About this time of year, you find someone, hopefully one of your students, asking, “If I’m finding where a function is increasing, is the interval open or closed?” Do you have an answer? This is a good time to teach some things about definitions and theorems. The place to start is to ask what it…
This is a re-post and update of the second in a series of posts from last year. It contains links to posts on this blog about the definition of the derivative for your reference in planning. Other updated post on the 2019 CED will come throughout the year, hopefully, a few weeks before you get…
This is a re-post and update of the second in a series of posts from last year. It contains links to posts on this blog about the definition of the derivative for your reference in planning. Other updated post on the 2019 CED will come throughout the year, hopefully, a few weeks before you get…
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:…
If you use your calculator or graphing program and zoom-in of the graph of a function (with equal zoom factors in both directions), the graph eventually looks like a line: the graph appears to be straight. This property is called Local Linearity. The slope of this line is the number called the derivative. (There are exceptions:…
Third in the graphing calculator series. In working up to the definition of the derivative you probably mention difference quotients. They are The forward difference quotient (FDQ): The backwards difference quotient (BDQ): , and The symmetric difference quotient (SDQ): Each of these is the slope of a (different) secant line and the limit of each as,…
Teaching Calculus 