Curriculum Overview: Defining the Derivative

This curriculum provides a foundational understanding of differential calculus, transitioning from the intuitive concept of a tangent line to the formal limit definition of the derivative function.

Prerequisites

Before starting this module, students should have mastery of the following:

• Algebraic Manipulation: Proficiency in simplifying complex fractions, factoring, and rationalizing numerators.
• Function Theory: Understanding of domain, range, and composite functions $f(g(x))$.
• Limits: Ability to evaluate limits using laws, including indeterminate forms ($0/0$).
• Continuity: Understanding the definition of continuity at a point and over an interval.

Module Breakdown

Module Objectives

1. Conceptualizing the Tangent

• Recognize the tangent line as the limit of secant lines as the interval $h$ approaches zero.
• Calculate the slope of a tangent line at a specific point $(a, f(a))$.

2. Formal Limit Definition

• Identify the derivative as the limit of a difference quotient: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$
• Apply the four-step process to find the derivative of linear, quadratic, and simple rational functions.

3. Graphical Analysis

• Sketch the graph of $f'(x) given the graph of f(x)$.
• Understand that $f'(x) > 0 when the function is increasing and f'(x) < 0$ when decreasing.

4. Differentiability vs. Continuity

• Prove that differentiability implies continuity ($f'(a)$\implies$cont. at a$).
• Identify non-differentiable points: Corners, Cusps, Vertical Tangents, and Discontinuities.

Success Metrics

To demonstrate mastery, students must be able to:

• Compute the derivative of $f(x)$= \sqrt{x}$$ or $f(x) = 1/x$ using the formal limit definition.
• Identify local extrema on a graph by finding where the derivative function crosses the x-axis ($f'(x) = 0$).
• Contrast average velocity vs. instantaneous velocity using mathematical notation.

[!IMPORTANT] Always remember: While every differentiable function is continuous, not every continuous function is differentiable (e.g., the absolute value function $f(x) = |x|$ at $x=0$).

Real-World Application

Physics: Motion Analysis

Derivatives allow us to move from position to velocity to acceleration. If $s(t)$ is position, then $v(t) = s'(t)$.

Economics: Marginal Analysis

In business, the derivative of a cost function $C(x)$ represents the Marginal Cost—the cost of producing one additional unit.

Geometric Representation