Schedule

Purpose of this Page

Here is a complete list of topics organized by week. As the semester progresses I add additional content, study suggestions, and descriptions. If you are in the course, you should bookmark this page and check back often.

For more information about the course, see our class homepage.

For a detailed schedule concerning official deadlines from the college, see the academic calendar.

Week 01 (08-21 to 08-27)

Course Introduction

Discussed the syllabus, and the goals of the course.

Homework 0: Send me a short email with the following information (due by 31 August at Midnight).

  • Who are you (preferred name, pronouns if you choose)? What is your major?
  • What brings you to my class? Is it your major? Are you just curious? Something else?
  • What do you hope to get out of this course?
  • Share one fun fact about yourself.

(My Fun Fact: I like to go fishing, and one time I caught a catfish that was this big! \*stretches arms really wide\*)

Calculus I Review

Here is a quick refresher on differential calculus. Also see our textbook for a brief overview of the ideas involved motivating the first half of this course!

Week 02 (08-28 to 09-03)

Antiderivatives

  • Definition of the antiderivative.
  • Turning the table of derivatives into a table of antiderivatives.
  • Simple substitution tricks.

We saw that the substitution technique is the anti-chain rule: \( \int f(u(x)) u'(x)\, dx = \int f(u)\, du \) provided we "unsubstitute" at the end.

See our textbook on antiderivatives for exercises and further explanations. In case those problems bore you, try this one too!

Challenge Problem: Compute \( \int x^5(x^3 + 4)^{20}\ dx \)

Riemann Sums

Approximating area (via left-hand endpoints).

Also see our textbook for exercises and further explanations.

Week 03 (09-04 to 09-10)

Homework 01 is due in class on Monday, 12 September 2022.

Note: The lecture on Friday, 9 September 2022 was asynchronous. I wrote videos (appearing below) to replace the usual lecture; office hours that day were cancelled.

The Riemann Integral

\( \displaystyle \int_{x = a}^b f\, dx = \lim_{n \to \infty} \sum_{k = 1}^n f\left(x_k\right) \, \Delta x \).

If we use the right-hand endpoints convention, this becomes: \( \displaystyle \int_{x = a}^b f\, dx = \lim_{n \to \infty} \sum_{k = 1}^n f\left(a + k \cdot \tfrac{b-a}{n}\right) \cdot \tfrac{b - a}{n} \).

Note: While this looks a bit much, we won't need to work directly with the definition for long.

I posted a video discussing the Riemann integral via its definition.

Fundamental Theorem of Calculus

If \( f \) is continuous on \( [a, b] \), then…

  1. the function \( F(x) = \int_{t = a}^x f(t)\, dt \) defined for \( a \leq x \leq b \) is an antiderivative of \( f \).
  2. for any antiderivative \( G \) of \( f \) we have \( \int_{x = a}^b f(x)\, dx = G(b) - G(a) \).

I wrote a video introducing the Fundamental Theorem of Calculus. Also see our textbook for exercises and further explanation on both part 1 and part 2 of the Fundamental Theorem of Calculus.

Note: Even though the textbook separates the Fundamental Theorem into two parts, it really is one big theorem. The two parts say that derivatives and integrals are "reverse" operations.

I wrote all the problems on the Fundamental Theorem of Calculus.

Week 04 (09-11 to 09-17)

Using the Fundamental Theorem of Calculus.

We discussed some uses of the Fundamental Theorem of Calculus in lecture. If you haven't already done so, have a look at our textbook. In case you're interested in seeing a bit more…

  • 3Blue1Brown explains the intuition behind the integral and Fundamental Theorem of Calculus.
  • Khan Academy does a good job explaining the Fundamental Theorem of Calculus.
  • Professor Dave Explains has a nice video explaining the Fundamental Theorem of Calculus.
  • The Organic Chemistry Tutor does a bunch of examples using the Fundamental Theorem of Calculus (in a slightly sus way).

I wrote all the problems on the Fundamental Theorem of Calculus.

Properties of the Definite Integral

  1. \( \int_a^b (f+g)\, dx = \int_a^b f\, dx + \int_a^b g\, dx \)
  2. \( \int_a^b cf\, dx = c\int_a^b f\, dx \) for all constants \( c \)
  3. \( \int_a^a f\, dx = 0 \)
  4. \( \int_a^b f\, dx = -\int_b^a f\, dx \)
  5. \( \int_a^b f\, dx = \int_a^r f\, dx + \int_r^b f\, dx \) for all \( a \leq r \leq b \)

Also see our textbook for exercises and further explanations.

Integration by Parts

We saw in class that this is the anti-product rule: \( \int u\, dv = uv - \int v\, du \). We also saw a bunch of examples and how finding a proper by-parts substitution often requires some ingenuity and thought.

