1. Use any graphing software to plot the following.
   1. Come up with a function and a closed interval [a,b] such that the area under the graph on this interval would be better approximated using the left endpoint method rather than the right endpoint method. Similarly, find a function that is better approximated using the right endpoint method versus the other method above. Justify why your function has this property. What key feature(s) must your functions have? (It may be useful to sketch rectangles along with graphs of functions. Also note that we are talking about the use of finitely many rectangles!)
   2. Is there a type of function that is always better approximated by the midpoint method when using finitely many rectangles? Justify your answer.
   3. Why might the above answers be useful? How would you apply this knowledge when encountering an unfamiliar function?
   4. For each of the functions you came up with, use Sage to approximate the area under your curve on [a,b] using n=100, 1000 and 10,000 subintervals of equal width. Do this for each of the three methods - right endpoint, left endpoint, and midpoint. What conclusions can you draw from this experiment?