Unit 4 · Contextual Applications of Differentiation
● Core concept · ○ Supporting concept
4.2 Straight-Line Motion: Connecting Position, Velocity, and Acceleration
Position function ● (core concept) — A function s(t) giving an object's location at time t along a line.
Velocity ● (core concept) — The instantaneous rate of change of position: v(t) = s'(t). Its sign indicates the direction of motion.
Acceleration ● (core concept) — The instantaneous rate of change of velocity: a(t) = v'(t) = s''(t).
Speed ● (core concept) — The magnitude of velocity, |v(t)|; speed is never negative.
4.4 Introduction to Related Rates
Related rates ● (core concept) — Problems in which several quantities change with time and are linked by an equation; differentiate implicitly with respect to time and substitute known values to find an unknown rate.
4.6 Approximating Values of a Function Using Local Linearity and Linearization
Local linearity ● (core concept) — The property that a differentiable function looks like a straight line when zoomed in close enough at a point.
Linearization (tangent line approximation) ● (core concept) — The approximation f(x) ≈ f(a) + f'(a)(x − a) for x near a, using the tangent line at a.
4.7 Using L'Hospital's Rule for Determining Limits of Indeterminate Forms
L'Hospital's Rule ● (core concept) — If lim f(x)/g(x) has indeterminate form 0/0 or ∞/∞, then the limit equals lim f'(x)/g'(x), provided that limit exists.
Indeterminate form ● (core concept) — A limit form such as 0/0 or ∞/∞ whose value cannot be determined from the form alone and requires further analysis.