Class 01: First Day Discussion

Instructor Information

Professor: Kris Hollingsworth (Student’s generally call me Dr. $\mathcal H$ )
Email: kristopher.hollingsworth@mnsu.edu
Office: Wissink Hall, 253 (See Building Floor Plans for campus building maps, but it is right across the hall from our classroom.)

Course Information

University: Minnesota State University, Mankato
Class: Math 122: Calculus 2 (4 credit-hours)
Location: Wissink Hall, 286A (See Building Floor Plans for campus building maps.)
Course Website: D2L at MNSU
- Syllabus, assignments, and other course documents will be found there.
- The site will link back to this page for lecture notes.

Additional Student Support Resources

A full schedule of support resources provided by the university will be posted in the first or second week of the semester once schedule’s are finalized. You can always find up to date information for each on their individual websites though.

Instructor’s Office Hours :: A survey will be given out the first week of class to aid in scheduling office hours when you can actually attend them. Note, this should be your primary resource throughout the semester to get real-time feedback on how you are doing with the content in the course.
MavPASS Support :: This course is supported by a MavPASS leader. Specific information about your leader this semester will be provided in class, including their session times and office hours. The MavPASS leader is a current undergraduate student at MNSU-Mankato who has taken the course previously and is present to support you and help you succeed. Get to know them, they are a valuable resource in the course!
Center for Academic Success :: The Center for Academic Success employs tutors in many topics, including mathematics, to support students in their academic journey. They have drop-in support in the Library regularly scheduled throughout the week, and also offer Zoom sessions which can be scheduled through their website.
Mathematics and Statistics Learning Center (MLC) :: The Mathematics & Statistics Learning Center offers tutoring on a walk-in basis for our undergraduate courses. Staffed by Math/Stat majors and graduate students, the center provides a welcoming place to work with either tutors or your peers on your assignments. The MLC is located in Wissink Hall 285.
TRIO Student Support Service :: TRIO is a federally funded program that can provide one-on-one peer tutoring. Your instructor was a TRIO tutor when they were an undergraduate. They primarily serve first-generation and PELL-Eligible students. Students that have joined their program can request a tutor for many courses. Choose 1-3 hours weekly with a trained one-on-one peer tutor who has been successful in the course.

Today’s Goals

Your turn (Roll Call of Rostered Students)
- Please let me know if your name was not called so we can ensure you are properly registered for the course. As students can freely add/drop classes throughout the first week, there might be a lot of variation over the first few days of the course, so just let me know if you join our section!
What is Calculus?
Course Description and Textbook Information
A Brief Timeline of Calculus
Why Calculus?
Discussion of how to learn.

Today’s Assignments

Note, you will have daily assignments for the first two weeks of the course. This is intended to help you build habits of daily practice in the course, as this will be essential to your overall success in the course. As a 4-credit hour course, you are expected to spend 8-12 hours outside of class each week in preparation and study for the course. After the first two weeks, you will be expected to shift your focus towards being prepared in class and completing the Learning Problem sets posted in D2L. Additionally, this allows for a significant amount of direct instructor feedback on your mathematical writing in the first week’s of the course to determine if you are communicating clearly as well as addressing any concerns in background knowledge.

First Assignment, Guided Practice 01: Student Survey and Review of Calculus I Memorized Facts. Due tomorrow.
Assignment 00: Support Hours Schedule Survey Due Friday. Take time to ensure you know your schedule before submitting!

What is Calculus?

You’re in your second semester of Calculus, so your 5-year old second cousin thrice removed (or other random relative) walks up and asks you >What is Calculus? Why are there 3 of them? You sure must like it a lot to take 3 of them! Maybe I’ll take it when I grow up like you! But what is it?

What do you tell them?

In Class Discussion

Think, pair, share. (30 seconds, 1 minute, 2 minutes)

Course Description

The official course description of Calculus II from the catalogue is: > Techniques of integration, applications of integration, improper integrals, the calculus of parametric curves, and infinite seires and sequences. > > Prerequisites: MATH 121 with ‘C’ or better or consent.

This isn’t particularly meaningful when you don’t know what any of those things mean yet. So roughly speaking, our semester is organized to study the following questions (in order): 1) How can the relationship between integrals and derivatives help us model real-world phenomena? Moreover, how can integration help us calculate useful quantities? 2) What can we do when the anti-derivative is not obvious when trying to calculate something from Question 1? 3) How do we work with curves that are not functions or perform calculus in other coordinate systems? 4) How do we deal with quantities involving infinity that somehow result in finite, meaningful answers? 5) Finally, when our techniques from answering Question 2 fail (or are impossible), can we still find meaningful answers to calculus questions? If so, how?

