Exploring Rolle's Theorem

Explore conditions of the Rolle's Theorem. The applet shows the graph of continuous differentiable function f(x) on a closed interval  [a, b]. Case 1.  f(x) < f(a) for some x inside the interval (a, b). Can you find the number c such that f'(c)=0? What did you notice about the point when f'(c)=0? Case 2.  f(x) > f(a) for some x inside the interval (a, b). Can you find the number  c such that f'(c)=0? What did you notice about the point when f'(c)=0? Can a given function have more than one number on a given interval such that f'(c)=0?

Tags: Calclulus

Parabolic Solar Cooker

Explore the relationship between f-number of a parabolic solar cooker and its sensitivity to change in angle of incoming light  

Tags: parabola, solar-cooker, reflection

N-Gon Approximating a Circle

This n-gon (regular polygon of n sides) has a variable number of sides. Changing the value for n demonstrates very visually how regular polygons of increasing numbers of sides, with the same radius, have a shape that increasingly approaches that of a circle.

Tags: N-gons, circles