How to draw a graph that is differentiable on some interval?

Question

The graph is continuous [0,3]. It is decreasing on [0,2] and increasing [2,3], differentiable at each point in(0,3) except at 1 where there is a vertical tangent.

Answer 1

Hi Eden,

  Take a look at this (crudely sketched) graph:

  https://www.wyzant.com/resources/files/465570/drawing_a_graph_continuous_differentiable

- continuous means has to be connected, although "kinks" would be allowed.

- decreasing and increasing are what we would expect (note that these always refer to how the curve changes as we move to the right.)

- differentiable means that the derivative must exist everywhere - so a "kink" etc is not allowed.

  (the derivative would be different on either side of a kink)

- a "vertical tangent" means the curve's slope is infinite, i.e., the derivative is infinite, it is "undefined".

- note that at x=2 it must be neither increasing nor decreasing, hence a horizontal tangent as shown.

 

- - -

 

Answer 2

Let's sketch this in the first quadrant by first drawing a vertical dashed line x=1; begin at (0,10) and draw an upward concave curve that moves downward until it curves sharply downward in order to be tangent to the dashed vertical line (to do this, the concavity of the curve will turn downward); at the instant of tangency, switch the concavity back to upward but continue its decreasing behavior, until when x = 2 the lowest point is reached, and the curve then moves upward. This ought to satisfy the conditions.