Exponential Properties
How to Use
- Drag the red-orange point left and right along the curve to explore any moment in time.
- Click anywhere on the plot to jump the point there.
- Toggle Rising / Falling to switch between a charging (RC) and decaying exponential.
- Toggle Ratio Marks to show or hide the constant-ratio visualization.
- Drag the τ slider to rescale the time constant and watch all properties update.
Three Key Properties
1. Tangent-Line Property (gold)
The tangent to the exponential at any point \(t\) always intersects the final value \(V_f\) at exactly \(t + \tau\):
\[
\text{slope at } t = \frac{dV}{dt}\bigg|_t = -\frac{V(t) - V_f}{\tau}
\quad\Rightarrow\quad
\text{tangent hits } V_f \text{ at } t + \tau
\]
Move the draggable point anywhere — the gold circle on \(V_f\) always stays exactly one \(\tau\) ahead.
2. Constant-Ratio Property (teal)
In every interval of length \(\tau\), the gap \(|V(t) - V_f|\) shrinks by the same factor \(e^{-1} \approx 0.368\):
\[
\frac{V(t+\tau) - V_f}{V(t) - V_f} = e^{-1} \approx 0.3679 \quad \text{(always)}
\]
3. Derivative Relationship (green)
The slope at any point is proportional to the remaining distance to the final value:
\[
\frac{dV}{dt} = -\frac{V(t) - V_f}{\tau}
\]
This is why the exponential is its own derivative — and why RC/RL circuits always follow this shape.