I wrote some problems on integration by parts. Also see our textbook for exercises and further explanations.

Week 05 (09-18 to 09-24)

The lecture on Wednesday, 21 September 2022 was a review for the first exam. For reasons unknown, I gave extra office hours on Thursday, 22 September 2022.

Office hours on Friday, 23 September 2022 are cancelled (so I can grade your exams).

More Integration by Parts

Some harder integrals solved via Integration by Parts.

I wrote some problems on integration by parts. You can still visit our textbook for exercises and further explanations.

Midterm Exam 1

The exam was during the usual lecture time on Friday, 23 September 2022. You will have the full lecture to complete the exam.

Week 06 (09-25 to 10-01)

Exam 1 was returned on Monday, 26 September 2022.

Inverse Functions

  • One-to-one functions.
  • Using the First Derivative Test to decide if a function is one-to-one.
  • Inverse functions.
  • Prop: A function is one-to-one exactly when it has an inverse function.
  • Prop: If function \( f \) is continuous and one-to-one, then its inverse function is continuous as well.

Inverse Function Theorem

Let \( f \) be a function with \( f(a) = b \), and suppose \( f \) is differentiable at \( a \) with \( f'(a) \neq 0 \). Then \( f^{-1} \) is differentiable at \( b \) and satisfies the formula \( \displaymath (f^{-1})'(b) = \frac{1}{f'(a)} \).

Note: Another way to write this is \( \displaymath (f^{-1})'(b) = \frac{1}{f'(f^{-1}(b))} \).

We discussed the natural logarithm and exponential in the context of the Inverse Function Theorem. I wrote a guide to logarithms and exponentials which builds the all logarithmic and exponential functions from the ground up.

See our textbook for more information and exercises on inverse functions.

Week 07 (10-02 to 10-08)

Inverse Trigonometric Functions

  • Derivations.
  • Derivatives.
  • Antiderivatives involving inverse trigonometric functions.

Trigonometric Integrals

To compute integrals of the form \( \displaystyle{\int \cos^m(x) \sin^n(x)\, dx} \), i.e., those involving only sine and cosine to (nonnegative) powers:

  • If either \( m \) or \( n \) is odd, make a substitution for "the other one" and then apply trigonometric identities to complete the substitution.
  • If both \( m \) and \( n \) are even, apply power reduction formulas and then reassess.

The trigonometric identities you will need are listed below.

Circle
\( \sin^2(x) + \cos^2(x) = 1 \).
Sine Addition Formula
\( \sin(x + y) = \sin(x)\cos(y) + \sin(y)\cos(x) \).
Double Angle
When \( x = y \) above we obtain \( \sin(2x) = 2\sin(x)\cos(x) \).
Cosine Addition Formula
\( \cos(x + y) = \cos(x)\cos(y) - \sin(x)\sin(y) \).
Double Angle
When \( x = y \) above we obtain \( \cos(2x) = \cos(x)^2 - \sin^2(x) \).
Power Reduction
Applying the circle identity and the cosine double angle formula, we obtain the following formulas for reducing powers.
Cosine Power Reduction
\( \cos^2(x) = \tfrac{1}{2}(1 + \cos(2x)) \)
Sine Power Reduction
\( \sin^2(x) = \tfrac{1}{2}(1 - \cos(2x)) \)

I wrote some problems on trigonometric substitution. See our textbook for more details and practice problems.

Week 08 (10-09 to 10-15)

Trigonometric Integrals

To compute integrals of the form \( \displaystyle{\int \sec^m(x) \tan^n(x)\, dx} \):

  • If \( m \) is even, try \( u = \tan(x) \) and apply identities to complete the substitution.
  • If \( n \) is odd, try \( u = \sec(x) \) and apply identities to complete the substitution.
  • If \( m \) is odd and \( n \) is even, be clever… Note that this type is often quite difficult to compute.

In addition, I suggest you learn the following as a rule:

  • \( \displaystyle{\int \sec(x)\, dx = \ln|\sec(x) + \tan(x)| + C} \).

I wrote some problems on trigonometric substitution. See our textbook for more details and practice problems.

Inverse Trigonometric Substitutions

  • Tangent substitutions.
  • Sine substitutions.
  • Secant substitutions.

I wrote some problems on inverse trigonometric substitution. See our textbook for more details and practice problems.

Week 09 (10-16 to 10-22)

There is no class on Monday, 17 October 2022 for Fall break. Do travel safely and enjoy the holiday!

Don't forget to view the homework page daily!

Partial Fractions

  • To integrate a function of the form \( \displaystyle{f(x) = \frac{n(x)}{d(x)}} \) where \( n, d \) are polynomials and \( \deg(d) < \deg(n) \)…
    1. Factor \( d(x) \) completely as a product of powers of irreducible polynomials.
    2. Set up a partial fractions equation using your factorization.
    3. Turn your partial fractions equation into a linear system of equations.
    4. Solve the linear system for the new coefficients.
    5. Integrate the result of the partial fractions equation using techniques from earlier in the course.

