Chapter 1: Prerequisites for Calculus – Lay the Foundation

This chapter focuses on filling gaps in foundational math knowledge. It uses clear definitions, step-by-step examples, and targeted practice to ensure you’re fully prepared for core calculus topics later.

1.1 Core Algebraic Operations

Algebra is the "language" of calculus. Mastering these operations ensures you won’t get stuck on calculations when learning derivatives or integrals.

1.1.1 Polynomial Factorization

Factorization breaks down polynomials into simpler multiplicative terms—critical for simplifying functions and solving limits.

• Method 1: Extract the Greatest Common Factor (GCF)
  Identify the largest term that divides all coefficients and variables, then factor it out.
  Example: Factor $6x^3 - 12x^2 + 18x$
  Step 1: Find the GCF of coefficients $(6, -12, 18) \rightarrow 6$
  Step 2: Find the GCF of variables $(x^3, x^2, x) \rightarrow x$
  Step 3: Factor out $6x$: $6x(x^2 - 2x + 3)$
  
• Method 2: Factor Quadratic Trinomials ($ax^2 + bx + c)$
  For $a = 1$: Find two numbers that multiply to $c$ and add to $b$.
  Example: Factor $x^2 + 5x + 6$
  Step 1: Find two numbers: $2$ and $3$ (since $2 \times 3 = 6$ and $2 + 3 = 5$)
  Step 2: Write as factors: $(x + 2)(x + 3)$
  
• Method 3: Difference of Squares ($a^2 + b^2$)
  Use the formula: $a^2 - b^2 = (a - b)(a + b)$
  Example: Factor $4x^2 - 9$
  Step 1: Rewrite as squares: $(2x)^2 - 3^2$
  Step 2: Apply the formula: $(2x - 3)(2x + 3)$
  

1.1.2 Rational Function Simplification

Rational functions are fractions with polynomials (e.g., $\frac{x^2 - 1}{x - 1}$). Simplifying them requires canceling common factors (note: exclude values that make the original denominator zero).

Example: Simplify $\frac{x^2 - 4}{x^2 + 2x}$

1. Factor numerator and denominator:

   • Numerator: $x^2 - 4 = (x - 2)(x + 2)$ (difference of squares)
   • Denominator: $x^2 + 2x = x(x + 2)$ (extract GCF)

2. Cancel the common factor $(x + 2)$ (where $x \neq -2$):

Simplified form: $\frac{x - 2}{x}$ (with $x \neq 0, -2$).

1.1.3 Exponent and Root Calculations

These operations appear frequently in calculus (e.g., derivatives of $x^{1/2}$). Memorize the key rules below:

Operation Type	Rule	Example
Exponent Multiplication	$a^m \cdot a^n = a^{m+n}$	$x^3 \cdot x^2 = x^5$
Exponent Division	$\frac{a^m}{a^n} = a^{m-n}$	$\frac{x^5}{x^2} = x^{3}$
Power of a Power	$(a^m)^n = a^{m \cdot n}$	$(x^2)^3 = x^{6}$
Negative Exponent	$a^{-n} = \frac{1}{a^n}$	$x^{-3} = \frac{1}{x^3}$
Root as Exponent	$\sqrt[n]{a} = a^{1/n}$	$\sqrt{x} = x^{1/2}$

Example: Simplify $2x^3 \cdot 3x^{-2} \cdot \sqrt{x}$

1. Convert roots to exponents: $\sqrt{x} = x^{1/2}$
2. Multiply coefficients: $2 \times 3 = 6$
3. Combine exponents: $x^3 \cdot x^{-2} \cdot x^{1/2} = x^{3 - 2 + 1/2} = x^{3/2}$
4. Final result: $6x^{3/2}$ (or $6x\sqrt{x}$)

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1.2 Function Basics

Calculus revolves around "functions"—relationships where each input has exactly one output. This section covers core concepts you’ll use daily.

1.2.1 Definition and Representation of Functions

A function $f(x)$ maps an input $x$ (from the domain) to an output $y = f(x)$ (from the range). Common representations:

• Formula: $f(x) = 2x + 3$ (linear), $g(x) = x^2 - 4$ (quadratic)
• Graph: A curve where no vertical line intersects it more than once (the "Vertical Line Test")

Example: Is $x^2 + y^2 = 9$ a function?
No. For $x = 0$, $y = 3$ and $y = -3$ — one input has two outputs, so it fails the Vertical Line Test.

