VI. Local Linear Approximation
VI. Local Linear Approximation
VI.1.1. Tangency to Transcendental Functions
VI.1.2. Basic Differentiation Rules
VI.1.3. Differentiation and Decomposition
VI.1.4. Newton’s Method
VI.2.1. Derivatives of Inverse Functions
VI.2.2. Implicitly Defined Functions and Their Derivatives
VI.2.3. Related Rates Problems
VI.3.1. Extreme Values
VI.3.2. Mean Value Theorem
VI.3.3. Antiderivatives
VI.3.4. L’Hopital’s Rule
VI.4.1. Sketching and Optimization with First Order Information
VI.4.2. The Second Derivative
VI.4.3. Concavity and Curve Sketching
VI.5.1. First Order Differential Equations
VI.5.2. Solving Simple Differential Equations
VI.5.3. Uniqueness of Solutions to Certain Differential Equations
VI.6.1. Particle Motion
VI.6.2. Curves on Simple Surfaces
VI.6.3. The Implicit Function Theorem
Worksheet VI.1: Justification of Newton’s Method
Worksheet VI.2: A Single Cylinder Engine
Worksheet VI.3A: Convergence of Series and the Mean Value Theorem
Worksheet VI.3B: Piecewise Linear Approximation
Worksheet VI.3C: Approximation and the Mean Value Theorem
Worksheet VI.3D: A Surgical Application of the Method of Undetermined Coefficients
Worksheet VI.4A: Layering Graphical Information
Worksheet VI.4B: Cauchy’s Mean Value Theorem and Second Order Control
Worksheet VI.5: Darboux’s Theorem
Worksheet VI.6A: Partial Derivatives
Worksheet VI.6B: A Proof of the Implicit Function Theorem for Planar Functions