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.
Interactively approximate definite integrals using left, right, and midpoint Riemann sums. Adjust the number of rectangles to watch the approximation converge to the true area under the curve.
Apply L'Hôpital's Rule to evaluate indeterminate forms like 0/0 and ∞/∞ by taking derivatives of the numerator and denominator. Visualize how lim[x→c] f(x)/g(x) = lim[x→c] f'(x)/g'(x) when the original limit produces an indeterminate form. Practice identifying when to apply the rule, handling repeated applications, and recognizing other indeterminate forms like 0·∞, ∞-∞, 0⁰, 1^∞, and ∞⁰.
Solve related rates problems where multiple quantities change with respect to time and are connected by an equation. Use implicit differentiation with respect to time to find how one rate of change relates to another. Visualize classic scenarios like ladder sliding down walls, water filling conical tanks, expanding circles, and moving shadows, applying the chain rule to connect dy/dt, dx/dt, and geometric relationships.
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.