1.2.2 Domain and Range Determination

• Domain: All valid $x$-values (exclude values that break the function: division by zero, square roots of negatives, logarithms of non-positives).
• Range: All possible $y$-values the function produces (found via graph analysis or algebra).

Example 1: Find the domain of $f(x) = \frac{1}{\sqrt{x - 2}}$

• Denominator can’t be zero: $\sqrt{x - 2} \neq 0 \rightarrow x \neq 2$
• Square root argument can’t be negative: $x - 2 \geq 0 \rightarrow x \geq 2$
• Combine conditions: Domain = $x > 2$ (or $(2, +\infty)$ in interval notation)

Example 2: Find the range of $g(x) = x^2 + 1$

• $x^2 \geq 0$ for all real $x$ (squares are non-negative)
• Add 1 to both sides: $x^2 + 1 \geq 1$
• Range = $y \geq 1$ (or $[1, +\infty)$)

1.2.3 Properties of Common Functions

You'll work with these 5 function types most often—memorize their key traits:

Function Type	General Form	Key Properties
Linear	$f(x) = mx + b$	- Graph: Straight line - $m$: Slope - $b$: Y-intercept
Quadratic	$f(x) = ax^2 + bx + c$	- Graph: Parabola - Vertex: $\left(-\frac{b}{2a}, f(-\frac{b}{2a})\right)$
Exponential	$f(x) = a^x (a>0,a\neq1)$	- Graph: Increasing if $a>1$, decreasing if $0<a<1$ - Y-intercept: $f(0)=1$
Logarithmic	$f(x) = \log_a x$	- Inverse of $a^x$ - Domain: $x > 0$
Trigonometric	$f(x) = \sin x, \cos x$	- Periodic (repeat every $2\pi$) - Range: $[-1,1]$

1.3 Preliminary Limit Awareness

Limits describe the value a function "approaches" as $x$ gets close to a number.

Key Intuitive Examples

Example 1: $f(x) = \frac{1}{x}$ as $x$ approaches infinity

• As $x$ gets larger, $f(x)$ gets closer to 0
• Intuitive limit: $\lim_{x \to +\infty} \frac{1}{x} = 0$

Example 2: $g(x) = \frac{x^2 - 1}{x - 1}$ as $x$ approaches 1

• $g(1)$ is undefined but simplifies to $g(x) = x + 1$ for $x \neq 1$
• As $x$ gets close to 1, $g(x)$ gets close to 2
• Intuitive limit: $\lim_{x \to 1} \frac{x^2 - 1}{x - 1} = 2$

Example 3: Piecewise function behavior
Let $h(x) = \begin{cases} x + 2 & \text{if } x < 0 \\ x^2 & \text{if } x \geq 0 \end{cases}$

• Left limit: approaches 2
• Right limit: approaches 0
• The limits don't match, so $\lim_{x \to 0} h(x)$ does not exist

1.4 Practice Problems (with Answer Hints)

1. Simplify: $3x^2y \cdot 2x^3y^{-2}$
   Hint: Multiply coefficients, combine exponents
2. Factor completely: $2x^3 - 8x$
   Hint: First extract the GCF, then use difference of squares
3. Find the domain of $f(x) = \log_2(x - 3) + \frac{1}{x - 5}$
   Hint: Logarithm requires $x - 3 > 0$; denominator requires $x - 5 \neq 0$
4. Find the range of $f(x) = -2x^2 + 4x + 1$
   Hint: Parabola opens downward; vertex is maximum point
5. Intuitively find: $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$
   Hint: Simplify the fraction first

Answer Hints

1. $6x^5y^{-1}$ (or $\frac{6x^5}{y}$)
2. $2x(x - 2)(x + 2)$
3. Domain = $x > 3$ and $x \neq 5$ (interval notation: $(3,5) \cup (5, +\infty)$)
4. Range = $y \leq 3$ (vertex at $x = 1$, $f(1) = 3$)
5. $4$ (simplifies to $x + 2$, so limit is $2 + 2 = 4$)