About this experiment
Move a point along a curve and inspect the tangent slope. A second point at x + h defines a secant. As you shrink h, compare its average rate of change with the exact derivative. The graph shows the function, tangent and secant together, so numerical values can be connected to local geometry.
Learning goals
- Interpret the derivative as a local rate of change and tangent slope
- Calculate a secant slope from two function values
- Investigate how the secant approaches the tangent as h decreases
How to use it
- 1Choose x squared and move the tangent point to x = 1. Compare the exact derivative and secant slope at h = 1.
- 2Use Halve h repeatedly and record how the slope error changes.
- 3Move to a point with negative derivative on the cubic curve and predict the tangent direction.
- 4Choose sine and compare the derivative near x = 0 and x = 1.57. Angles are in radians.
The model
Assumptions
- All three functions are smooth on the displayed interval, with sine evaluated in radians.
- The orange tangent uses an analytic derivative. The blue secant is calculated directly from two function values.
- Only positive nonzero h is shown. The slider stops at 0.01 to keep two distinct points visible.
Try this in class
- Q1For x squared, how does the secant slope differ from the derivative at a fixed x?
- Q2Where does the cubic function have a horizontal tangent?
- Q3Why does agreement from positive h alone not establish differentiability for an arbitrary function?
Frequently asked questions
Why does the derivative remain exact when h changes?
The derivative is evaluated from the analytic formula, independently of h. The finite secant provides a separate approximation that can be compared against it.
Is the tangent always above or below the curve?
No. Its position depends on curvature. For the convex function x squared the tangent lies below the curve, but other functions can cross their tangent.