Calculus and continuous transformation — derivatives, integrals, ODEs, Taylor series, and optimization. Computes rates of change, accumulates quantities over intervals, solves differential equations, and classifies critical points. Use when computing derivatives or integrals, solving ordinary differential equations, performing Taylor series approximations, finding critical points, or analyzing continuous change and motion.
Part III: Moving — Chapters 8, 9, 10 — Plane Position: (0, -0.2) radius 0.4 — 58 Primitives
Derivative (definition): The derivative of f at x is f'(x) = lim_{h->0} [f(x+h) - f(x)]/h, when this limit exists. It represents the instantaneous rate of change of f at x, and the slope of the tangent line to the graph at (x, f(x)).
Definite Integral (definition): The definite integral of f from a to b is integral_a^b f(x)dx = lim_{n->inf} sum_{i=1}^{n} f(x_i*)*Delta_x, when this limit exists. It represents the signed area between f and the x-axis over [a,b].
Ordinary Differential Equation (definition): An ordinary differential equation (ODE) is an equation involving a function y(x) and its derivatives: F(x, y, y', y'', ..., y^(n)) = 0. The order is the highest derivative present. A solution is a function that satisfies the equation on an interval.
Limit (definition): The limit of f(x) as x approaches a is L, written lim_{x->a} f(x) = L, if for every epsilon > 0 there exists delta > 0 such that 0 < |x - a| < delta implies |f(x) - L| < epsilon.
Continuity (definition): A function f is continuous at a point a if: (1) f(a) is defined, (2) lim_{x->a} f(x) exists, and (3) lim_{x->a} f(x) = f(a). f is continuous on an interval if continuous at every point in the interval.
Higher-Order Derivatives (definition): The nth derivative f^(n)(x) is obtained by differentiating f n times. f''(x) = d^2f/dx^2 (acceleration, concavity). f^(n)(x) = d^n f/dx^n. A function is C^n if its first n derivatives are continuous.
Critical Point (definition): A critical point of f is a value c in the domain of f where f'(c) = 0 or f'(c) does not exist. Critical points are candidates for local maxima, local minima, or inflection points.
Antiderivative (Indefinite Integral) (definition): An antiderivative of f is a function F such that F'(x) = f(x). The indefinite integral integral f(x)dx = F(x) + C represents the family of all antiderivatives, where C is an arbitrary constant.
Power Rule (theorem): For any real number n: d/dx(x^n) = n*x^(n-1). This holds for integer, rational, and real exponents (where the function is defined).
Chain Rule (theorem): If g is differentiable at x and f is differentiable at g(x), then the composite function f(g(x)) is differentiable at x with: d/dx[f(g(x))] = f'(g(x)) * g'(x).