Cubic Symmetry Some things are fairly obvious. For example, if you look at the graphs of a few cubic equations, you might think that each is symmetric to a point and on closer inspection the point of symmetry is the point of inflection. This is true and easy to prove. You can find the point…
Given equations that define a region in the plane students are asked to find its area and the volume of the solid formed when the region is revolved around a line or used as a base of a solid with regular cross-sections. This standard application of the integral has appeared every year since 1969 on…
The Free-response Questions The free-response questions fall into 10 general categories or types. The multiple-choice questions fall largely into the same categories plus some straight-forward questions asking students to find limits, derivatives, and integrals. Often two or more type are combined into one question. The types are the following. Rate and Accumulation Linear motion Graph Analysis…
Density, as an application of integration, has snuck onto the exams. It is specifically not mentioned in the “ Curriculum Framework” chapter of the new Course and Exam Description. There is an example (#12 p. 58) in the AB sample exam question section of Course and Exam Description.The first time this topic appeared was in…
The sixth in the Graphing Calculator / Technology Series Both graphing calculators and CAS calculators allow students to evaluate definite integrals. In the sections of the AP Calculus that allow calculator use students are expected to use their calculator to evaluate definite integrals. On the free-response section, students should write the integral on their paper,…
Sometimes I think textbooks are too rigorous. Behind every Riemann sum is a definite integral. So, authors routinely show how to solve an application of integration problem by developing the method starting from the Riemann sum and proceeding to an integral that give the result that is summarized in a “formula.” There is nothing wrong with…
Today’s question is not a good questions. It’s a bad question. But sometimes a bad question can become a good one. This one leads first to a discussion of units, then to all sorts of calculus. Here’s the question a teacher sent me this week taken from his textbook: The normal monthly rainfall at the…