I wrote some problems on partial fractions for you to try. See our textbook for more details and practice problems.

Week 10 (10-23 to 10-29)

Because I am foolishly kind, I offered to give extra office hours on Thursday, 27 October 2022 (details communicated on email).

More Practice with Recent Techniques

Don't forget to view the homework page daily! The problems page also exists, so you should look at that one maybe?

I know at this point that telling you about the textbook is a losing battle, but it really is a good resource.

Midterm Exam 2 (Friday, 28 October)

The exam is during the usual lecture time on Friday, 28 October 2022. You will have the full lecture to complete the exam.

Week 11 (10-30 to 11-05)

Infinite Sequences

  • Definition of sequence
  • Many example sequences
  • Limits of sequences
  • Convergence and divergence
  • Let \( (a_n) \), \( (b_n) \), and \( (c_n) \) be sequences. If \( a_n \leq b_n \leq c_n \) for all \( n \) and \( a_n, c_n \to L \) as \( n \to \infty \), then \( b_n \to L \) as \( n \to \infty \).
  • A sequence \( a_n \) is…
    • bounded above when there is an \( M \) such that \( a_n \leq M \) for all \( n \).
    • bounded below when there is an \( m \) such that \(m \leq a_n \) for all \( n \).
    • bounded when it is both bounded above and bounded below.
  • A sequence \( a_n \) is…
    • monotone increasing when \( a_n \leq a_{n+1} \) for all \( n \).
    • monotone decreasing when \( a_n \geq a_{n+1} \) for all \( n \).
    • monotone when it is either monotone increasing or monotone decreasing.
  • Let \( (a_n) \) be a sequence.
    1. If \( (a_n) \) is monotone increasing and bounded above, then \( (a_n) \) converges.
    2. If \( (a_n) \) is monotone decreasing and bounded below, then \( (a_n) \) converges.

See Apex Calculus for introductory material on sequences. Also see the problems page for practice problems, and please practice your limits!

Introduction to Infinite Series

  • Convergence and Divergence
  • Geometric Series
  • Telescoping Series

Look at Apex Calculus for more discussion and exercises. Also see the textbook for further examples and exercises.

Finally, this video has a nice geometric interpretation of the geometric series.

Week 12 (11-06 to 11-12)

Divergence Test

Divergence Test
If \( \sum a_n \) converges, then \( a_n \to 0 \) as \( n \to \infty \).
(no term)
We usually apply the Divergence Test in the following way (the "contrapositive"):
Divergence Test
If \( \lim_{n \to \infty} a_n \neq 0 \), then \( \sum a_n \) diverges.
(no term)
The Divergence Test makes no promises when \( \lim_{n \to \infty} a_n = 0 \).

Look at Apex Calculus for more discussion and exercises. Also see the textbook for further examples and exercises.

Improper Integrals

An improper integral is a definite integral \( \int_a^b f\, dx \) where the interval \( [a, b] \) contains finitely many values not in the domain of \( f \). This can mean one of two things:

Finite Type
There may be a \( c \) with \( a < c < b \) and \( c \) is not in the domain of \( f \).
Infinite Type
One of the endpoints may be infinite.

For the purposes of our course, we will focus on the infinite type. However, everything we say here applies equally well for finite type.

We can use the basic properties of definite integrals to break an improper integral into pieces having one non-domain point each, at one end of the interval. Then, we define the value of an improper integral to be the limit along the interval towards the non-domain point. In particular, if \( b \) is the non-domain point, then we define \( \displaystyle{\int_a^b f\, dx = \lim_{t \to b} \int_a^t f\, dx} \). For example, \( \displaystyle{\int_1^\infty \frac{1}{x}\, dx} = \lim_{t \to \infty} \int_1^t \frac{1}{x}\, dx \).

See Apex Calculus for more discussion and exercises.

Integral Test

Integral Test
If \( f \) is a weakly decreasing function on \( [a, \infty) \) and \( f(x) \geq 0 \) for all \( x \), then the improper integral \( \displaystyle{\int_a^\infty f\, dx} \) converges exactly when the infinite series \( \displaystyle{\sum_{n=a}^\infty f(n)}\) converges.
(no term)
Note: The Integral Test does not tell us what the series converges to (if it does). It is purely a convergence test…

By using the Integral Test, we can deduce the following (as we did in class).

\( p \)-Series Test
If \( p > 1 \), then \( \sum_{n = 1}^\infty \frac{1}{n^p} \) converges. If \( p \leq 1 \), then \( \sum_{n = 1}^\infty \frac{1}{n^p} \) diverges.

