Lesson 1.3

Algebraic Evaluation of f(x+h)

The single most critical algebraic skill for Differential Calculus. Mastering this substitution now makes the derivative definition trivial later.

Introduction

The single most critical algebraic skill for Differential Calculus. Mastering this substitution now makes the derivative definition trivial later.

Past Knowledge

In Lesson 1.2, you learned that means replacing every with "stuff." Today, that "stuff" is specifically the binomial .

Today's Goal

We are focusing on expanding terms like and correctly within a function definition. This requires flawless execution of distribution laws.

Future Success

The definition of the derivative is . If you cannot correctly calculate , you cannot do Calculus.

Key Concepts

The Expansion Rules

Squaring a Binomial

Do NOT write . You will miss the middle term .

Cubing a Binomial

Memorize this pattern (Pascal's Triangle row 3: 1, 3, 3, 1).

Distributing Negatives

The negative sign distributes to BOTH terms inside the parentheses.

The 4-Step Process

Skeleton: Write with empty parentheses.
Plug In: Write into every hole.
Expand: Foil exponents like .
Distribute: Multiply coefficients (especially negatives) into the expanded terms.

Worked Examples

Level: Basic

Example 1: Linear Function

Given , find and simplify .

Step 1: Skeleton

Step 2: Substitute (x+h)

Step 3: Distribute

Level: Intermediate

Example 2: Quadratic Function

Given , find simplified .

Step 1 & 2: Skeleton & Sub

Step 3: Expand the Square

Step 4: Distribute the -4

Note: None of these terms are "like terms," so you cannot combine them further.

Level: Advanced (Calculus Prep)

Example 3: Setup for Rational Functions

Given , setup the expression and combine into a single fraction.

Identify Parts

Step 1: Set up subtraction

Step 2: Common Denominator

Multiply left by and right by .

Step 3: Combine and Distribute Negative

Answer:

Common Pitfalls

The Freshman's Dream Error:
Writing . This is arguably the most common mistake in all of Algebra. You MUST distribute: to get the middle term .
Forgetting to distribute the negative:
In , students often write instead of . The coefficient applies to EVERYTHING in the parentheses.

Real-Life Applications

Economics: Marginal Cost Analysis

In economics, if is the cost to produce items, then is the cost to produce items plus more.

C(100) = Cost to make 100 cars
C(101) = C(100 + 1) = Cost to make 101 cars
C(101) - C(100) = Marginal cost of the 101st car

Calculating the difference allows businesses to determine if expanding production is profitable.