I: The Beginning

The Approach

Some of mathematics works like this: you get some rules and properties, and you are expected to see when and how to use them accurately.

Problems can be extremely complicated and it can be hard to identify when to use what property.

To accurately do this in calculus, you need to be able to identify terms and factors.

Terms and Factors

Terms are entities separated by subtraction and addition.
Factors are entities separated by multiplication.

When identifying terms and factors, you need to specify which context you are working with.

Identify if you have terms or factors in the global context.
  1. $x + 2$
  2. $(x+2)(x+4)$
  3. $(x+2)(x+4) - 3$
In what context would $(x+4)$ be considered a factor in the expression \[(x+2)(x+4) - 3\]

For fractional expressions, only consider the global context of just the numerator and denominator when labeling terms/factors.

Identify if you have terms or factors in the global context.
  1. $\dfrac{(x^2 + 1)(x + 3) + 4}{x + 3}$
  2. $\dfrac{-x(x-2)(2x+3)}{2x + 3}$

Many common mistakes are due to not identifying terms/factors, therefore applying the wrong property or even applying the property incorrectly.

Let's see why!

Dealing with Fractional Expressions

Here are the five ways to deal with Fractions.

These are the only ways to manipulate fractions.

Properties of Fractions
Suppose $a,b,c,d$ are real numbers.
  1. $\dfrac{a}{b}\cdot \dfrac{c}{d} = \dfrac{ac}{bd}$
  2. $\dfrac{a}{b}\div \dfrac{c}{d} = \dfrac{a}{b}\cdot \dfrac{d}{c}$
  3. $\dfrac{a}{c} + \dfrac{b}{c} = \dfrac{a + b}{c}$
  4. $\dfrac{a}{b} + \dfrac{c}{d} \ \ $ first find the LCD, then use property 3.
  5. $\dfrac{ac}{bc} = \dfrac{a}{b}$

We can already see the importance of terms and factors.

In Fraction Law #5, $c$ must be a factor in the global context of the numerator and denominator.

Can I cancel out the $x^2$ in \[\dfrac{x^2 + 1}{x^2 + 2}\] Why or why not?
Can you cross out the $x-1$ in \[\dfrac{(x-1)(x+2) + 3x^2}{(x-1)(x+3)}\] Why or why not?
Can you cross out the $x$ in \[\dfrac{x(x-1)(x+1)}{2x + 1}\] Why or why not?
Use the proper fraction law to simplify the following:
  1. $\dfrac{1}{6} + \dfrac{1}{8}$
  2. $\dfrac{1}{x + 1} + \dfrac{1}{x + 2}$
  3. $\dfrac{(x-1)(x+2)(x-3)}{x-1}$
  4. $\dfrac{x+1}{\frac{x+1}{3x - 4}}$
We will fully deal with all fractional expressions in a few days.

Properties of Real Numbers

Typically you'll be given this list of properties of real numbers.

Let $a, b, c$ be real numbers.

How to think about commutative property: Say you have the expression \[(x-1)(x+2) + 3x^2(x-3) + 4\]

The addition property says you can switch the positions of the terms around.

The multiplication property says within each term, you can switch the positions of each factor around.

Using terms and factors, describe what the distributive law \[a(b + c)\] is describing.
Use the distributive property to expand $(x - 1)(x + 2)$.

Negatives

A negative expression $-(x-2)$ has a very specific meaning: $(-1) \cdot (x-2)$.

This means a negative is really a factor of $-1$ in front of the $x - 2$.

Properties of Negatives
Suppose $a,b$ are real numbers.
  1. $(-1)a = -a$
  2. $-(-a) = a$
  3. $-(a+b) = -a-b$
Use the properties of negatives to simplify the expression:
  1. $4-(x+2)$
  2. $-(-(-3))$
  3. $-(x+y-z)$

The Real Line

The rest of this lecture has a recorded video, which you can follow along here.

The real numbers can be represented with a horizontal line, drawn in class.

Sets and Intervals

A set is a collection of objects. These objects are called elements of the set.

For example, the set of all real numbers is denoted $\mathbb{R}$. If $a$ is a real number, we write $a \in \mathbb{R}$ meaning $a$ is an element of the real numbers.

Other common sets are

  1. $\mathbb{N}$, the set of natural numbers
  2. $\mathbb{Z}$, the set of integers
  3. $\mathbb{Q}$, the set of all rational numbers

There are two ways to describe a set.

The first is to list out all the elements: \[A = \{1,2,3,4,5,6\}\]

The second is to use set-builder notation: \[A = \{\text{objects} \ : \ \text{condition}\}\]

Set $A$ above could also be described as \[A = \{x : x \in \mathbb{Z} \text{ and } 0 < x < 7\}\]

If $S$ and $T$ are sets, there are two operations:

  1. union, denoted $S \cup T$, which is the set that contains all elements that are in $S$ or $T$
  2. intersection, denoted $S \cap T$, which is the set that contains all elements that are in $S$ and $T$

If \[S = \{1,2,3,4,5\} \qquad T = \{4,5,6,7\} \qquad V = \{6,7,8\}\] find
  1. $S \cup T$
  2. $S \cap T$
  3. $S \cap V$

Sets of real numbers are called intervals. I will describe them in class.

Draw the real line representation of the following sets:
  1. $(1,3) \cup [4,7]$
  2. $(-\infty,-3) \cup (3, \infty)$
  3. $(-\infty, -2) \cup (0, 1) \cup (1, \infty)$
Use set builder notation to describe the set "all real numbers except for 1 and 4."