Geometrically, the derivative of a function f at a point (a,f(a)) is interpreted as the slope of the line tangent to the function's graph at x = a. This applet is designed to help you better understand that the output (y-value) of the derivative of a function f (at x = a) is the same as the slope of the tangent line drawn to the graph of f at x = a. [br][br]The following applet allows you to trace out the [color=#c51414]FIRST DERIVATIVE (its trace is shown in red)[/color] and the [color=#1551b5]SECOND DERIVATIVE (its trace is shown in blue)[/color] each of the six functions shown below. Simply select which function you'd like to examine and select which derivative(s) of that function you'd like to trace out. Then, drag the blue point along the original function to observe traces of the specific deriviative(s) you've selected. [br][br]It is suggested that you trace out the first derivative of a selected function FIRST. Then, proceed to trace out the second derivative. [br]It is also suggested that you sketch the graph of each original function on a separate sheet of paper first. After doing so, see if you can create rough sketches of the graphs of its first and second derivatives (in that order) would look like BEFORE graphing them on the applet below. [br][br]When switching from function to function, you'll need to clear your previous traces. To clear traces ([color=#c51414]red dots[/color] and [color=#1551b5]blue dots[/color]), simply hit "Control" + "F" on your keyboard. Or, you can go to the "View" menu and click "Refresh Views". [br][br]BE SURE TO ZOOM IN CLOSE TO x = 0 for function j !!!

1) What feature(s) about the graph of an original function can cause its derivative not to exist at a certain point? Explain. [br][br]2) Is the derivative of a continuous function always a continuous function itself? Explain why or provide a counterexample to show why not.