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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).
Labs / Mathematics / Slope of a Curve
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).
Why it matters
Why explore this?
Mathematics helps us describe patterns precisely and predict what comes next. This model turns an abstract relationship into something you can change and observe.
Explore
Try the model
h 0.000Δy/Δx 0.000f'(x) 0.000
Investigate
Questions to test
Predict first, try it in the model, then use what you observe to answer.
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?
Explain
Make sense of what happened
State the pattern in words, test it with another example, and then connect it to the mathematical rule or representation shown in the model.
Check
Before you leave
What pattern did you observe?
Does the pattern work for another example?
If you can answer all three using evidence from the model, you understand the main idea. If not, return to Explore and change one setting at a time.