3.6.2 Derivatives as Rates of Change

Growth Models – Using Derivatives as Rates of Change

Key Ideas

• “Growth model” ⇢ any function that represents how a quantity (population, revenue, users, etc.) increases or decreases with time.
• Derivative of the model gives an instantaneous growth rate (slope of the tangent).
• Slope of the secant line between two points gives an average growth rate over an interval.

Internet-User Example (2000–2015)

• Data set: Worldwide Internet users, measured every February.
• Best-fit polynomial provided by instructor (exact constants not written in transcript).
  – Let t = years after February 2000 (so 2005 ⇢ t = 5, 2010 ⇢ t = 10, 2011 ⇢ t = 11, 2015 ⇢ t = 15, 2020 ⇢ t = 20).
  – Model: P(t) (millions of users).
  – Instructor’s computed derivative: \frac{dP}{dt}=12t+98.

A. Average growth rate (2005 → 2010)

• Formula: \text{Average rate}=\frac{P(10)-P(5)}{10-5}.
• Given values:
  P(10)=2{,}011.2\text{ million},\qquad P(5)=1{,}071.2\text{ million}.
• Calculation:
  \frac{2{,}011.2-1{,}071.2}{5}=188\;\text{million users·year}^{-1}.
• Interpretation: During 2005-2010, world Internet adoption grew by ≈188 million new users each year (on average).

B. Instantaneous growth rate (February 2011)

• Need the derivative value at t=11:
  P'(11)=12(11)+98 = 230.
• Units: million users per year.
  → In Feb 2011, the user base was expanding at roughly 230 million per year at that instant.

C. Shape of the growth-rate graph (P'(t))

• Graph of P'(t)=12t+98 is a straight line with positive slope 12.
  – Always positive ⇒ adoption never shrinks in 2000-2015 window.
  – Increasing slope ⇒ rate of adoption is itself accelerating each year.

D. Extrapolation beyond 2015 (to 2020)

• Instructor assumes same quadratic model valid to t=20.
  – “Plug in 20” gives P(20)\approx7{,}700\text{ million} users (≈ worldwide population).
  – Caveat: Extrapolating a fitted quadratic many years past data can be unreliable, but this demonstrates the method.

Cost Analysis – Average vs. Marginal Cost

Setup

• Cost function C(x) = total dollars to manufacture the first x items.
  (Transcript gives) C(x) = -0.2x^{2}+50x+100.

Definitions

• Average cost per item: \bar{C}(x)=\frac{C(x)}{x}.
• Marginal cost (approximate cost of one additional item after the first x): C'(x).

Derivations

1. Average cost function:
   \bar{C}(x)=\frac{-0.2x^{2}+50x+100}{x}= -0.2x + 50 + \frac{100}{x}.
2. Marginal cost function (take the derivative):
   C'(x)=\frac{d}{dx}\bigl(-0.2x^{2}+50x+100\bigr)=-0.4x+50.

Numerical Evaluations & Interpretations

Observations

• \bar{C}(x) decreases as x grows ⇒ strong economies of scale.
• C'(x) is linear & decreasing; eventually marginal cost could become negative (mathematical artifact of the quadratic model, not realistic). Practical takeaway: producing more lowers incremental expense up to some physical capacity limit.

Conceptual Connections & Practical Significance

• Derivatives link change to value: population growth, production cost, spread of technology, etc.
• Secant-line slopes (average rates) guide broad planning; tangent slopes (instantaneous rates) guide fine-tuned decisions and forecasting.
• For businesses, marginal-cost curves inform pricing, capacity expansion, and profit-maximizing output.
• In demographic studies, a positive and increasing P'(t) warns of accelerating demand on infrastructure, market size, and policy needs.

Equations & Formulas Recap

• Average growth rate over [t1,t2]:
  \text{AGR}=\frac{P(t2)-P(t1)}{t2-t1}.
• Instantaneous growth rate: P'(t).
• Average cost: \bar{C}(x)=\frac{C(x)}{x}.
• Marginal cost: C'(x)=\frac{dC}{dx}.
• Quadratic marginal-cost example: C'(x)=-0.4x+50. (Slope −0.4 represents $0.40 cost reduction per extra unit produced.)

Ethical & Real-World Implications Discussed

• Technological diffusion: rapid, accelerating Internet growth implies issues of digital equity, infrastructure, and global access.
• Economic scaling: decreasing marginal cost encourages mass production, which can boost accessibility but also raises sustainability and labor-practice concerns if not managed responsibly.

Quick Study Checklist

• Can you compute both average & instantaneous rates of change from a numeric table or formula?
• Given C(x), can you derive \bar{C}(x) and C'(x), then interpret them economically?
• Do you recognize the limitations of extrapolating a fitted model far beyond its data range?