Curriculum Overview: Exponential Growth and Decay

This curriculum provides a structured pathway from the algebraic foundations of transcendental functions to the application of differential calculus in modeling real-world growth and decay phenomena. Students will explore how the natural base $e$ governs biological, chemical, and economic systems.

Prerequisites

Before engaging with this module, students should possess a strong mastery of the following:

• Algebraic Foundations: Proficiency in the laws of exponents (e.g., $b^m \cdot b^n = b^{m+n}$) and the definition of a function.
• Transcendental vs. Algebraic: Understanding that exponential functions $f(x) = b^x are distinct from power functions f(x) = x^b$.
• Limits & Continuity: Basic understanding of limits at infinity, specifically how $e^x \to 0$ as $x \to -\infty$.
• The Chain Rule: Ability to differentiate composite functions, as most growth models involve $e^{f(x)}$.

---

Module Breakdown

---

Learning Objectives per Module

Module 1: Functional Forms

• Objective: Distinguish between linear, power, and exponential growth.
• Example: Identifying that while $y=x^2$ grows, $y=2^x$ eventually dominates any polynomial.

Module 2: The Derivative of Transcendental Functions

• Objective: Apply differentiation rules to exponential functions.
• Formula: $\frac{d}{dx}(e^{u}) = e^u \frac{du}{dx}$.

Module 3: Growth Applications

• Objective: Solve for "doubling time" using natural logarithms.
• Real-World Case: Modeling the population of Toledo, Ohio, starting at 500,000 with a 5% annual increase.

Module 4: Decay Applications

• Objective: Determine the remaining mass of an isotope given its half-life.
• Real-World Case: Calculating the decay rate of Erbium-12 (half-life of 12 hours).

---

Visual Anchors

Model Decision Logic

Comparing Function Behaviors

---

Success Metrics

Students are considered to have mastered this curriculum when they can:

1. Construct Models: Given a 2000 population and a 5% rate, correctly write $P(t) = 500,000(1.05)^t$.
2. Calculate Relative Change: Use the formula $$100 \cdot \frac{f'(x)}{f(x)}$%$ to describe growth speed.
3. Solve Inverse Problems: Use $$\ln(x)$to find the exact time t$ when a population doubles or a substance reaches 10% of its initial mass.
4. Analyze Asymptotes: Identify $y=0$ as the horizontal asymptote for decaying functions.

---

Real-World Application

[!TIP] Why this matters for your career:

• Pharmacology: Half-life determines how long a life-saving drug stays in a patient's bloodstream.
• Data Science: Understanding Gompertz growth is vital for predicting how a new viral app or product saturates a market.
• Environmental Science: Modeling the decay of pollutants or the growth of invasive species in an ecosystem.

▶Deep Dive: The Gompertz Function

In advanced biology, populations don't grow forever. The Gompertz function P(x) = ae^{-b \cdot e^{-cx}} is used to model systems that have a "carrying capacity" or a point where growth slows down as resources are depleted.

Formula Reference