📈 Derivatives as Secant-to-Tangent Limits
Drag the cyan $x_0$ handle or rose $x_0 + \Delta x$ handle. Watch the secant line morph into the instantaneous tangent as $\Delta x \to 0$. The synchronized bottom canvas traces $f'(x)$.
📊 Definite Integrals & Riemann Slices
Drag the $a$ (Amber) and $b$ (Cyan) handles along the axis. Increase slices $N$ from 1 to 100 or click "Melt Slices" to watch rectangles dissolve into continuous area.
🔄 Fundamental Theorem of Calculus
Watch the instantaneous height $f(x)$ on the top curve directly govern the tangent slope of the accumulated area function $F(x) = \int_0^x f(t)dt$ on the bottom curve: $F'(x) = f(x)$.
🏔 2D Gradient Descent Optimization
Explore optimization landscapes. Downhill gradient vectors $-\nabla f(x, y)$ show the direction of steepest descent. Drag the loss ball $\theta$ or adjust learning rate $\eta$ and momentum $\beta$.
〰️ Taylor Series Approximations
Increase the degree $N$ to watch the polynomial wrap around the function. Drag the center point $a$ to shift the approximation.
🌀 Volume of Revolution
Rotate a function around an axis. Watch the 2D area sweep out a 3D solid.
🎯 Projectile Kinematics & Target Hoop Game
Real-time velocity vector decomposition $(v_x, v_y)$, constant-time strobe markers ($\Delta t = 0.25\text{s}$), and planetary gravity presets. Drag the target hoop to challenge your trajectory precision!
⏱ Damped Harmonic Pendulum & Phase Space
Watch kinetic energy $E_k$ and gravitational potential energy $E_p$ trade off continuously. The synchronized phase space portrait $(\theta, \omega)$ reveals inward damping spirals vs closed conservative orbits.
🌈 Optics, Refraction & Total Internal Reflection
Interact with incident light rays crossing medium boundaries. When $n_1 > n_2$ and the incidence angle exceeds critical angle $\theta_c$, observe the transition to Total Internal Reflection (TIR).
🌊 Dual-Source Wave Interference & Ripple Tank
Observe 2D wave superpositions from coherent sources $S_1$ and $S_2$. Notice constructive interference maxima and destructive nodal hyperbola lines projecting onto the screen.
🪐 Orbital Mechanics & Kepler's Laws
Simulate a planet orbiting a star. Watch the velocity vector change and see equal areas swept in equal times (Kepler's 2nd Law).
🪀 Spring-Mass Harmonic Oscillator
Drag the mass to stretch the spring. Watch the continuous exchange of Kinetic and Potential Energy.
🔲 2D Matrix Transformations & Grid Warping
Experience linear algebra in the style of 3Blue1Brown. Drag the basis vector handles $\hat{i}$ (Cyan) and $\hat{j}$ (Rose). Watch the entire coordinate space warp, determinant area flip orientation, and eigenvector span axes remain invariant.
📐 Quadratic Functions & Discriminant Gauge
Adjust parameters $a, b, c$. The live discriminant gauge $\Delta = b^2 - 4ac$ reveals why $\Delta > 0$ yields two real roots, $\Delta = 0$ touches the axis once, and $\Delta < 0$ branches into complex conjugates.
⭕ Unit Circle & Sine Wave Unwinding
Rotate angle $\theta$. The cyan horizontal projection bar reflects $\cos\theta$, while the rose vertical projection bar reflects $\sin\theta$, seamlessly unwinding into an animated running sine wave graph.
✖ Systems of Linear Equations & Cramer's Rule
Visualize the geometric intersection of two lines. Cramer's determinant $D = a_1 b_2 - a_2 b_1$ provides instantaneous classification into Unique, Parallel (Inconsistent), or Coincident (Dependent) states.
⚡ Complex Plane & Euler's Formula
Visualize $e^{i\theta} = \cos(\theta) + i\sin(\theta)$. Drag the angle $\theta$ to see the relationship between circular motion, real, and imaginary components.
🔀 Eigenvectors & Eigenvalues
A matrix transformation scales space. Eigenvectors are the vectors that remain on their original span. Drag the basis vectors to discover their eigenvectors!