We’ll spend about the next 16 weeks working on answering these questions, even though we’ll only have about 48 hours together in this room.

Recommended Textbook and Other Resources

Below is a list of official, recommended, and supplemental resources (with links):

Official Textbook: Stewart’s Essential Calculus
- Note: This text is recommended primarily if you plan to take Calculus III, as it is commonly required for that course as well, so you can get familiar with it now.
Free Online Resource: Paul’s Online Notes
- A widely used and student-friendly resource that closely follows the structure and content of Stewart’s Calculus. I sometimes take exam questions for examples on this website.
Low-Cost Alternative Textbook: APEX Calculus
- An Open Educational Resource (OER), meaning it is freely available online or as a downloadable PDF. Inexpensive print copies are also available (approximately $15).

In practice, most standard calculus textbooks will work well for this course. The primary consideration when using a resource other than the official textbook is that the order of topics may differ. As a result, you may occasionally encounter concepts or computations that seem unfamiliar at first—we will cover them later.

Keep in mind that all courses are context-sensitive. When in doubt, refer back to these lecture notes to track what material has and has not yet been covered.

Why is this worthwhile?

Point 1: From the Perspective of Human History

A (Brief) Timeline of Calculus

A (Brief) Timeline of Calculus
Approximate Date	Milestone in the Development of Calculus
c. 5th century BCE	Discovery of irrational numbers (Pythagoreans): Recognition that not all quantities can be expressed as ratios of integers, challenging earlier notions of number and continuity.
c. 3rd century BCE	Archimedes’ Method of Exhaustion: A precursor to integration, used to compute areas and volumes by approximating with inscribed polygons and taking limits conceptually.
9th century CE	Thābit ibn Qurra: Computed results equivalent to integrating functions like $\sqrt{x}$ , extending Greek geometric methods into early integral-like reasoning.
11th–13th centuries	Development of time measurement (minutes/seconds): Increasingly precise timekeeping enabled later scientific notions of instantaneous velocity and acceleration.
Early 17th century (c. 1640s)	Grégoire de Saint-Vincent: Advanced early ideas related to limits while studying areas (notably the hyperbola), helping move toward logarithmic and limiting processes.
Mid 17th century (c. 1650s–60s)	John Wallis: Introduced modern power notation, worked with infinite series, and developed general formulas for integrals of polynomial-like expressions.
Late 17th century (1660s–1680s)	Newton and Leibniz: Independently developed infinitesimal calculus; Leibniz’s notation $\left(\int, \frac{d}{dx}\right)$ ultimately became standard.
Late 17th–early 18th century	Bernoulli family: Studied early differential equations and applied calculus to physical problems (e.g., brachistochrone problem).
18th century	Leonhard Euler: Systematized calculus, popularized function notation $f(x)$ , and expanded the use of infinite series and differential equations.
Early 19th century (1820s–30s)	Augustin-Louis Cauchy: Provided rigorous foundations for limits, continuity, and convergence; extended analysis into the complex plane.
Mid 19th century (1850s–70s)	Karl Weierstrass (and others): Formalized the $\varepsilon$ - $\delta$ definition of limits and continuity, removing reliance on intuition about infinitesimals.
Mid 19th century (1854 onward)	Bernhard Riemann: Developed the formal definition of the definite integral (Riemann integral), forming the basis for integration theory studied in this course.

Your Intellectual Inheritance

Mathematics (including calculus) has developed in more or less continuous form since the Ancient Greeks. While its foundations have been refined and expanded, core results from antiquity remain meaningful tools in modern study rather than historical curiosities.

This continuity makes mathematics one of the most enduring intellectual traditions in human history—an inheritance stretching back millennia.

By contrast, most other scientific disciplines have undergone fundamental paradigm shifts that effectively replaced earlier worldviews. What was once “correct” is often now only of historical interest. A few defining revolutions include:

Medical Sciences: Germ Theory of Disease (replacing miasma theory and humoral medicine)
Astronomy: The Copernican Revolution (heliocentrism replacing geocentrism)
Physics: Quantum Theory (discrete probabilistic behavior replacing classical determinism at small scales)
Physics: Einstein’s Theory of Relativity (space and time as relative rather than absolute)
Biology: Evolution by Natural Selection (Darwin and Wallace replacing static species doctrine)
Genetics: DNA as Hereditary Material (molecular basis of inheritance replacing blending theories)
Chemistry: Atomic Theory and the Periodic Table (matter composed of discrete atoms with structured relationships)
Geology: Plate Tectonics (dynamic Earth replacing fixed-continent models)
Psychology: Cognitive Revolution (information processing models replacing strict behaviorism)

In most sciences, entire frameworks have been rebuilt from the ground up, with old models being completely thrown out. Mathematics is unusual in that it accumulates rather than replaces—older ideas are rarely discarded, only deepened, generalized, or reinterpreted.

