Derivative of a linear piecewise function

On the left is a piecewise linear function. On the left is the derivative of that piecewise linear function.

1) In the left window, drag points A, B, C, D, and E to see how the derivative changes. 2) How is the graph formed? 3) Why are there open circles in the derivative? Should they be connected? Explain.

Accumulating Penguins

In a math twitter discussion about calculus, the idea was brought up of penguins entering a room. [br][br]If this many entered each minute, how many penguins?

The limit of a sequence

The limit of the sequence ([math]a_n[/math]) is [math]L[/math] if given any [math]\varepsilon>0[/math], there is an [math]N>[/math]0 such that [math]|a_n-L|<\varepsilon[/math] for all [math]n\ge N[/math]. The yellow band is [math]L±\varepsilon[/math] and the blue region correspons to those [math]n\ge N[/math]. The green dots are within [math]\varepsilon[/math] of [math]L[/math] and the red dots and red x's are not. The limit is [math]L[/math] if for any given [math]\varepsilon[/math] you are able to move the blue region far enough to the right that there are no red x's. Note: To move the window, hold the shift key down while you click and drag. To zoom in or out, hold down the shift key and scroll the mouse.