesday

• Topic: Section 9.3: Convergence of Series, continued. We continue our discussion from last time.
• Class materials: Lecture notes, worksheet
• After class:

Friday

• Topic: Section 9.4: Tests for Convergence. 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:

Chapter 9

Monday

• Topic: Section 9.4: Tests for Convergence, 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:

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. Along the way we also introduce the Absolute Convergence Test.
• Class materials: Lecture notes, worksheet
• After class:

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:

Chapter 9

Monday

• Topic: Section 9.4: Tests for Convergence, continued. 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:

Wednesday

• 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, worksheet
• After class:

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, worksheet
• After class:

TBA

Monday

• Topic: TBA
• Class materials: Lecture notes, worksheet
• After class:

Wednesday

• Topic: Review.
• Class materials: Lecture notes, worksheet
• After class:

Friday

• Topic: Exam 2.
• After class:

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, worksheet
• After class:

Wednesday

• Topic: Section 10.2: Taylor series. We discuss power series that come about by considering Taylor polynomials with infinitely many terms.
• Class materials: Lecture notes, worksheet
• After class:

Friday

• Topic: Section 10.3: Finding and Using Taylor Series. 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:

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!