Point 2: From the standpoint of the Academy

The general importance to scientific inquiry is well illustrated by the number of courses and programs requiring it to progress. The first is a list of classes requiring a ‘C’ or better, and the second is a list of degree programs with the same requirement as ‘Common Core’.

Courses listing Calc II as a Prerequisite

AST 225 — Astronomy and Astrophysics II
CHEM 101 — General Chemisty Skills
CHEM 190 — Learning in Chemistry Applications
CHEM 191 — Chemistry Applications
CHEM 446 — Physical Chemistry II
ME 281 — Computing Techniques For Mechanical Engineering Analysis
ME 291 — Engineering Analysis
ENGR 240 — Linear Circuit Analysis and Design
MATH 223 — Calculus III
MATH 247 — Linear Algebra I
MATH 290 — Foundations of Mathematics
MATH 321 — Ordinary Differential Equations
MATH/STAT 354 — Concepts of Probability & Statistics
MATH 461 / 561 — Mathematical Theory of Interest
MATH 470 / 570 — Numerical Analysis I
MATH 528 — Linear Optimization Methods
PHYS 335 — Modern Physics I
PHYS 457 — Optics
PHYS 465 / 565 — Computer Applications in Physics

List of Degree Programs

Automotive Engineering Technology (BS)
Biochemisty (BS)
Chemistry - ACS Approved (BS)
Civil Engineering (BSCE)
Computer Engineer (BSEC)
Computer Science (BS)
Electrical Engineering (BSEE)
Integrated Engineering (BSE)
Manufacturing Engineering Technology (BS)
Mathematics (BS)
Mathematics Teaching (BS)
Mechanical Engineering (BSME)
Physics (BS)
Software Engineering (BS)
Statistics (BS)
Statistics (BS), Actuarial Track

Point 3: From a personal, cognitive development viewpoint.

The following is a summary of one of the common models of cognitive development in students.

Bloom’s Revised Taxonomy (Cognitive Domain)

Remember

Retrieve relevant knowledge from long-term memory.
Examples: list, define, recall, identify

Understand

Construct meaning from instructional messages.
Examples: summarize, explain, interpret, classify

Apply

Use knowledge in new or concrete situations.
Examples: solve, compute, demonstrate, use

Analyze

Break material into parts and determine relationships or structure.
Examples: compare, organize, differentiate, examine

Evaluate

Make judgments based on criteria and standards.
Examples: justify, critique, assess, defend

Create

Put elements together to form a new whole or original product.
Examples: design, construct, develop, formulate

Brief Discussion of How to Learn this Semester

Learning mathematics requires active engagement—working through ideas in real time, testing your understanding, and responding to feedback. One of the simplest ways to check your progress is to verify your answers and carefully review your work to identify mistakes. This process—attempt, check, revise—is central to improvement.

At some point, however, you will encounter problems where you cannot move forward on your own. When this happens, it is essential to seek real-time feedback from another person. This is how gaps in understanding are identified and corrected effectively.

While tools such as videos or chatbots can be helpful supplements, they can also create the illusion of understanding without requiring you to fully process the material. There is no shortcut around grappling with confusion—the struggle itself is a necessary part of learning and building long-term retention.

You will get stuck.

If you have worked on a problem for about 4–5 minutes without meaningful progress, mark it and move on. After completing the rest of your work, bring those questions to office hours or MavPASS. Be prepared to show what you attempted and where you became stuck.

This approach—focused effort followed by targeted feedback—is far more effective than passively “getting through” problems. It builds the ability to recall, understand, apply, analyze, evaluate, and create ideas independently, which is exactly what you will need for exams, group work, future courses, and in your careers.

Some Videos on Learning

How to Make it Through Calculus (Neil deGrasse Tyson) > Famous science communicator and physicist Neil deGrasse Tyson talks about how to make it through Calculus and provides it as an allegory for all learning.

Never Leave the Basics > A consultant for the NFL talking about the importance of the fundamentals.

Jamie Foy practicing > Professional skateboarder Jamie Foy working on a long nosegrind.

Prompt

Why are the second two videos relevant to your learning in this class?

Next Class

Go over course schedule, assessment, and grading. Then we’ll start Section 7.7) Ordinary Differential Equations.