Match each of the derivatives shown below with the corresponding
functions. Which functions shown below are continuous? Which derivatives
shown below are continuous? Explain what shapes continuous functions must
have to produce jump discontinuities in the derivative and asymptotes in the
derivative. Draw a continuous function whose derivative has a jump
discontinuity at $x=1$ and an asymptote at $x=5$; draw the derivative.
Tags: diff, continuity, abstract-function, creating-examples, t1, c2 This gives students experience with visualizing features of a graph that produce discontinuities in the derivative. Make sure they go through the explanation part carefully, so that they are able to describe the features they think they see.
This is an excellent problem that makes students wrestle with the geometric interpretation of derivatives. Because it is straight forward and purely geometric, students are able to easily enter into a discussion about it without first having to work out a lengthy computation or puzzle over what to do.