Limits & Continuity
Explore the foundational concepts of limits and continuity that underpin all of calculus. Visualize one-sided limits, two-sided limits, and limits at infinity. Understand the three conditions for continuity at a point: f(c) is defined, lim[x→c] f(x) exists, and lim[x→c] f(x) = f(c). Practice identifying discontinuities (removable, jump, and infinite) and applying limit laws to evaluate complex expressions.
Taylor Series
Explore Taylor and Maclaurin series, which approximate functions as infinite polynomials using derivatives at a single point. Visualize how f(x) ≈ f(a) + f'(a)(x-a) + f''(a)(x-a)²/2! + ... converges to the original function. Understand how adding more terms improves accuracy, and learn common series for e^x, sin(x), cos(x), and ln(1+x). Practice finding intervals of convergence and estimating error bounds.
Fundamental Theorem of Calculus
Explore the Fundamental Theorem of Calculus, which connects differentiation and integration as inverse operations. Visualize Part 1: if F(x) = ∫[a to x] f(t)dt, then F'(x) = f(x), and Part 2: ∫[a to b] f(x)dx = F(b) - F(a) where F is any antiderivative of f. Understand how this theorem enables efficient calculation of definite integrals and reveals the deep relationship between rates of change and accumulation.