Unit 5 covers the application of derivatives to the analysis of functions and graphs. Reasoning and justification of results are also important themes in this unit. (CED – 2019 p. 92 – 107). These topics account for about 15 – 18% of questions on the AB exam and 8 – 11% of the BC questions.…
The new 2019 AP Calculus AB and BC Course and Exam Description is now available. New and experienced AP Calculus teachers should download a copy and read it carefully. (A paper copy with binder can be ordered here – it’s FREE.) The main sections of the book are here with notes on each. Part 1: General…
Why “scaling” is necessary No teacher can make two tests on the same topics equal in difficulty. No two teachers, even if they collaborate, can make two tests on the same topic equal in difficulty. No two teachers in different schools, districts, or states can make two tests on the same subject equal in difficulty.…
I got to thinking about grading the other day after seeing a question on Facebook. We’ll get to that question in a minute, but first I want to try to outline a grading scheme I used towards the end of my teaching career. It is based on how free-response question on AP Calculus exams are…
Some thoughts on pacing and planning your year’s work for AP Calculus AB or BC. The ideas are my own and are only suggestions for you to consider. Almost all textbooks provide an AP pacing guide among their ancillary material. You can consult the guide for your book for specific suggestions for the number of days…
Unit 4 covers rates of change in motion problems and other contexts, related rate problems, linear approximation and L’Hospital’s Rule. (CED – 2019 p. 82 – 90). These topics account for about 10 – 15% of questions on the AB exam and 6 – 9% of the BC questions. Topics 4.1 – 4.6 Topic 4.1 Interpreting…
Being a believer in the Rule of Four, I have been trying for years to find a good visual (graphical) illustration of why or how the Chain Rule for derivatives works. This very simple example is the best I could come up with. Consider the function . (See figure 1. A tangent segment at is drawn.)…
Unit 3 covers the Chain Rule, differentiation techniques that follow from it, and higher order derivatives. (CED – 2019 p. 67 – 77). These topics account for about 9 – 13% of questions on the AB exam and 4 – 7% of the BC questions. Topics 3.1 – 3.6 Topic 3.1 The Chain Rule. Students learn…
An important theorem concerning derivatives is this: If a function f is differentiable at x = a, then f is continuous at x = a. The proof begins with the identity that for all And therefore, Since both sides are finite, the function is continuous at x = a. The converse of this theorem is…
Unit 2 contains topics rates of change, difference quotients, and the definition of the derivative (CED – 2019 p. 51 – 66). These topics account for about 10 – 12% of questions on the AB exam and 4 – 7% of the BC questions. Topics 2.1 – 2.4: Introducing and Defining the Derivative Topic 2.1:…
Since it’s soon time to start derivatives, today we look at one way people found maximums and minimums before calculus was invented. Pierre de Fermat (1607 – 1655) was a French lawyer. His hobby, so to speak, was mathematics. He is considered one of the people who built the foundations of calculus. Along with Descartes,…
Recently, a number of questions about the limit of composite functions have been discussed on the AP Calculus Community bulletin board and also on the AP Calc TEACHERS – AB/BC Facebook page. The theorem that we would like to apply in these cases is this: If f is continuous at b and , then .…
