Calculus I Lesson 0: Welcome!

1. Introduction
2. Warm-up
3. Administrative Notes
   1. Class Format
   2. Office hours
   3. Studying / workload
   4. Study groups
   5. Grading
4. Functions
   1. Describing Functions
   2. Side Note
5. Classes of Functions
   1. Bounded above
   2. Bounded Below
   3. Neither
   4. “Hole”
   5. Asymptote
6. Classes of Functions
   1. Behavior examples
7. Other Behaviors?
8. Homework

To miss out on mathematics is to live without an opportunity to play with beautiful ideas and see the world in a new light. To grasp mathematical beauty is a unique and sublime experience that everyone should demand. All of us—no matter who you are or where you’re from—can cultivate mathematical affection.

(Francis Su, Mathematics for Human Flourishing)

Introduction

Welcome to Calculus I! Let’s start with a warm-up activity:

Warm-up

Link to warm up activity

Administrative Notes

Now for the syllabus…

Class Format

Somewhat of a hybrid format:

• Some lessons: traditional lectures.
• Some lessons: no lecture, but there will be notes, textbook readings, videos, problems to go through on your own.
• In lieu of lectures: those days we will have “small group meetings”. I will make groups and set times for people to come.
• First set of meetings: this Thursday. I will post groups by Thursday morning on BrightSpace.

Office hours

Mondays and Thursdays, 9:15 AM

• Don’t feel like you’re on your own in this class.
• Be in touch with me often

Studying / workload

• 4 credits: 4 hours in class, 8 hours out of class (individual studying, homework, etc)
• Lecture videos, reading notes, textbook all help, but most importantly: do lots of practice problems.

Study groups

• Having a social structure around you helps you stay on task and stay motivated
• Form study groups with other students in the class
  • Quiz each other about lectures / notes
  • Keep each other on track
  • Share work on exercises
  • Help each other with homework

Grading

The syllabus, posted on BrightSpace, lists out the grading scheme. Pay attention to this breakdown: there will be many different forms of assessing your work:

• Homework and quizzes (30%)
• Classwork / participation / group work (5%)
• Exams: best 2/3 x 10%, plus final exam (25%)
• Problem presentations (2 x 5%)
• Final paper (10%)

Functions

Function: A relationship between two variables, denoted $x$ and $y$, where different $y$-values cannot be related to the same $x$-value

\(f : X \to Y\) is read “f is a function with domain $X$ and codomain $Y$”. In Calc 1 - 2: usually sets of real numbers (usually intervals).

Describing Functions

How do we describe functions? (At least 4 ways):

1. Algebraically: with an equation / algebraic expression
2. Visually: with a graph
3. Numerically: with a table
4. Verbally: “the function which divides its input by 2 and then squares the result”

Other ways?

Side Note

Most functions cannot be described efficiently.

…but it’s usually easier if we do have some formula.

Q: What do scientists / engineers do with real data sets?

Classes of Functions

Exercise: Come up with examples of the following:

1. A function which is “bounded above” (its graph does not keep going “up” to positive $\infty$)
2. A function which is “bounded below” (its graph does not keep going “down” to negative $\infty$)
3. A function which is neither bounded above nor below
4. A function with a “hole”
5. A function with an “asymptote”

Bounded above

$y = 4 - x^2$

Bounded Below

$y = x^4$

Neither

$y = x^3 - x$

“Hole”

$y = \frac{x^2-1}{x-1}$

Asymptote

$y = \frac{x^2+1}{x-1}$

Classes of Functions

The functions we have seen were all either polynomials or rational functions.

• Polynomials: sums of functions of the form $a x^n$ ($a$ is a constant real, $n$ a non-negative integer)
• Rational functions: quotients of polynomials

They give us good examples of “well-behaved” functions (functions whose behavior follows certain rules).

Behavior examples

• if the degree of a polynomial is even, it will either be bounded above or below. (Determined by leading coefficient)
• if the degree of a polynomial is odd, it will be unbounded above and below
• a rational function has an asymptote if …
• a rational function has a hole if …

Other Behaviors?

• $y = \sin(x)$
• $y = e^x$
• $y = \arctan(x)$

Homework

• DeltaMath: create your account, first homework is already posted
• BrightSpace: short responses + video