Calculus I: Single-Variable Differential Calculus — Curriculum Overview

This document outlines the structured path for mastering single-variable differential calculus. This course bridges the gap between static algebra and dynamic mathematical modeling by introducing the concepts of limits, rates of change, and accumulation.

## Prerequisites

Before beginning this curriculum, students should have a strong foundation in the following areas:

• Algebra II & Pre-Calculus: Proficiency in manipulating algebraic expressions, solving polynomial equations, and understanding function notation $f(x)$.
• Trigonometry: Knowledge of the six basic trigonometric functions, radian measure, and fundamental identities (e.g., $\sin^2\theta + \cos^2\theta = 1$).
• Geometry: Understanding of slopes, areas of basic shapes, and the Cartesian coordinate system.
• Function Analysis: Ability to determine domain and range, and recognize transformations (shifts, stretches, reflections) of parent functions.

[!NOTE] This curriculum is designed to accommodate both Early Transcendental and Late Transcendental approaches. Exponential and logarithmic functions are introduced early but can be explored rigorously in later modules.

## Module Breakdown

The Conceptual Pipeline

## Learning Objectives per Module

Module 2: Limits and Continuity

• Estimate limits using numerical tables and graphical analysis.
• Evaluate limits using algebraic Limit Laws and the Squeeze Theorem.
• Define Continuity: Determine if a function is continuous at a point $a using the three-part test: f(a)$ exists, $\lim_{x \to a} f(x)$ exists, and they are equal.
• Infinite Limits: Identify vertical and horizontal asymptotes through end-behavior analysis.

Module 3: Derivatives

• Formal Definition: Calculate $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$.
• Mastery of Rules: Apply Power, Product, Quotient, and Chain Rules to differentiate complex expressions.
• Implicit Differentiation: Solve for $\frac{dy}{dx}$ in equations where $y is not isolated (e.g., x^2 + y^2 = 25$).

Module 4: Applications of Derivatives

• Optimization: Model real-world scenarios (e.g., maximizing profit or minimizing material) as functions and find extrema.
• L’Hôpital’s Rule: Use derivatives to solve indeterminate limits of the form $0/0$ or $\infty/\infty$.
• Graph Analysis: Use the First and Second Derivative Tests to find intervals of increase/decrease and concavity.

## Success Metrics

To demonstrate mastery of this curriculum, a student must be able to:

1. Algebraic Fluency: Differentiate any combination of polynomial, trigonometric, exponential, and logarithmic functions without reference materials.
2. Graphical Interpretation: Sketch a function's graph given only its derivative properties ($f'$ and $f''$ signs).
3. Modeling Proficiency: Translate a word problem (like a "Related Rates" scenario) into a solvable calculus equation.
4. Rigorous Proof: Construct a formal $\epsilon-\delta$ proof for a basic linear limit.

Visualizing the Tangent Problem

Below is a representation of the Secant line approaching the Tangent line as $h \to 0$.

## Real-World Application

Calculus is the language of change and is used to solve high-stakes problems across industries:

• Physics & Engineering: Calculating instantaneous velocity and acceleration; determining the hydraulic force against structures like the Hoover Dam.
• Economics: Finding the Marginal Cost and Marginal Revenue to optimize business production levels.
• Biology: Modeling population growth rates and the spread of diseases using differential equations.
• Seismology: Using logarithmic scales to compare the relative intensity of earthquakes.

[!IMPORTANT] The "Big Idea" of this course is that by looking at infinitely small intervals, we can understand the behavior of systems at a single, precise moment.