Unfortunately, this YouTube channel cannot help you understand \( p \)-series.

Look at Apex Calculus for more discussion and exercises.

Week 13 (11-13 to 11-19)

Comparison Tests

Look at Apex Calculus for more discussion and exercises. Also see the textbook for further examples and exercises.

Direct Comparison

If \( 0 \leq a_n \leq b_n \) for all \( n \), then \( \sum a_n \leq \sum b_n \). In particular, if \( \sum b_n \) converges, then \( \sum a_n \) converges.

Note that this also means that if \( a_n \) diverges, then \( b_n \) diverges! The converse of the test fails. As with all tests, a failure means we cannot conclude anything!

We use this test when we understand the convergence/divergence of one of the series in question via some other test. Here are a few nice use-cases.

  • The series \( \sum_{n = 1}^\infty \frac{1}{n^2 + n + 1} \) satisfies \( 0 \leq \frac{1}{n^2 + n + 1} \leq \frac{1}{n^2} \) for all \( n \geq 1 \) (why?). Because \( \sum_{n = 1}^\infty \frac{1}{n^2} \) converges by the \( p \)-series test, we conclude that \( \sum_{n = 1}^\infty \frac{1}{n^2 + n + 1} \) converges by the Direct Comparison Test.
  • The series \( \sum_{n = 1}^\infty \frac{1}{n - \ln(n)} \) satisfies \( 0 \leq \frac{1}{n} \leq \frac{1}{n - \ln(n)} \) for all \( n \geq 1 \) (why?). Because \( \sum_{n = 1}^\infty \frac{1}{n} \) diverges by the \( p \)-series test, we conclude that \( \sum_{n = 1}^\infty \frac{1}{n - \ln(n)} \) diverges by the Direct Comparison Test.

Limit Comparison

Suppose \( 0 \leq a_n \) and \( 0 \leq b_n \) for all \( n \). If \( \lim_{n \to \infty} \frac{a_n}{b_n} = L \) is convergent and \( L \neq 0 \), then \( \sum a_n \) and \( \sum b_n \) either both converge or both diverge.

The Limit Comparison Test fails if \( L = 0 \) or \( L = \infty \). The converse of the test also fails. As with all tests, a failure means we cannot conclude anything!

We use this test when we understand the convergence/divergence of one of the series in question via some other test, but a direct comparison would be difficult. As practice, repeat the two examples above using the Limit Comparison Test instead of the Direct Comparison Test.

Every Limit Comparison can be converted into a Direct Comparison, but not conversely.

  • Use a Direct Comparison to show that \( \sum_{n = 1}^\infty \frac{1}{\ln(n+1)} \) diverges, comparing with \( \sum_{n=1}^\infty \frac{1}{n} \).
  • Show that the corresponding Limit Comparison fails!

Alternating Series

  • Alternating Series Test
  • Absolute convergence
  • Conditional convergence

Look at Apex Calculus for more discussion and exercises.

Week 14 (11-20 to 11-26)

There are no classes on Wednesday, 23 November 2022 and Friday, 25 November 2022 for Thanksgiving Break. We used the lecture on Monday, 21 November 2022 for an interactive problem session.

Make sure to see the homework page.

Week 15 (11-27 to 12-03)

Root and Ratio Tests

  • Root Test
  • Ratio Test

Look at Apex Calculus for more discussion and exercises. Also see the textbook for further examples and exercises.

Power Series

  • Power series
  • Radius of convergence
  • Interval of convergence

Look at Apex Calculus for more discussion and exercises. Also see the textbook for further examples and exercises.

Week 16 (12-04 to 12-10)

There is a review session for the final exam on the reading day (Thursday, 8 December 2022 at 14:00–Thursday, 8 December 2022 at 16:00 in Woods 121). This is entirely optional; I gave this option because I am extremely kind.

Maclaurin–Taylor Series

  • Functions represented by power series
  • Taylor polynomials
  • Taylor–Maclaurin series

Look at Apex Calculus for more discussion of Taylor polynomials and Taylor–Maclaurin series, as well as exercises. See the textbook for further examples and exercises.

Final Exam

The final exam is to be given on the following days and times. See the official final exam schedule; in the event of a disagreement, that page supersedes this one!

At the time of the final, students must submit the study guide assigned over the course of the last two weeks. This includes the assigned items below.

  • List of topics
  • List of important propositions
  • Collection of worked examples

Section A (Meetings on MWF at 9am)

The exam will be Saturday, 10 December 2022 at 09:00, for courses in Canonical Hour B.

Section B (Meetings on MWF at 10am)

The exam will be Monday, 12 December 2022 at 09:00, for courses in Canonical Hour C.

Course Completed

Grades have been posted, and the course has finished. Thanks for a great semester!

This page is no longer updated.

Last update: Thursday, 15 December 2022.