Transcendental functions and tangent circles

Historically, transcendental functions have required numerical approximation methods to identify specific values and solutions. There was no way, for example, to compute ln 0.25. Through clever methods such as infinite series expansions, however, mathematicians identified numerical techniques that allowed them to provide increasingly more accurate approximations for such values. Although students of today may not appreciate Napier's tables of logarithms--instead associating logarithms with a calculator button--exposing students to modern-day examples in which they can navigate transcendental functions can be exciting. It may even help them appreciate all the work that has led up to having these values appear at the press of a button. Here, Neel demonstrates some issues that arise with transcendental functions, with the added benefit of being able to visualize the situation geometrically.