produce a carefully written (and word-processed-typed ) 2-page paper
including small pictures. The goal is to demonstrate that you have mastered the primary relationships
between the graph of a function f and the properties of the first and second derivatives of f.
Choose one of the real-world situations below, and begin with a brief introduction of the topic. Make
observations about the shape of the graph of f and what it tells you about f
0 and f
00. Be particularly
careful and precise in your use of terms such as increasing function, positive slope, and increasing slope. If
you want to include what you think are possible graphs of f
0 and f
00, this may be helpful; but it is not
Another part of the task is to discuss how your observations about f, f0
, and f
00 can be interpreted in the
stated real-world situation (the context). What does the rate of change mean in the given context? If the
second derivative is positive throughout an interval, what sense does that make in the given context?
Would it be equally sensible if the second derivative were negative on that interval instead? If there is an
input where the second derivative is zero, how can you describe that point in terms of the shape of the
graph? What would this point mean in the given context? Why might the population of sick mice have
this particular shape? Why might the cost per yard of fabric increase as we produce larger amounts? Why
might it increase as we produce very small amounts? Why might blood alcohol concentration rise rapidly
and fall slowly?