Understanding Local Linearity in Calculus

Overview

This lecture explains the concept of local linearity in calculus, demonstrating how derivatives enable linear approximations and predictions for functions near a specific point.

Local Linearity and Differentiability

• Local linearity means a differentiable function closely resembles a straight line near any given point.
• A function's derivative at a point gives the slope of the tangent line at that point.
• Predictions using the derivative are most accurate for small changes in the input (delta x).
• The smaller the interval, the better the linear approximation between the curve and its tangent.

Visualizing Local Linearity

• Even nonlinear functions (e.g., sine) appear linear when zoomed in sufficiently close to a point where the derivative exists.
• Highly oscillatory functions can still show local linearity away from problematic points, as shown by zooming in at specific regions.

Linear Approximation Formula

• The change in function value (delta y) for a small change in input (delta x) is approximately delta y ≈ f '(x) × delta x.
• Slope (f '(x)) represents the instantaneous rate of change at point x.
• This formula generalizes the rise over run concept to estimate changes for small intervals.

Derivative Notation

• Slope can be written as f '(x) or in Leibniz notation as dy/dx, linking instantaneous and average rates of change.
• Using the derivative, we approximate changes in output for small input changes: delta y ≈ f '(x) × delta x.

Application Example

• For revenue models: if f '(a) is the rate of revenue change with respect to advertising, then a small increase in advertising yields delta revenue ≈ f '(a) × delta a.
• Example: With f '(a) = $1.8$ (thousand dollars per thousand dollars advertising) and an increase in advertising by $2,000$, revenue increases by approx. $3,600$.

Key Terms & Definitions

• Derivative (f '(x)) — Instantaneous rate of change of a function at point x.
• Local linearity — The property of a differentiable function to resemble a straight line when observed very close to a point.
• Tangent line — The straight line that touches a curve at one point, matching its slope there.
• Delta x (Δx) — Small change in the input variable x.
• Delta y (Δy) — Approximate change in the output, calculated by derivative times input change.

Action Items / Next Steps

• Practice applying the local linearity formula to additional real-world problems.
• Review derivative notations and their interpretations.
• Complete assigned homework on linear approximations and predictions using derivatives.