By Lew Ludwig (Editor-in-Chief), Denison University
As the semester begins and we prep for classes, the practice of backward course design is a powerful way to get the most of the learning experience for our students. With this in mind, I thought I would share one of my favorite exam questions for a first semester calculus course, which appears below. I call it “the one question calculus final.” Now of course this is tongue-in-cheek, as the one question has over 15 questions. Nonetheless, this one question tour de force covers the full range of a first semester calc course. To substantiate this bold claim, I found a comprehensive list of typical topics in such a course at Wolfram Mathworld Classroom. The chart below cross-references each alphabetically listed topic with its specific question. While some questions touch on a range of topics, the cross-referencing refers to the primary reference.
Why I like this question:
I really like this question because it requires students to problem-solve, not just memorize a procedure. For example, instead of providing a typical composite function and asking for the derivative, my students must understand that the function in question (f) is a composition, know how to apply the chain rule, then read the graph to fill in the missing values. While I admit that question (m) may be a stretch for a Riemann sum, question (p) helps students realize the a definite integral is just a question about area. I especially like questions such as (j) and (k) that help students intuitively use important results like the intermediate value theorem or the mean value theorem. Finally, I like this question because no piece of technology can do the work for you. I can safely permit graphing calculators during my final without the fear of some CAS (computer algebra system) making short work of my exam.
How I use this question:
Please do not unleash this question on your students without prior exposure! My students have been working with this type of question for the whole semester. The beginning of the semester would focus more on limit questions like (a)-(e). By the second test of the semester, my students work on questions like (f) and (g) to understand the with mechanics of differentiation. By the third test, we get practice with applications of the derivative with questions such as (h)-(l). And by the end of the semester, questions of the type (m)-(p) test students’ understanding of the definite and indefinite integral. To make sure students do not forget prior material, my tests include questions from previous tests. Cognitive psychologists refer to this technique as interleaving.
How I grade this question:
Since this question has so many parts, I only count each sub-question for one point out of a 100-point final exam. Okay, I do ask other questions beside this one! I grade each question as right or wrong, no partial credit. While this may seem harsh, by giving each question a small point value, a student can miss a few of these questions without serious detriment to the overall grade. Moreover, past tests have shown a student’s exam score tracks fairly closely with performance on this question.
How to modify this question:
Of course there is a myriad of ways this question could be modified. For one, change the graph. When I initially developed this question, I would make sketches of the graph by hand. Now the online graphing program Desmos helps me produce graphs that are easy to read and export into LaTex or word processing programs. Students can contribute by creating their own questions for a graph you provide. Or you can turn that on end and have students provide a graph based on questions you provide. However you use it, you will find this focused cumulative approach will help deepen your students understanding of calculus.