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 .…
