Search

A basketball player attempting a free throw has two parameters under his or her control: the angle of elevation and the force with which the ball is thrown. We compute upper and lower bounds for the initial velocity for suitable values of the angle of elevation, generating a subset of the configuration space of all successful free throws. A computer-assisted search of this configuration space yields a free throw shot most forgiving of error hence optimal.

It's well known that slicing a cone with a plane and then allowing the plane to rotate through all possible angles of inclination yields the conic sections. What paths then do the foci of these conics trace out as this cutting plane passes through the different angles? In this article, we derive formulae for these trajectories and generate the associated curves. Of particular interest is how they articulate when the conic shifts from ellipse to parabola to hyperbola.

The article focuses on teaching proofs. Three main components to teaching proofs include the recognition of proof, the appreciation of proofs, and the production of proofs. Techniques and resources that standardize certain procedures are presented in relation to the impediments to these goals. The statement "its proof is the very thought process that led to that discovery" is explained in the workings of the theorem and the proof.