textbook-theory · English
Best Affine Approximations, Derivatives, and Rates of Change
Section 3.2 of Dan Sloughter's open calculus text 'Difference Equations to Differential Equations' develops the concept of the derivative as the slope of the best affine (linear) approximation to a function at a point, establishing the formal limit definition. The section covers differentiability, the relationship between differentiability and continuity, Leibniz notation, and the interpretation of the derivative as an instantaneous rate of change, illustrated with worked examples including polynomial, radical, and absolute-value functions as well as a projectile-motion problem. A problem set of 20 exercises extends the material to finding derivatives, best affine approximations, tangent lines, and applied rate-of-change problems.