Tangent approximation

Parametric equations

Secant slope:

m = \frac{f ( x _{P} + Δ x ) - f ( x _{P} )}{Δ x}

As $Δ x \to 0$ , this converges to the derivative. Point-slope form:

y - f (x_{P}) = m (x - x_{P})

The function:

f (x) = 0.25 sin (2 π k x + t) - 0.4 (x - 0.5)^{2}

The point of tangency $x_{P} = 0.5 + 0.1 sin (0.6 t)$ drifts a little with time. Δx is the spacing, on the $x$ -axis, of the two ends of the secant. k controls how much the curve bends.

Interaction

Target span Δx: shrink the secant’s spacing in x and watch the secant approach the tangent
Wave frequency k: change how much the curve bends; the tangent slope changes with the position
Time speed ω: move the secant endpoints along the curve

What to notice

The secant slope $Δ y /Δ x$ approaches the tangent slope as $Δ x \to 0$ .
On a more curved stretch, the same $Δ x$ leaves a larger gap between the secant and the tangent.
The derivative is an instantaneous rate of change, not an average rate of change.

Parametric equations

Interaction

What to notice

Further reading

Comments

Parametric equations

Interaction

What to notice

Related

Further reading

Comments