Showing 12 results
Optimization Problems
Solve optimization problems by finding absolute and relative extrema using calculus techniques. Learn to identify constraints, write objective functions, take derivatives, find critical points using f'(x) = 0, and apply the first or second derivative test. Explore real-world applications including maximizing area, minimizing cost, optimizing volume, and finding shortest distances in geometry, physics, and economics.
Related Rates Visualizer
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.
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.
Volumes w/ Known Cross Sections
Calculate volumes of solids with known cross-sectional shapes perpendicular to an axis using integration. Visualize how V = ∫[a to b] A(x)dx sums infinitely many cross-sectional areas—squares, rectangles, semicircles, equilateral triangles, and isosceles right triangles. Understand how the base region determines the limits of integration and how the cross-section shape determines the area function A(x).
Volumes of Solids of Revolution
Calculate volumes of three-dimensional solids formed by rotating regions around axes using disk, washer, and shell methods. Visualize the disk method V = π∫[a to b] [R(x)]²dx for solids without holes, the washer method V = π∫[a to b] ([R(x)]² - [r(x)]²)dx for solids with holes, and the shell method V = 2π∫[a to b] x·h(x)dx for rotation around vertical axes. Master choosing the most efficient method for each problem.
Solid of Revolution 3D Visualizer
Interactive Calculus visualizer converting 2D function areas into 3D volumes of revolution using Disk and Washer methods.
Limits & Continuity Explorer
Interactive limit evaluation (left-hand, right-hand, absolute) and 3-step continuity logical check tool simulating removable, jump, infinite, and oscillating discontinuities.
Definition of the Derivative
Interactive secant-to-tangent limit convergence visualizing the formal definition of the derivative with dynamic dx collapsing.
Chain Rule Visualizations
Mechanical gear visualization translating the abstract composite derivative multiplication f'(g(x)) * g'(x) into physical rotational interlocking.
Implicit Differentiation
Implicit differentiation analyzer for non-functions (Circles, Ellipses, Foliums). Evaluates dy/dx dynamically, reacting to both x and y inputs to map vertical and horizontal tangents.
Related Rates Visualizer
Related rates visualization using implicit chain rule integration with respect to time (dt). Demonstrates the geometric parameter shift paradox of Constant dV/dt impacting spherical radii and conical fill heights dynamically.
MVT & Rolle's Explorer
MVT interactive module letting users sweep endpoints to compute the secant, subsequently automatically identifying all internal $c$ points where tangent perfectly parallels secant. Built with Rolle's edge case and cusp logic breakers.