Labs / Mathematics
Slope of a Curve
Two points sit on a curve, joined by a straight secant line. Drag the second point toward the fixed amber point and the gap h shrinks — watch the secant swing until it rests exactly on the tangent. That limiting slope is the derivative, f'(x).
h 0.000Δy/Δx 0.000f'(x) 0.000
What to try
- As you shrink h, does Δy/Δx settle on a single number? What number is that?
- Drag the fixed point to where the parabola bottoms out. Why does the tangent go flat there?
- On the sine curve, where is the slope steepest, and where does it vanish?
- Turn on f'(x). How does the height of that curve relate to the steepness of the original one?
- On the cubic, can you find the two places where f'(x) = 0, and what is the curve doing there?