CalcGospel 國際數學圖譜

Secant to Tangent Explorer

Let the second point approach A and watch a secant become a tangent

Gradient from first principles

What happens as approaches zero?

Keep fixed. Changing moves the second point along the curve .

Secant and tangent graph

Secant approaching a tangent The curve y equals x squared, a fixed point A, a moving point P, the secant AP, and the tangent at A.
Secant Tangent at

The calculation behind the picture

The difference between the two gradients is exactly . Therefore, as approaches zero from either side, the secant approaches the tangent.