Welcome to Calculus I. For many of you, this is your first math course in college, and I want to start by acknowledging that. It's completely normal to have forgotten some concepts over the summer, or to feel that you never quite mastered them in the first place. That's okay. The goal here is to build a strong foundation together.
To that end, this first week will be a review of essential pre-calculus topics. However, this review won't be comprehensive. A key part of learning at the college level is taking ownership of your education. After this week, you are responsible for filling in any gaps in your pre-calculus knowledge as they arise. I am always here to point you to resources, but the initiative must be yours.
Doing college math is different than in high school. We stress deep understanding over formula memorization. You will not be successful if you simply try to memorize formulas and homework problems without understanding the 'why' behind them. My goal is to help you become a critical thinker and a problem solver.
Each class is 110 minutes long. We will typically spend 30 minutes on a new topic, 20 minutes on examples, and the remaining 60 minutes on an in-class active learning assignment where you'll work on problems in groups.
When you are in class, you must participate. Using your phone or talking about non-math topics is disrespectful to your classmates who are trying to learn. It can make them feel inferior or that their efforts are not valued. This is an open, equitable classroom where everyone should feel comfortable learning. Please, be considerate of your fellow students.
We begin with the building blocks of calculus: the real numbers. This is the set of all numbers that can be represented on a continuous line. We denote this set with the symbol $\mathbb{R}$, or sometimes $\mathbb{R}^1$ to emphasize it's one-dimensional.
| Integers | Rational Numbers | Irrational Numbers |
|---|---|---|
| -5 | $\frac{1}{2}$ | $\pi$ |
| 0 | -0.75 | $\sqrt{2}$ |
| 7 | $\frac{10}{3}$ | $e$ |
| -100 | 2.0 | $\sqrt{17}$ |
The symbols $\infty$ (infinity) and $-\infty$ (minus infinity) are not numbers themselves. They are concepts representing the idea that the real numbers continue without end in the positive and negative directions.
We can visualize the entire set of real numbers using a number line.
One of the most fundamental tools for calculus is the line. The concept of a tangent line is central to differential calculus.
Lines have the general form $y = mx + b$, where $m$ is the slope of the line, and $b$ is the y-intercept.
The slope of a line determines its steepness and direction.
There are different ways to find the equation of a line. While the slope-intercept form is useful, it's more common in calculus to need to find the equation of a line given two points.
In the Cartesian coordinate system ($\mathbb{R}^2$), each point is an ordered pair $(x, y)$. For example, the point $P(3,2)$ is located at $x=3$ and $y=2$.
To find the equation of the line passing through points Q and P, we first compute the slope, which measures the steepness. A key property of lines is that the slope between any two points on the line is always the same.
The slope $m$ is the "rise over run", or the change in y ($\Delta y$) divided by the change in x ($\Delta x$).
$$ m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 1}{3 - 1} = \frac{1}{2} $$
With the slope, we can use the point-slope formula, $y - y_1 = m(x - x_1)$, to find the equation.
$y - 2 = \frac{1}{2}(x - 3)$
$y - 2 = \frac{1}{2}x - \frac{3}{2}$
$y = \frac{1}{2}x + \frac{1}{2}$
$y - 1 = \frac{1}{2}(x - 1)$
$y - 1 = \frac{1}{2}x - \frac{1}{2}$
$y = \frac{1}{2}x + \frac{1}{2}$
As you can see, both points yield the same equation, as expected.